Some basic aspects of probability theory, using some basic analysis

Stephen Semmes Rice University

Preface

A fundamental question to be addressed in these informal notes is

What is a probability space?

A slight variant of this question that one may ask is

What is a probability measure?

In fact, a probability measure is included in the definition of a probability space. More precisely, a probability space is defined to be a measurable space with a probability measure. We shall discuss these and related matters in some detail here.

Of course, these questions are related to measure theory and integration more broadly. The reader is not necessarily expected to be familiar with measure theory, and indeed part of the aim here is to give an introduction to some of the relevant notions and results. However, this is not intended to be a detailed treatment of measure theory, and instead we would like to emphasize some aspects related to probability theory.

The reader is expected to be familiar with some basic analysis, along the lines of metric spaces, sequences and series, continuous mappings, and compactness. Some topics may be reviewed briefly, as needed.

Some familiarity with topological structures and spaces could be helpful in some places, although it is not required. In particular, measures are often defined on metric or topological spaces, and this can be relevant for the corresponding collections of measurable sets, as well as some regularity properties of the measures. There are some analogies between measurability and continuity of functions, although there are some additional properties of measurable functions too. Cartesian products of measure spaces are very important, and this is related to products of metric and topological spaces.

There are many connections with Fourier analysis and functional analysis as well, and we shall see some aspects of this here. The reader is not necessarily expected to be familiar with these topics either.

There are certainly connections with geometric measure theory too, although we do not really get into this too much here. In particular, we do at least mention some techniques commonly used to get regularity properties of Borel measures on metric spaces, for instance.

There are also connections with ergodic theory, that are not discussed here, other than the definition of a measure-preserving transformation. Silva’s book [317] includes an introduction to measure theory, and the reader may be interested in [9, 17, 122, 146, 147, 221, 266, 318, 363] as well.

Halmos’ survey [144] may be particularly interesting for many readers, and Fréchet’s paper [120] has some interesting perspectives too. A question came up about references related to counterexamples, and the reader may be interested in [28, 127, 128, 290, 308, 330, 338, 378] in this regard.

Contents

1 Some basic notions 1

1.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Some remarks about sequences of real numbers . . . . . . 2 1.1.2 A topology on the set of extended real numbers . . . . . . 3

1.2 More on extended real numbers . . . . . . . . . . . . . . . . . . . 3 1.2.1 Suprema and infima of functions . . . . . . . . . . . . . . 4 1.2.2 Infinite series with nonnegative terms . . . . . . . . . . . 4

1.3 Upper and lower limits of sequences in R . . . . . . . . . . . . . 5 1.3.1 Defining upper and lower limits . . . . . . . . . . . . . . . 6 1.3.2 Some properties of the upper and lower limits . . . . . . . 6 1.3.3 Some variants and related properties . . . . . . . . . . . . 7 1.3.4 Another approach to upper and lower limits . . . . . . . . 8

1.4 Indicator functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.1 Some upper and lower limits . . . . . . . . . . . . . . . . 9 1.4.2 Upper and lower limits of sequences of sets . . . . . . . . 9 1.4.3 More on sequences of subsets of X . . . . . . . . . . . . . 10

1.5 Algebras and σ-algebras of sets . . . . . . . . . . . . . . . . . . . 10 1.5.1 σ-Algebras of sets . . . . . . . . . . . . . . . . . . . . . . 11 1.5.2 Rings of sets . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.3 More on rings of sets . . . . . . . . . . . . . . . . . . . . . 12 1.5.4 σ-Rings of sets . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Some remarks about intervals . . . . . . . . . . . . . . . . . . . . 13 1.6.1 Some elementary subsets of R . . . . . . . . . . . . . . . 14 1.6.2 Intervals in R^{n} . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6.3 Some elementary subsets of R^{n} . . . . . . . . . . . . . . . 15 1.6.4 Some more rings and algebras of sets . . . . . . . . . . . . 15

1.7 Finitely and countably additive measures . . . . . . . . . . . . . 16 1.7.1 Countable additivity . . . . . . . . . . . . . . . . . . . . . 16 1.7.2 Measurable and measure spaces . . . . . . . . . . . . . . . 17 1.7.3 Some basic examples of measures . . . . . . . . . . . . . . 17 1.7.4 Some more examples of measures . . . . . . . . . . . . . . 18

1.8 Measures and nonnegative sums . . . . . . . . . . . . . . . . . . . 19 1.8.1 Nonnegative sums over arbitrary sets . . . . . . . . . . . . 19 1.8.2 Comparison with infinite series of nonnegative numbers . 19 1.8.3 Summable functions on X . . . . . . . . . . . . . . . . . . 20 1.8.4 Some properties of the sum . . . . . . . . . . . . . . . . . 20 1.8.5 Monotone convergence for nonnegative sums . . . . . . . 21 1.8.6 Sums over countable unions of subsets of X . . . . . . . . 22 1.8.7 Measures defined using these sums . . . . . . . . . . . . . 23

1.9 Some basic properties of measures . . . . . . . . . . . . . . . . . 23 1.9.1 A partial converse . . . . . . . . . . . . . . . . . . . . . . 24 1.9.2 Decreasing sequences of measurable sets . . . . . . . . . . 24 1.9.3 Another partial converse . . . . . . . . . . . . . . . . . . . 25 1.9.4 A counterexample with μ(C_{1}) = +∞ . . . . . . . . . . . . 25

1.10 Subadditivity and outer measures . . . . . . . . . . . . . . . . . . 25 1.10.1 Finite subadditivity . . . . . . . . . . . . . . . . . . . . . 26 1.10.2 Countable subadditivity . . . . . . . . . . . . . . . . . . . 26 1.10.3 A criterion for countable additivity . . . . . . . . . . . . . 27 1.10.4 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . 27

1.11 More on outer measures . . . . . . . . . . . . . . . . . . . . . . . 27 1.11.1 Measurability with respect to μ . . . . . . . . . . . . . . . 28 1.11.2 Metric outer measures . . . . . . . . . . . . . . . . . . . . 28 1.11.3 Some constructions of outer measures . . . . . . . . . . . 29 1.11.4 A very nice extension theorem . . . . . . . . . . . . . . . 30

1.12 Some regularity conditions for measures . . . . . . . . . . . . . . 31 1.12.1 An outer regularity condition . . . . . . . . . . . . . . . . 31 1.12.2 Two inner regularity conditions . . . . . . . . . . . . . . . 31 1.12.3 Some examples concerning inner regularity . . . . . . . . 32 1.12.4 Regularity and elementary sets . . . . . . . . . . . . . . . 32

1.13 Regularity conditions and countable additivity . . . . . . . . . . 32 1.13.1 Volumes of elementary sets . . . . . . . . . . . . . . . . . 34 1.13.2 Regularity of the volume . . . . . . . . . . . . . . . . . . . 34

1.14 Another outer regularity condition . . . . . . . . . . . . . . . . . 34 1.14.1 Using this condition to get countable additivity . . . . . . 35 1.14.2 More on the n-dimensional volume on E_{1}(R^{n}) . . . . . . . 36

1.15 Lebesgue outer measure on R^{n} . . . . . . . . . . . . . . . . . . . 36 1.15.1 Lebesgue measure on R^{n} . . . . . . . . . . . . . . . . . . 37

2 More on measures and σ-algebras 38

2.1 Getting an outer measure from a measure . . . . . . . . . . . . . 38 2.1.1 The infimum in the definition of \overline{μ}(E) is attained . . . . . 38 2.1.2 When E is σ-finite with respect to \overline{μ} . . . . . . . . . . . . 39 2.1.3 Complete measure spaces . . . . . . . . . . . . . . . . . . 39 2.1.4 More on σ-finiteness . . . . . . . . . . . . . . . . . . . . . 40

2.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.1 Borel measures . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Two basic families of examples of Borel sets . . . . . . . . 41 2.2.3 Some examples concerning intervals . . . . . . . . . . . . 42

2.3 Separability and countable bases . . . . . . . . . . . . . . . . . . 42 2.3.1 Separable metric and topological spaces . . . . . . . . . . 43 2.3.2 Bases and Borel sets . . . . . . . . . . . . . . . . . . . . . 43 2.3.3 Some more examples of bases using intervals . . . . . . . 44

2.4 Limits of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1 One-sided limits on R . . . . . . . . . . . . . . . . . . . . 45 2.4.2 Monotonically increasing functions on R . . . . . . . . . . 45

2.5 Lengths of intervals with respect to α . . . . . . . . . . . . . . . 46 2.5.1 Lebesgue–Stieltjes outer measures . . . . . . . . . . . . . 47 2.5.2 When α is continuous and strictly increasing . . . . . . . 48 2.5.3 Some properties of monotone functions . . . . . . . . . . . 48 2.5.4 Another approach to Lebesgue–Stieltjes outer measures . 49

2.6 Monotone classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6.1 Monotone classes generated by algebras . . . . . . . . . . 50 2.6.2 Monotone classes and finite measures . . . . . . . . . . . 51 2.6.3 Monotone classes and some sets of finite measure . . . . . 51 2.6.4 Using a σ-finiteness condition on μ, ν . . . . . . . . . . . 52

2.7 Regular outer measures . . . . . . . . . . . . . . . . . . . . . . . 52 2.7.1 A nice property of regular outer measures . . . . . . . . . 53 2.7.2 Regularity and σ-finiteness . . . . . . . . . . . . . . . . . 54 2.7.3 Borel regular outer measures . . . . . . . . . . . . . . . . 54

2.8 Using a collection C of subsets of X . . . . . . . . . . . . . . . . 54 2.8.1 Some regular outer measures . . . . . . . . . . . . . . . . 55 2.8.2 Outer regular outer measures . . . . . . . . . . . . . . . . 56

2.9 More on the extension theorem . . . . . . . . . . . . . . . . . . . 56 2.9.1 Showing that ν_{1} ≤ μ_{1} . . . . . . . . . . . . . . . . . . . . 57 2.9.2 When μ_{1}(E) < +∞ . . . . . . . . . . . . . . . . . . . . . 57 2.9.3 When E is σ-finite with respect to μ_{1} . . . . . . . . . . . 58

2.10 Some conditions concerning finite measure . . . . . . . . . . . . . 59 2.10.1 Local compactness . . . . . . . . . . . . . . . . . . . . . . 59 2.10.2 σ-Compact sets . . . . . . . . . . . . . . . . . . . . . . . . 60 2.10.3 σ-Compactness and inner regularity . . . . . . . . . . . . 60 2.10.4 Another basic regularity property . . . . . . . . . . . . . . 61

2.11 Some nice families of sets . . . . . . . . . . . . . . . . . . . . . . 61 2.11.1 Another characterization of Borel sets . . . . . . . . . . . 62 2.11.2 Regularity of finite Borel measures . . . . . . . . . . . . . 62 2.11.3 An analogous argument for inner regularity of μ . . . . . 63 2.11.4 Inner regularity for Borel sets of finite measure . . . . . . 63

2.12 More on Borel measures . . . . . . . . . . . . . . . . . . . . . . . 63 2.12.1 Outer regularity and open sets of finite measure . . . . . 63 2.12.2 More on inner regularity . . . . . . . . . . . . . . . . . . . 65 2.12.3 Some more regularity results . . . . . . . . . . . . . . . . 65

2.13 Locally finite Borel measures on the real line . . . . . . . . . . . 66 2.13.1 Some normalizing conditions on α . . . . . . . . . . . . . 66 2.13.2 Getting monotonically increasing functions . . . . . . . . 67

2.14 Some more families of rings and algebras of sets . . . . . . . . . . 69 2.14.1 Some remarks related to the Lindelöf property . . . . . . 69 2.14.2 Zero-dimensional spaces . . . . . . . . . . . . . . . . . . . 70

2.14.3 Some remarks about countable bases . . . . . . . . . . . . 70 2.15 Ultrametrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.15.1 Open balls in ultrametric spaces . . . . . . . . . . . . . . 71 2.15.2 Closed balls in ultrametric spaces . . . . . . . . . . . . . . 72

3 Measurable functions and integration 73

3.1 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.1 When Y is a metric or topological space . . . . . . . . . . 73 3.1.2 Measurable mappings into measurable spaces . . . . . . . 74 3.1.3 Compositions of measurable mappings . . . . . . . . . . . 74 3.1.4 Measurability and bases for topologies . . . . . . . . . . . 74

3.2 Inclusion mappings and induced σ-algebras . . . . . . . . . . . . 75 3.2.1 More on induced σ-algebras . . . . . . . . . . . . . . . . . 75 3.2.2 Borel sets in the set of extended real numbers . . . . . . . 76

3.3 Extended real-valued measurable functions . . . . . . . . . . . . 76

3.3.1 Some additional remarks about measurability . . . . . . . 77 3.4 Some mappings into finite products . . . . . . . . . . . . . . . . . 78

3.4.1 Continuity of mappings into Y . . . . . . . . . . . . . . . 78 3.4.2 Measurability of mappings into Y . . . . . . . . . . . . . 78 3.4.3 Measurability of complex-valued functions . . . . . . . . . 79

3.5 More on measurable functions . . . . . . . . . . . . . . . . . . . . 80 3.5.1 More on real-valued functions . . . . . . . . . . . . . . . . 80 3.5.2 The set where two measurable functions are equal …. 81

3.6 Some more measurability properties . . . . . . . . . . . . . . . . 81 3.6.1 Measurability of pointwise limits . . . . . . . . . . . . . . 82 3.6.2 Measurability of monotone functions . . . . . . . . . . . . 82

3.7 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.7.1 Approximation by simple functions . . . . . . . . . . . . . 84 3.8 Integrating nonnegative functions . . . . . . . . . . . . . . . . . . 84

3.8.1 Defining the Lebesgue integral . . . . . . . . . . . . . . . 84 3.8.2 Conditions when the integral is equal to 0 . . . . . . . . . 85 3.8.3 Using finite partitions of X . . . . . . . . . . . . . . . . . 86

3.9 More on integrating nonnegative functions . . . . . . . . . . . . . 86 3.9.1 Integrating some bounded measurable functions . . . . . . 87 3.9.2 Two basic classes of examples . . . . . . . . . . . . . . . . 88

3.10 Some properties of these integrals . . . . . . . . . . . . . . . . . . 88 3.10.1 When the integral is 0 . . . . . . . . . . . . . . . . . . . . 89 3.10.2 When the integral is finite . . . . . . . . . . . . . . . . . . 89 3.10.3 The monotone convergence theorem . . . . . . . . . . . . 89

3.11 Using the monotone convergence theorem . . . . . . . . . . . . . 90 3.11.1 Some integrals of sums . . . . . . . . . . . . . . . . . . . . 91 3.11.2 Another result about monotone convergence . . . . . . . . 91 3.11.3 Fatou’s lemma . . . . . . . . . . . . . . . . . . . . . . . . 92

3.12 Real and complex-valued functions . . . . . . . . . . . . . . . . . 93 3.12.1 Some remarks about complex numbers . . . . . . . . . . . 93 3.12.2 Integrating complex-valued functions . . . . . . . . . . . . 94

3.12.3 More on integrals over measurable subsets of X . . . . . . 95 3.13 The dominated convergence theorem . . . . . . . . . . . . . . . . 96

3.13.1 Convergence almost everywhere . . . . . . . . . . . . . . . 97 3.13.2 The bounded convergence theorem . . . . . . . . . . . . . 97 3.13.3 Some more integrals of sums . . . . . . . . . . . . . . . . 97

3.14 Connection with Riemann integrals . . . . . . . . . . . . . . . . . 98 3.14.1 Lebesgue integrals on R . . . . . . . . . . . . . . . . . . . 99 3.14.2 Invariance under translations . . . . . . . . . . . . . . . . 99

3.15 Some measures from integrals . . . . . . . . . . . . . . . . . . . . 99 3.15.1 Integrals with respect to ν . . . . . . . . . . . . . . . . . . 100 3.15.2 Some additional properties of ν . . . . . . . . . . . . . . . 100

4 More on functions and measures 102

4.1 Absolutely continuous measures . . . . . . . . . . . . . . . . . . . 102 4.1.1 The Radon–Nikodym theorem . . . . . . . . . . . . . . . 102 4.1.2 More on absolute continuity . . . . . . . . . . . . . . . . . 103

4.2 Mutually singular measures . . . . . . . . . . . . . . . . . . . . . 103 4.2.1 The Lebesgue decomposition theorem . . . . . . . . . . . 103 4.2.2 More on the Lebesgue decomposition . . . . . . . . . . . . 104

4.3 Differentiating monotone functions . . . . . . . . . . . . . . . . . 104 4.3.1 A simpler version of α′ . . . . . . . . . . . . . . . . . . . . 104 4.3.2 Integrating α′ on an interval . . . . . . . . . . . . . . . . 105 4.3.3 Equality in the estimate for the integral of α′ . . . . . . . 106

4.4 Absolutely continuous functions . . . . . . . . . . . . . . . . . . . 106 4.4.1 Absolute continuity and differentiability . . . . . . . . . . 107 4.4.2 Integrating integrable functions on [a, b] . . . . . . . . . . 107 4.4.3 A criterion for absolute continuity . . . . . . . . . . . . . 108

4.5 More on absolutely continuous functions . . . . . . . . . . . . . . 108 4.5.1 More on differentiating monotone functions . . . . . . . . 109 4.5.2 One way to organize results like these . . . . . . . . . . . 110 4.5.3 Another way to organize results like these . . . . . . . . . 110 4.5.4 The derivative as a limit in an integral sense . . . . . . . 111

4.6 The L^{1} norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6.1 Convergence in L^{1} . . . . . . . . . . . . . . . . . . . . . . 113 4.6.2 Absolutely convergent series in L^{1} . . . . . . . . . . . . . 113 4.6.3 Convergence in L^{1} of Cauchy sequences in L^{1} . . . . . . . 114 4.6.4 Pointwise convergent subsequences . . . . . . . . . . . . . 114

4.7 Convergence in measure . . . . . . . . . . . . . . . . . . . . . . . 114 4.7.1 Convergence in L^{1} implies convergence in measure …. 115 4.7.2 Pointwise convergence and Egorov’s theorem . . . . . . . 115 4.7.3 More on pointwise convergent subsequences . . . . . . . . 116 4.7.4 Cauchy sequences in measure . . . . . . . . . . . . . . . . 116

4.8 More on bounded convergence . . . . . . . . . . . . . . . . . . . . 116 4.9 Some more integral distance functions . . . . . . . . . . . . . . . 117

4.9.1 Some properties of d, d̃ . . . . . . . . . . . . . . . . . . . 118 4.9.2 Connection with convergence in measure . . . . . . . . . . 118

4.9.3 Cauchy conditions associated to d and d̃ . . . . . . . . . . 119 4.10 Absolute convergence with respect to d, d̃ . . . . . . . . . . . . . 119

4.10.1 Getting pointwise convergence almost everywhere . . . . . 120 4.10.2 Convergence of the sum with respect to d . . . . . . . . . 120 4.10.3 Convergence of Cauchy sequences with respect to d …. 121

4.11 Pushing measures forward . . . . . . . . . . . . . . . . . . . . . . 121 4.11.1 Probability distributions and random variables . . . . . . 122 4.11.2 Compositions and isomorphisms . . . . . . . . . . . . . . 122 4.11.3 Compositions and integrals . . . . . . . . . . . . . . . . . 122 4.11.4 Pushing outer measures forward . . . . . . . . . . . . . . 123

4.12 Some measures from measurable sets . . . . . . . . . . . . . . . . 123 4.13 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . 124

4.13.1 The Lebesgue–Stieltjes measure associated to F . . . . . 125 4.13.2 Density functions . . . . . . . . . . . . . . . . . . . . . . . 125 4.13.3 A slightly different type of distribution function . . . . . . 126

4.14 Approximate inverses of monotone functions . . . . . . . . . . . . 126 4.14.1 Another approximate inverse ψ_{2} . . . . . . . . . . . . . . 127 4.14.2 Some properties of ψ_{1}, ψ_{2} . . . . . . . . . . . . . . . . . . 127

4.15 Some more properties of ψ_{1} . . . . . . . . . . . . . . . . . . . . . 128 4.15.1 When α is right continuous on R . . . . . . . . . . . . . . 128 4.15.2 When α is right continuous and α(−∞) = 0 . . . . . . . . 129 4.15.3 More on ψ_{1} and ψ_{2} . . . . . . . . . . . . . . . . . . . . . . 130

5 The L^{p} norm, product measures, and more 131

5.1 The L^{p} norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.1 Sums of pth-power integrable functions . . . . . . . . . . 131 5.1.2 Minkowski’s inequality . . . . . . . . . . . . . . . . . . . . 132 5.1.3 A related inequality when p < 1 . . . . . . . . . . . . . . 132 5.1.4 The corresponding metrics and norms . . . . . . . . . . . 132

5.2 Convergence in L^{p} . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2.1 Another version of Chebyshev’s inequality . . . . . . . . . 133 5.2.2 Absolutely convergent series in L^{p} . . . . . . . . . . . . . 134 5.2.3 Convergence of the partial sums in L^{p} . . . . . . . . . . . 135 5.2.4 Convergence in L^{p} of Cauchy sequences in L^{p} . . . . . . . 135

5.3 Hölder’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.1 A bit of the proof . . . . . . . . . . . . . . . . . . . . . . 137 5.3.2 An extension of Hölder’s inequality . . . . . . . . . . . . . 137 5.3.3 Using Hölder’s inequality when μ(X) is finite . . . . . . . 137 5.3.4 Another extension of Hölder’s inequality . . . . . . . . . . 138

5.4 Essentially bounded functions . . . . . . . . . . . . . . . . . . . . 138 5.4.1 More on the L^{∞} norm . . . . . . . . . . . . . . . . . . . . 139 5.4.2 Convergence in L^{∞} . . . . . . . . . . . . . . . . . . . . . . 139 5.4.3 More on essentially bounded functions . . . . . . . . . . . 139

5.5 Integral inner products . . . . . . . . . . . . . . . . . . . . . . . . 140

5.5.1 Orthogonal functions . . . . . . . . . . . . . . . . . . . . . 141 5.6 Products of outer measures . . . . . . . . . . . . . . . . . . . . . 141

5.6.1 Countable measurability and σ-finiteness . . . . . . . . . 142 5.7 Elementary families of sets . . . . . . . . . . . . . . . . . . . . . 142

5.7.1 Products of two measurable spaces . . . . . . . . . . . . . 143 5.7.2 The obvious mappings from X × Y into X, Y . . . . . . . 143 5.7.3 More on measurable rectangles . . . . . . . . . . . . . . . 144 5.7.4 Monotone classes and measurable rectangles . . . . . . . . 144

5.8 More on A_{X×Y} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.8.1 When X and Y are metric or topological spaces . . . . . 145 5.8.2 Measurability of mappings into X × Y . . . . . . . . . . . 145 5.8.3 Sections of subsets of X × Y . . . . . . . . . . . . . . . . 146 5.8.4 Sections of functions on X × Y . . . . . . . . . . . . . . . 146

5.9 Product measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.9.1 Another approach to μ × ν . . . . . . . . . . . . . . . . . 148 5.9.2 Integration on X × Y . . . . . . . . . . . . . . . . . . . . 148 5.9.3 Uniqueness of the product measure . . . . . . . . . . . . . 149

5.10 Products of n measurable spaces . . . . . . . . . . . . . . . . . . 149 5.10.1 Collections of subsets of X_{j} that generate A_{X_{j}} . . . . . . 150 5.10.2 When the X_{j}’s are metric or topological spaces . . . . . . 150 5.10.3 Measurability of mappings into X . . . . . . . . . . . . . 151 5.10.4 Products of n measurable sets . . . . . . . . . . . . . . . . 151

5.11 Some related Cartesian products . . . . . . . . . . . . . . . . . . 152 5.12 Products of n measures . . . . . . . . . . . . . . . . . . . . . . . 153

5.12.1 Some iterated integrals . . . . . . . . . . . . . . . . . . . . 154 5.12.2 Extending μ to A_{X} . . . . . . . . . . . . . . . . . . . . . . 154 5.12.3 Uniqueness of μ on A_{X} . . . . . . . . . . . . . . . . . . . 154 5.12.4 Another approach to the product measure on X . . . . . 155

5.13 Sequences of measurable spaces . . . . . . . . . . . . . . . . . . . 155 5.13.1 Cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.13.2 Collections of subsets of the X_{j}’s generating A_{X_{j}} . . . . . 156 5.13.3 Borel sets in X . . . . . . . . . . . . . . . . . . . . . . . . 156 5.13.4 Measurable mappings into X . . . . . . . . . . . . . . . . 157

5.14 A helpful family of related Cartesian products . . . . . . . . . . . 157 5.14.1 σ-Algebras of measurable sets in these products . . . . . . 158 5.14.2 Some remarks about products of measurable sets . . . . . 158

5.15 Sequences of probability spaces . . . . . . . . . . . . . . . . . . . 159 5.15.1 Using the standard mapping from X onto X(n) . . . . . . 160 5.15.2 Defining μ on AC_{X}, and on A_{X} . . . . . . . . . . . . . . . 161 5.15.3 Some more properties of these product measures . . . . . 161 5.15.4 Some other approaches to product measures like these . . 161

6 Independence in probability theory 163

6.1 Some terminology in probability theory . . . . . . . . . . . . . . 163 6.1.1 Independent pairs of events . . . . . . . . . . . . . . . . . 163 6.1.2 Independence of n events . . . . . . . . . . . . . . . . . . 164 6.1.3 Independent sequences of events . . . . . . . . . . . . . . 164 6.1.4 Independence and complements . . . . . . . . . . . . . . . 165 6.2 Infinite products of real numbers . . . . . . . . . . . . . . . . . . 165

6.2.1 Intersections of independent events . . . . . . . . . . . . . 166 6.2.2 Products of positive real numbers . . . . . . . . . . . . . . 167 6.2.3 More on products with 0 ≤ t_{j} ≤ 1 for each j . . . . . . . . 167

6.3 The Borel–Cantelli lemma . . . . . . . . . . . . . . . . . . . . . . 169 6.3.1 Two proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.3.2 When the A_{j}’s are independent . . . . . . . . . . . . . . . 170

6.4 Independent random variables . . . . . . . . . . . . . . . . . . . . 171 6.4.1 Indicator functions of independent events . . . . . . . . . 171 6.4.2 Some independent random variables . . . . . . . . . . . . 172 6.4.3 Some more independent random variables . . . . . . . . . 172 6.4.4 Compositions with other functions . . . . . . . . . . . . . 172

6.5 Rademacher functions and related topics . . . . . . . . . . . . . . 173 6.6 More on pushing measures forward . . . . . . . . . . . . . . . . . 174

6.6.1 Measurable mappings into R^{n} . . . . . . . . . . . . . . . 175 6.6.2 Sequences of random variables . . . . . . . . . . . . . . . 175

6.7 Pushing some product measures forward . . . . . . . . . . . . . . 176

6.7.1 Some mappings between product spaces . . . . . . . . . . 176 6.8 Some identifications with finite products . . . . . . . . . . . . . . 177

6.8.1 Another Cartesian product . . . . . . . . . . . . . . . . . 178 6.8.2 Associated product σ-algebras . . . . . . . . . . . . . . . 178 6.8.3 Associated product measures . . . . . . . . . . . . . . . . 179

6.9 Some combinations of random variables . . . . . . . . . . . . . . 179

6.9.1 The measure β_{∗}(μ) on R^{N} . . . . . . . . . . . . . . . . . 180 6.10 The multiplication theorem . . . . . . . . . . . . . . . . . . . . . 180

6.10.1 Reducing to the case of product measures . . . . . . . . . 181 6.10.2 Using nonnegative functions on the real line . . . . . . . . 182 6.10.3 Using other functions on the real line . . . . . . . . . . . 182 6.10.4 A more direct approach . . . . . . . . . . . . . . . . . . . 183

6.11 Some remarks about finite moments . . . . . . . . . . . . . . . . 183 6.11.1 The variance of f . . . . . . . . . . . . . . . . . . . . . . . 184 6.11.2 Another description of the variance . . . . . . . . . . . . . 184 6.11.3 More on the variance . . . . . . . . . . . . . . . . . . . . . 185

6.12 The variance of a sum . . . . . . . . . . . . . . . . . . . . . . . . 185 6.12.1 The variance of an average . . . . . . . . . . . . . . . . . 186 6.12.2 Identically distributed random variables . . . . . . . . . . 187

6.13 A criterion for convergence to 0 a.e. . . . . . . . . . . . . . . . . 187 6.13.1 Some L^{4} norms of sums . . . . . . . . . . . . . . . . . . . 188 6.13.2 Using independence and mean 0 . . . . . . . . . . . . . . 188 6.13.3 Using a bound on ∥f_{j}∥_{4} . . . . . . . . . . . . . . . . . . . 189 6.13.4 Another convergence property . . . . . . . . . . . . . . . . 190

6.14 An inequality of Kolmogorov . . . . . . . . . . . . . . . . . . . . 190 6.14.1 Some measurable sets A_{l} . . . . . . . . . . . . . . . . . . 191 6.14.2 Some related independence properties . . . . . . . . . . . 191 6.14.3 Using Chebyshev’s inequality another way . . . . . . . . . 192

6.15 Some more convergence results . . . . . . . . . . . . . . . . . . . 193 6.15.1 Some convergent series . . . . . . . . . . . . . . . . . . . . 193 6.15.2 Kolmogorov’s strong law . . . . . . . . . . . . . . . . . . . 194 6.15.3 Some sequences and series . . . . . . . . . . . . . . . . . . 194 6.15.4 Khinchine’s strong law . . . . . . . . . . . . . . . . . . . . 194

7 Weak^{∗} limits and Fourier transforms 196

7.1 Some spaces of continuous functions on R^{n} . . . . . . . . . . . . 196 7.1.1 Compact support in R^{n} . . . . . . . . . . . . . . . . . . . 196 7.1.2 Vanishing at infinity on R^{n} . . . . . . . . . . . . . . . . . 197 7.1.3 More on functions with compact support in R^{n} . . . . . . 197 7.1.4 Nonnegative linear functionals on C_{com}(R^{n}) . . . . . . . . 198

7.2 The complex exponential function . . . . . . . . . . . . . . . . . 199 7.2.1 The Fourier transform of an integrable function on R . . 199 7.2.2 The Fourier transform of a finite Borel measure on R . . 200 7.2.3 The Fourier transform of a finite Borel measure on R^{n} . . 201

7.3 Characteristic functions of random variables . . . . . . . . . . . . 201

7.3.1 Joint characteristic functions . . . . . . . . . . . . . . . . 202 7.4 Convolutions of finite Borel measures on R . . . . . . . . . . . . 203

7.4.1 Sums of independent random variables . . . . . . . . . . . 205 7.5 Weak^{∗} convergence for Borel measures on R^{n} . . . . . . . . . . . 205

7.5.1 Weak^{∗} convergence and measures of open sets in R^{n} … 206 7.5.2 Weak^{∗} convergence and measures of compact sets in R^{n} . 207

7.6 Weak^{∗} convergence for Borel measures on R . . . . . . . . . . . . 208 7.6.1 When μ({a}) = μ({b}) = 0 . . . . . . . . . . . . . . . . . 208 7.6.2 The set of x ∈ R with μ({x}) > 0 is at most countable . . 209 7.6.3 Some remarks about step functions . . . . . . . . . . . . . 209 7.6.4 A criterion for weak^{∗} convergence on R . . . . . . . . . . 209

7.7 Other types of weak^{∗} convergence on R^{n} . . . . . . . . . . . . . . 210 7.7.1 The uniform boundedness principle . . . . . . . . . . . . . 210 7.7.2 Convergence with respect to C_{b}(R^{n}) . . . . . . . . . . . . 211

7.8 A related condition on sequences of measures . . . . . . . . . . . 211 7.8.1 A reformulation using continuous functions . . . . . . . . 212 7.8.2 A connection with weak^{∗} convergence . . . . . . . . . . . 212 7.8.3 Necessity of this condition . . . . . . . . . . . . . . . . . . 213 7.8.4 More on convergence with respect to C_{b}(R^{n}) . . . . . . . 213

7.9 Multi-indices and the Schwartz class . . . . . . . . . . . . . . . . 214 7.9.1 The Schwartz class . . . . . . . . . . . . . . . . . . . . . . 214 7.9.2 The Fourier transform on S(R^{n}) . . . . . . . . . . . . . . 215 7.9.3 More on the Fourier transform on S(R^{n}) . . . . . . . . . 216 7.9.4 The multiplication formula . . . . . . . . . . . . . . . . . 216

7.10 More on sequences of finite measures on R^{n} . . . . . . . . . . . . 216 7.10.1 Getting pointwise convergence of {μ̂_{j}}_{j=1}^{∞} . . . . . . . . . 217 7.10.2 Using pointwise convergence of {μ̂_{j}}_{j=1}^{∞} . . . . . . . . . . 217

7.11 Weak convergence of random variables . . . . . . . . . . . . . . . 218 7.11.1 When f is constant . . . . . . . . . . . . . . . . . . . . . . 219 7.11.2 When {f_{j}}_{j=1}^{∞} converges pointwise . . . . . . . . . . . . . 219

7.11.3 When {f_{j}}_{j=1}^{∞} converges in measure . . . . . . . . . . . . 220 7.12 The derivative of exp(a t) . . . . . . . . . . . . . . . . . . . . . . 220

7.12.1 Differentiating the Fourier transform . . . . . . . . . . . . 221 7.12.2 Differentiating ν̂ some more . . . . . . . . . . . . . . . . . 221 7.12.3 More on Fourier transforms of integrable functions on R . 222

7.13 Gaussian distributions . . . . . . . . . . . . . . . . . . . . . . . . 222 7.14 A convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . 223

7.14.1 The central limit theorem . . . . . . . . . . . . . . . . . . 224 7.14.2 Some remarks about logarithms . . . . . . . . . . . . . . . 224

Appendix 226

A Metrics, ultrametrics, and product spaces 226

A.1 Some remarks related to compactness . . . . . . . . . . . . . . . 226 A.1.1 The limit point property . . . . . . . . . . . . . . . . . . . 226 A.1.2 Sequential compactness . . . . . . . . . . . . . . . . . . . 227 A.1.3 Countable compactness . . . . . . . . . . . . . . . . . . . 227

A.2 Totally bounded sets . . . . . . . . . . . . . . . . . . . . . . . . . 228 A.2.1 Totally bounded metric spaces . . . . . . . . . . . . . . . 228 A.2.2 Another characterization of compactness . . . . . . . . . . 228

A.3 Some basic examples . . . . . . . . . . . . . . . . . . . . . . . . . 229

A.3.1 Some examples of ultrametrics . . . . . . . . . . . . . . . 229 A.4 Products of n metric spaces . . . . . . . . . . . . . . . . . . . . . 230

A.4.1 Some remarks about open and closed balls in X . . . . . 230 A.4.2 Some remarks about open sets in X . . . . . . . . . . . . 231 A.4.3 Some remarks about closures and closed sets in X …. 231 A.4.4 Sequences in X . . . . . . . . . . . . . . . . . . . . . . . . 232

A.5 Diameters in metric spaces . . . . . . . . . . . . . . . . . . . . . 232 A.5.1 Diameters in finite products . . . . . . . . . . . . . . . . . 233 A.5.2 Total boundedness in finite products . . . . . . . . . . . . 234

A.6 Compactness in finite products . . . . . . . . . . . . . . . . . . . 234

A.6.1 Using total boundedness and completeness . . . . . . . . 235 A.7 Some remarks about truncating metrics . . . . . . . . . . . . . . 235

A.7.1 Some new metrics on a metric space . . . . . . . . . . . . 235 A.7.2 Another new metric on Y . . . . . . . . . . . . . . . . . . 236 A.7.3 Some more easy inequalities . . . . . . . . . . . . . . . . . 236 A.7.4 Open and closed balls with respect to d_{Y,t} . . . . . . . . . 237 A.7.5 Compatibility of d_{Y} and d_{Y,t} . . . . . . . . . . . . . . . . 237 A.7.6 Sequences of mappings into Y . . . . . . . . . . . . . . . . 237 A.7.7 Mappings to and from other metric spaces . . . . . . . . . 237

A.8 Sequences of metric spaces . . . . . . . . . . . . . . . . . . . . . . 238 A.8.1 Some metrics on countable products . . . . . . . . . . . . 238 A.8.2 When d_{X_{j}} is an ultrametric on X_{j} for each j . . . . . . . 238 A.8.3 Closed balls in X are products of closed balls . . . . . . . 239 A.8.4 Some open sets in X . . . . . . . . . . . . . . . . . . . . . 239 A.8.5 Products of closed sets are closed sets in X . . . . . . . . 239 A.8.6 Convergence of sequences in X . . . . . . . . . . . . . . . 239 A.8.7 Diameters of products in X . . . . . . . . . . . . . . . . . 240

A.9 When {b_{j}}_{j=1}^{∞} converges to 0 . . . . . . . . . . . . . . . . . . . . 240 A.9.1 Open balls in X are products of open balls . . . . . . . . 241 A.9.2 More on convergence of sequences in X . . . . . . . . . . 241 A.9.3 Total boundedness in infinite products . . . . . . . . . . . 241 A.9.4 Compactness in infinite products . . . . . . . . . . . . . . 242

A.10 Snowflake metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 A.10.1 A nice inequality . . . . . . . . . . . . . . . . . . . . . . . 243 A.10.2 Some properties of d_{X}^{a} . . . . . . . . . . . . . . . . . . . . 243

A.11 Lipschitz mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 244 A.11.1 Using snowflake metrics . . . . . . . . . . . . . . . . . . . 245 A.11.2 Real-valued Lipschitz functions . . . . . . . . . . . . . . . 245

A.12 The set of binary sequences . . . . . . . . . . . . . . . . . . . . . 246 A.12.1 Another description of d_{B,θ}(x, w) . . . . . . . . . . . . . . 246 A.12.2 The usual mapping from B onto [0, 1] . . . . . . . . . . . 247 A.12.3 A mapping from B onto the Cantor set . . . . . . . . . . 247 A.12.4 Mapping the Cantor set onto the unit interval . . . . . . . 248

A.13 More on B and ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.13.1 The restrictions of ϕ to A_{0}, A_{1}, and B \ A . . . . . . . . . 250 A.13.2 Continuity of ϕ^{−1} on [0, 1] \ E . . . . . . . . . . . . . . . 250

A.14 More on product spaces . . . . . . . . . . . . . . . . . . . . . . . 251 A.14.1 Some bases for the topology of X . . . . . . . . . . . . . . 251 A.14.2 Dense sets in countable products . . . . . . . . . . . . . . 252 A.14.3 Some bases for the topology of X again . . . . . . . . . . 252 A.14.4 Countability of these bases for the topology of X . . . . . 253

Bibliography 254

Index 277

Chapter 1

Some basic notions

1.1 Extended real numbers

The set of extended real numbers is defined to be the set R of real numbers together with two additional elements, +∞ and −∞, such that −∞ < x < +∞ (1.1.1) for every x ∈ R. When we use extended real numbers here, we shall typically only be concerned with nonnegative extended real numbers.

It is customary to put x + (+∞) = (+∞) + x = +∞ (1.1.2) when −∞ < x ≤ +∞, and x + (−∞) = (−∞) + x = −∞ (1.1.3) when −∞ ≤ x < +∞. Similarly, we put x (+∞) = (+∞) x = +∞, x (−∞) = (−∞) x = −∞ (1.1.4) when 0 < x ≤ +∞, and x (+∞) = (+∞) x = −∞, x (−∞) = (−∞) x = +∞ (1.1.5) when −∞ ≤ x < 0.

We also put x/(+∞) = x/(−∞) = 0 when x ∈ R. Although 1/0 is not normally defined, it may be appropriate to interpret it as being +∞ when dealing with nonnegative extended real numbers.

The product of 0 and ±∞ is normally not defined. However, in measure theory and integration, it turns out to be convenient to interpret it as being equal to 0.

1.1.1 Some remarks about sequences of real numbers

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers. We say that x_{j} tends to +∞ as j → ∞, or x_{j} → +∞ as j → ∞, if for every nonnegative real number R there is a positive integer L such that

(1.1.6) x_{j} > R for every j ≥ L.

Similarly, we say that x_{j} tends to −∞ as j → ∞, or x_{j} → −∞ as j → ∞, if for every nonnegative real number R there is a positive integer L such that

(1.1.7) x_{j} < −R for every j ≥ L.

Suppose that x_{j} → x as j → ∞ for some extended real number x, and similarly that {y_{j}}_{j=1}^{∞} is a sequence of real numbers with y_{j} → y as j → ∞ for some extended real number y. If x + y is defined as an extended real number, then x_{j} + y_{j} → x + y as j → ∞. (1.1.8)

Of course, this is well known when x, y ∈ R. If x = +∞, for instance, then one can check that x_{j} + y_{j} → +∞ as j → ∞ (1.1.9)

when the set of y_{j}’s has a lower bound in R. In particular, this holds when y_{j} → y as j → ∞, with y > −∞.

Similarly, if x y is defined as an extended real number, then

x_{j} y_{j} → x y as j → ∞. (1.1.10)

This does not include the case where one of x or y is 0 and the other is ±∞. If x, y ∈ R, then (1.1.10) is well known, as before. If x = +∞ and y > 0, for example, then x_{j} y_{j} → +∞ as j → ∞. (1.1.11)

More precisely, this works when there is a positive real number a such that y_{j} ≥ a for all but finitely many j.

If {x_{j}}_{j=1}^{∞} is a sequence of nonzero real numbers such that

|x_{j}| → +∞ as j → ∞, (1.1.12)

then it is easy to see that

1/x_{j} → 0 as j → ∞. (1.1.13)

If {x_{j}}_{j=1}^{∞} is a sequence of positive real numbers that converges to 0, then one can verify that 1/x_{j} → +∞ as j → ∞. (1.1.14)

1.1.2 A topology on the set of extended real numbers

One might like to have a metric on the set of extended real numbers so that sequences of real numbers tending to +∞ or −∞ correspond to convergent sequences with respect to that metric. It is not too difficult to do that, although one cannot simply extend the standard metric on the real line to the set of extended real numbers in a convenient way. Instead one may consider metrics on the set of extended real numbers whose restriction to R is equivalent to the standard metric in a suitable sense, so that they determine the same convergent sequences in R in particular.

It is somewhat simpler to define an appropriate topology on the set of extended real numbers, if one is familiar with the notion of a topological space. This topology is determined by a metric on the set of extended real numbers, as in the preceding paragraph. One can choose this topology so that the corresponding induced topology on the real line is the same as the standard topology on R. One can also choose this topology so that sequences of real numbers tending to +∞ or −∞ correspond to convergent sequences with respect to the topology.

It is sometimes convenient to consider sequences of extended real numbers as well. One can define what it means for a sequence of extended real numbers to tend to +∞ or to −∞ in the same way as for sequences of real numbers. A sequence of extended real numbers can converge to a real number only when all but finitely many terms in the sequence are real numbers, so that convergence is basically the same as for sequences of real numbers.

1.2 More on extended real numbers

If A is any set of extended real numbers, then the notions of upper and lower bounds for A in the set of extended real numbers may be defined in the usual way. Similarly, the notions of supremum or least upper bound and infimum or greatest lower bound for A in the set of extended real numbers may be defined as usual. In fact, the supremum and infimum of A in the set of extended real numbers always exist, as follows.

If +∞ ∈ A, or if A ∩ R has no upper bound in R, then sup A = +∞. If +∞ ∉ A, and if A ∩ R is nonempty and has an upper bound in R, then the supremum of A is the same as the supremum of A∩R in R. Otherwise, if A = ∅ or A = {−∞}, then sup A = −∞.

The infimum of A may be characterized analogously. Note that

inf A ≤ sup A (1.2.1)

when A ≠ ∅.

Let t be a positive real number, and put

t A = {t x : x ∈ A}. (1.2.2)

One can check that sup(t A) = t (sup A), (1.2.3)

and similarly for the infimum of t A.

1.2.1 Suprema and infima of functions

Let E be a set, and let f be a function on E with values in the set of extended real numbers. We may use the notation

sup_{x∈E} f(x) = sup f(E) (1.2.4)

and

inf_{x∈E} f(x) = inf f(E). (1.2.5)

If t is a positive real number, then

sup_{x∈E}(t f(x)) = t ( sup_{x∈E} f(x)) (1.2.6)

and

inf_{x∈E}(t f(x)) = t ( inf_{x∈E} f(x)), (1.2.7)

as in (1.2.2) and (1.2.3), with A = f(E), respectively.

Let g be another function on E with values in the extended real numbers. One can check that

(1.2.8)  sup_{x∈E}(f(x) + g(x)) ≤ ( sup_{x∈E} f(x)) + ( sup_{x∈E} g(x))

and

(1.2.9)  inf_{x∈E}(f(x) + g(x)) ≥ ( inf_{x∈E} f(x)) + ( inf_{x∈E} g(x)).

Suppose now that f, g ≥ 0 on E. One can verify that

(1.2.10)  sup_{x∈E}(f(x) g(x)) ≤ ( sup_{x∈E} f(x)) ( sup_{x∈E} g(x))

and

(1.2.11)  inf_{x∈E}(f(x) g(x)) ≥ ( inf_{x∈E} f(x)) ( inf_{x∈E} g(x)).

1.2.2 Infinite series with nonnegative terms

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers that is monotonically increasing, so that x_{j} ≤ x_{j+1} for each j. If the set of x_{j}’s has an upper bound in R, then it is well known that {x_{j}}_{j=1}^{∞} converges to the supremum of this set. Otherwise, if the set of x_{j}’s does not have an upper bound in R, then it is easy to see that x_{j} → +∞ as j → ∞. In both cases, if we put

x = sup_{l≥1} x_{l}, (1.2.12)

then we have that x_{j} → x as j → ∞.

Consider an infinite series ∑_{j=1}^{∞} a_{j} of nonnegative real numbers. Of course, the corresponding sequence of partial sums ∑_{j=1}^{n} a_{j} is monotonically increasing. If the sequence of partial sums has an upper bound in R, then ∑_{j=1}^{∞} a_{j} converges, with

∑_{j=1}^{∞} a_{j} = sup_{n≥1} ∑_{j=1}^{n} a_{j}, (1.2.13)

as in the preceding paragraph. Otherwise, if the sequence of partial sums does not have an upper bound on R, then we may interpret the value of ∑_{j=1}^{∞} a_{j} as being +∞.

If t is a positive real number, then

∑_{j=1}^{n} t a_{j} = t ∑_{j=1}^{n} a_{j} (1.2.14)

for each positive integer n. This implies that

∑_{j=1}^{∞} t a_{j} = t ∑_{j=1}^{∞} a_{j}. (1.2.15)

More precisely, this also works when the sums are equal to +∞. This works when t = 0 too, with the right side interpreted as being equal to 0 even when ∑_{j=1}^{∞} a_{j} = +∞, as in the previous section.

Let ∑_{j=1}^{∞} b_{j} be another infintie series of nonnegative real numbers, and note that

∑_{j=1}^{n}(a_{j} + b_{j}) = ∑_{j=1}^{n} a_{j} + ∑_{j=1}^{n} b_{j} (1.2.16)

for every positive integer n. One can use this to get that

∑_{j=1}^{∞}(a_{j} + b_{j}) = ∑_{j=1}^{∞} a_{j} + ∑_{j=1}^{∞} b_{j}. (1.2.17)

This also works with suitable interpretations when any of the sums are equal to +∞, as before.

1.3 Upper and lower limits of sequences in R

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers, and let E be the set of extended real numbers x such that there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} with

x_{j_{l}} → x as l → ∞. (1.3.1)

It is well known that E ≠ ∅. (1.3.2)

Indeed, if the set of x_{j}’s has upper and lower bounds in R, then E ∩ R ≠ ∅, because closed intervals in R are sequentially compact. If the set of x_{j}’s does not have an upper bound in R, then it is not too difficult to show that +∞ ∈ E. Similarly, if the set of x_{j}’s does not have a lower bound in R, then −∞ ∈ E.

1.3.1 Defining upper and lower limits

The upper limit or limit superior of {x_{j}}_{j=1}^{∞} may be defined by

(1.3.3)  lim sup_{j→∞} x_{j} = sup E,

as on p56 of [299]. Similarly, the lower limit or limit inferior of {x_{j}}_{j=1}^{∞} may be defined by

(1.3.4)  lim inf_{j→∞} x_{j} = inf E,

as in [299]. Some other equivalent formulations are sometimes used, and we shall say more about that soon.

Note that

(1.3.5)  lim inf_{j→∞} x_{j} ≤ lim sup_{j→∞} x_{j}.

If there is an extended real number x such that x_{j} → x as j → ∞, then

E = {x}, (1.3.6)

and

(1.3.7)  lim sup_{j→∞} x_{j} = lim inf_{j→∞} x_{j} = x.

One can check that

(1.3.8)  lim sup_{j→∞} x_{j} ≤ sup_{j≥1} x_{j}

and

(1.3.9)  lim inf_{j→∞} x_{j} ≥ inf_{j≥1} x_{j},

directly from the previous definitions, or using some of the other characterizations of the upper and lower limits that we shall discuss. It is easy to give examples where these inequalities are strict.

1.3.2 Some properties of the upper and lower limits

Put y = lim sup_{j→∞} x_{j} and u = lim inf_{j→∞} x_{j}. It is not too difficult to show that (1.3.10) if z ∈ R and y < z, then x_{j} < z for all but finitely many j.

Otherwise, there would be a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that

x_{j_{l}} ≥ z (1.3.11) for each l. This would imply that there is an element of E greater than or equal to z, using a subsequence of {x_{j_{l}}}_{l=1}^{∞} that tends to an extended real number that is necessarily greater than or equal to z. This corresponds to part (b) of Theorem 3.17 on p56 of [299].

Similarly,

(1.3.12) if t ∈ R and t < u, then x_{j} > t for all but finitely many j.

This can be obtained using an analogous argument, or by reducing to the previous case, for {−x_{j}}_{j=1}^{∞} in place of {x_{j}}_{j=1}^{∞}.

We also have that

(1.3.13) if w ∈ R and w < y, then x_{j} > w for infinitely many j.

Indeed, if w < y, then w is not an upper bound for E, by definition of the supremum, so that there is an x ∈ E with w < x. One can use this to get that x_{j} > w for infinitely many j. Alternatively, if x_{j} ≤ w for all but finitely many j, then one can check that w is an upper bound for E.

Similarly,

(1.3.14) if v ∈ R and u < v, then x_{j} < v for infinitely many j.

One can check that y is uniquely determined by (1.3.10) and (1.3.13). Similarly, u is uniquely determined by (1.3.12) and (1.3.14).

1.3.3 Some variants and related properties

Part (a) of Theorem 3.17 on p56 of [299] says that

y ∈ E. (1.3.15)

This can be obtained from (1.3.10) and (1.3.13). Note that (1.3.15) implies (1.3.14).

Another part of Theorem 3.17 on p56 of [299] states that y is uniquely determined by (1.3.10) and (1.3.15). This is essentially the same as before.

Similarly, u ∈ E, (1.3.16) and u is uniquely determined by (1.3.12) and (1.3.16).

Suppose for the moment that

(1.3.17)  lim sup_{j→∞} x_{j} = lim inf_{j→∞} x_{j},

and let x be their common value. One can check that x_{j} → x as j → ∞, using (1.3.10) and (1.3.12).

If {a_{j}}_{j=1}^{∞} and {b_{j}}_{j=1}^{∞} are sequences of real numbers, then it is well known that

(1.3.18)  lim sup_{j→∞}(a_{j} + b_{j}) ≤ lim sup_{j→∞} a_{j} + lim sup_{j→∞} b_{j},

as long as the right side is defined as an extended real number, so that it is not a sum of +∞ and −∞. This corresponds to part (b) of Exercise 4 on p32 of [298], and to Exercise 5 on p78 of [299].

1.3.4 Another approach to upper and lower limits

Let {x_{j}}_{j=1}^{∞} be a sequence of extended real numbers, and consider

sup_{j≥l} x_{j} (1.3.19)

for each positive integer l. This is a monotonically decreasing sequence of extended real numbers, and the upper limit of {x_{j}}_{j=1}^{∞} is sometimes defined by

(1.3.20)  lim sup_{j→∞} x_{j} = inf_{l≥1} ( sup_{j≥l} x_{j}),

as in Definition 1.13 on p14 of [298].

Similarly,

inf_{j≥l} x_{j} (1.3.21)

is a monotonically increasing sequence of extended real numbers, and the lower limit of {x_{j}}_{j=1}^{∞} is sometimes defined by

(1.3.22)  lim inf_{j→∞} x_{j} = sup_{l≥1} ( inf_{j≥l} x_{j}),

as in [298]. This is essentially the way that upper and lower limits of sequences of real numbers are defined in Section 2.9 beginning on p46 of [135].

If (1.3.19) is finite for each l, then this sequence tends to the infimum of its terms as l → ∞. This also works when (1.3.19) may be ±∞, with suitable interpretations. Similarly, the sequence (1.3.21) tends to the supremum of its terms as l → ∞, with suitable interpretations when these terms may be ±∞.

It is mentioned on p14 of [298] that (1.3.20) satisfies the same conditions as the previous definition of the upper limit, so that the two approaches are equivalent. It is perhaps a bit simper to check that (1.3.20) satisfies (1.3.10) and (1.3.13), and similarly that (1.3.22) satisfies (1.3.12) and (1.3.14). This corresponds to Theorem 2.9 L on p50 of [135] for bounded sequences of real numbers. It is mentioned just after the proof that (1.3.20) and (1.3.22) are characterized by these properties.

The fact that (1.3.20) is an element of the set E defined earlier is shown in the proof of Theorem 2.9 M on p51 of [135] for bounded sequences of real numbers. The analogous statement for (1.3.22) is the first part of Exercise 3 on p51 of [135]. Exercise 2 on p51 of [135] states that (1.3.20) is an upper bound for E, and the second part of Exercise 3 on p51 of [135] is the same as saying that (1.3.22) is a lower bound for E.

1.4 Indicator functions

Let X be a set, and let A be a subset of X. The indicator function of A with respect to X is defined by

(1.4.1)  1_{A}(x) = 1 when x ∈ A
= 0 when x ∈ X \ A.

This is also known as the characteristic function of A, and denoted χ_{A}. However, this terminology is often used for something else.

1.4.1 Some upper and lower limits

Let {x_{j}}_{j=1}^{∞} be an infinite sequence such that for each j, x_{j} = 0 or 1. One can check that

(1.4.2) lim sup_{j→∞} x_{j} = 1 when x_{j} = 1 for infinitely many j
= 0 when x_{j} = 0 for all but finitely many j.

Similarly,

(1.4.3) lim inf_{j→∞} x_{j} = 0 when x_{j} = 0 for infinitely many j
= 1 when x_{j} = 1 for all but finitely many j.

1.4.2 Upper and lower limits of sequences of sets

Let {A_{j}}_{j=1}^{∞} be a sequence of subsets of X. The upper limit or limit superior of {A_{j}}_{j=1}^{∞} may be defined as a subset of X by

(1.4.4)  lim sup_{j→∞} A_{j} = {x ∈ X : x ∈ A_{j} for infinitely many j},

as in Definition 2.12 B on p65 of [135].

Similarly, the lower limit or limit inferior of {A_{j}}_{j=1}^{∞} may be defined as a subset of X by

(1.4.5) lim inf_{j→∞} A_{j} = {x ∈ X : x ∈ A_{j} for all but finitely many j},

as in [135]. Note that

(1.4.6)  lim inf_{j→∞} A_{j} ⊆ lim sup_{j→∞} A_{j},

as in Exercise 1 on p65 of [135].

If x ∈ X, then

(1.4.7)  lim sup_{j→∞} 1_{A_{j}}(x) = 1_{lim sup_{j→∞} A_{j}}(x),

as mentioned on p65 of [135]. Similarly,

(1.4.8)  lim inf_{j→∞} 1_{A_{j}}(x) = 1_{lim inf_{j→∞} A_{j}}(x),

as in [135]. This uses the remarks in the previous subsection.

1.4.3 More on sequences of subsets of X

One can show that

lim sup_{j→∞} A_{j} = ⋂_{l=1}^{∞} ( ⋃_{j=l}^{∞} A_{j}), (1.4.9)

as in part (b) of Exercise 3 on p66 of [135]. Similarly,

lim inf_{j→∞} A_{j} = ⋃_{l=1}^{∞} ( ⋂_{j=l}^{∞} A_{j}), (1.4.10)

as in part (b) of Exercise 4 on p66 of [135].

If A_{j} ⊆ A_{j+1} for each j, then

(1.4.11)  lim sup_{j→∞} A_{j} = lim inf_{j→∞} A_{j} = ⋃_{j=1}^{∞} A_{j},

as in part (a) of Exercise 5 on p66 of [135]. Similarly, if A_{j+1} ⊆ A_{j} for each j, then

(1.4.12)  lim sup_{j→∞} A_{j} = lim inf_{j→∞} A_{j} = ⋂_{j=1}^{∞} A_{j},

as in part (b) of Exercise 5 on p66 of [135].

If

(1.4.13)  lim sup_{j→∞} A_{j} = lim inf_{j→∞} A_{j},

then one may say that {A_{j}}_{j=1}^{∞} converges, with limit

lim_{j→∞} A_{j} (1.4.14)

equal to (1.4.13). This means that

lim_{j→∞} 1_{A_{j}}(x) = 1_{lim_{j→∞} A_{j}}(x) (1.4.15)

for each x ∈ X. Monotonic sequences of subsets of X converge, as in the preceding paragraph.

1.5 Algebras and σ-algebras of sets

We are often interested here in defining something like the “measure” or “probability” of a subset of a set X. However, we may only want to do this for some collection of subsets of X. In this section, we consider some types of collections of subsets of X that can be helpful for this.

Let A be a nonempty collection of subsets of X. We say that A is an algebra of subsets of X if it has the following two properties. First, if A, B ∈ A, then

A ∪ B ∈ A. (1.5.1)

1.5. ALGEBRAS AND σ-ALGEBRAS OF SETS 11

Second, if A ∈ A, then X \ A ∈ A. (1.5.2)

This corresponds to part of Definition 2.1 on p11 of [20], parts of 11.1 on p118 of [158] and Definition 1.11 on p4 of [159], with some minor differences in the formulation, and it is also mentioned on p21 of [118], p10 of [298], and p270 of [335].

If A is an algebra of subsets of X and A, B ∈ A, then it is easy to see that

A ∩ B ∈ A. (1.5.3)

This uses the fact that

X \ (A ∩ B) = (X \ A) ∪ (X \ B). (1.5.4)

We also have that X ∈ A, (1.5.5)

because X = A ∪ (X \ A) for every A ⊆ X and A ≠ ∅, by hypothesis. Similarly, the empty set is an element of A.

1.5.1 σ-Algebras of sets

We say that A is a σ-algebra of subsets of X if (1.5.2) holds for every A ∈ A, and if for every sequence A_{1}, A_{2}, A_{3}, … of elements of A, we have that

⋃_{j=1}^{∞} A_{j} ∈ A. (1.5.6)

This corresponds to Definition 2.23 on p26 of [14], part of Definition 2.1 on p11 of [20], parts of 11.1 on p118 of [158] and Definition 1.13 on p4 of [159], and to part (a) of Definition 1.3 on p9 of [298], and it is mentioned on p21 of [118] and p23, 263 of [335].

If A is a σ-algebra of subsets of X, then A is an algebra of subsets of X, and (1.5.5) holds in particular. In this case, if A_{1}, A_{2}, A_{3}, … is a sequence of elements of A, then

⋂_{j=1}^{∞} A_{j} ∈ A, (1.5.7)

because

X \ ( ⋂_{j=1}^{∞} A_{j}) = ⋃_{j=1}^{∞}(X \ A_{j}). (1.5.8)

The collection of all subsets of X is a σ-algebra, as is the collection consisting only of X and ∅.

A σ-algebra of subsets of X is called a Borel family in Definition 2.2.1 on p59 of [113]. The σ-algebra of Borel sets in a metric or topological space will be discussed in Section 2.2.

12 CHAPTER 1. SOME BASIC NOTIONS

1.5.2 Rings of sets

We say that A is a ring of subsets of X if for every A, B ∈ A we have that (1.5.1) holds, and also A \ B ∈ A. (1.5.9) This is mentioned in Exercise 1 on p24 of [118], and it corresponds to parts of 11.1 on p118 of [158] and Definition 1.11 on p4 of [159] again, as well as Definition 11.1 on p301 of [299], although the condition that A be nonempty does not appear to be included in [299]. The nonemptiness of A implies that ∅ ∈ A, (1.5.10) as mentioned in [159].

If A is a ring of subsets of A and A, B ∈ A, then (1.5.3) holds, because A ∩ B = A \ (A \ B), (1.5.11) as mentioned in [159, 299]. If X ∈ A, then A is an algebra of subsets of X, as in [159].

If A and B are subsets of X, then their symmetric difference is the subset of X defined by A△B = (A \ B) ∪ (B \ A). (1.5.12) If A is a ring of subsets of X and A, B ∈ A, then A△B ∈ A, (1.5.13) as mentioned on p4 of [159].

Conversely, suppose that (1.5.3) and (1.5.13) hold for all A, B ∈ A, and let us check that A is a ring of subsets of X. If A, B are any subsets of X, then A \ B = A△(A ∩ B). (1.5.14) This implies that (1.5.9) holds when A, B ∈ A. We also have that (A \ B)△B = ((A \ B) \ B) ∪ (B \ (A \ B)) = (A \ B) ∪ B = A ∪ B. (1.5.15) It follows that (1.5.1) holds when A, B ∈ A.

1.5.3 More on rings of sets

If A and B are any subsets of X, then 1_{A}(x) 1_{B}(x) = 1_{A∩B}(x) (1.5.16) for every x ∈ X. We also have that 1_{A}(x) + 1_{B}(x) = 1_{A∪B}(x) + 1_{A∩B}(x) = 1_{A△B}(x) + 2 1_{A∩B}(x) (1.5.17) for every x ∈ X. This uses the fact that

A△B = (A ∪ B) \ (A ∩ B). (1.5.18)

In particular, this means that

1_{A}(x) + 1_{B}(x) ≡ 1_{A△B}(x) modulo 2 (1.5.19)

for every x ∈ X.

It is well known that {0, 1} is a field with respect to addition and multiplication modulo 2. The collection of all functions on X with values in {0, 1} may be considered as a commutative ring with respect to pointwise addition and multiplication of functions module 2. Of course, these functions on X are the same as indicator functions of subsets of X.

This means that the collection of all subsets of X may be considered as a commutative ring with respect to symmetric differences as addition and intersections as multiplication. Rings of subsets of X correspond to subrings of this ring, as discussed in [377].

1.5.4 σ-Rings of sets

A ring A of subsets of X is said to be a σ-ring of subsets of X if (1.5.6) holds for every sequence A_{1}, A_{2}, A_{3}, … of elements of A, as in Exercise 1 on p24 of [118], 11.1 on p118 of [158], Definition 1.13 on p4 of [159], and Definition 11.1 on p301 of [299]. If X ∈ A, then it follows that A is a σ-algebra of subsets of X. This condition is included in the definition of a measurable space on p310 of [299].

If A is a σ-ring of subsets of X and A_{1}, A_{2}, A_{3}, … is a sequence of elements of A, then (1.5.7) holds, as mentioned on p301 of [299]. This uses the observation that

⋂_{j=1}^{∞} A_{j} = A_{1} \ ( ⋃_{j=1}^{∞}(A_{1} \ A_{j})). (1.5.20)

The collection of all finite subsets of X is a ring of subsets of X. If X has infinitely many elements, then this is neither an algebra of subsets of X, nor a σ-ring of subsets of X. This example is mentioned on p4 of [159], when X is the set Z_{+} of positive integers.

Similarly, the collection of all subsets of X with only finitely or countably many elements is a σ-ring of subsets of X. If X is uncountable, then this is not a σ-algebra of subsets of X, as mentioned on p4 of [159].

1.6 Some remarks about intervals

Let a and b be extended real numbers with a < b. We may use (a, b) for the set of real numbers x such that a < x < b. (1.6.1)

Similarly, we may use [a, b] for the set of extended real numbers x such that

a ≤ x ≤ b. (1.6.2)

We may also use this notation when a = b, so that the set contains only one element. In the same way, we may use the notation [a, b) for the set of extended real numbers x such that a ≤ x < b, (1.6.3)

and the notation (a, b] for the set of extended real numbers x such that

a < x ≤ b. (1.6.4)

A subset of the real line any of these four types may be described as an interval. The length of such an interval is defined to be b − a. Thus the length of the interval is finite exactly when a, b ∈ R, and we may say that the interval is bounded in this case. The empty set may be considered as an interval in R as well, with length 0.

1.6.1 Some elementary subsets of R

Let us say that a subset of R is a bounded elementary set if it can be expressed as the union of finitely many bounded intervals. This corresponds to an elementary subset of R as defined on p303 of [299]. Let E(R) be the collection of all bounded elementary subsets of R. This is a ring of subsets of R, and not a σ-ring, as in (12) on p303 of [299]. Note that E(R) is not an algebra of subsets of R.

Let us say that a subset of the real line is a possibly unbounded elementary set if it can be expressed as the union of finitely many intervals that are not necessarily bounded. Let E_{1}(R) be the collection of all possibly unbounded elementary subsets of R. This is an algebra of subsets of R that is not a σalgebra.

1.6.2 Intervals in R^{n}

Let n be a positive integer, and let A_{1}, …, A_{n} be n sets. The Cartesian product of A_{1}, …, A_{n} may be denoted

A_{1} × A_{2} × ⋯ × A_{n} or ∏_{j=1}^{n} A_{j}, (1.6.5)

and is the set of n-tuples a = (a_{1}, …, a_{n}) with a_{j} ∈ A_{j} for j = 1, …, n.

In particular, the space R^{n} of n-tuples of real numbers is the same as the Cartesian product of n copies of the real numbers. A subset of R^{n} may be called a bounded interval if it is the Cartesian product of n bounded intervals in R. This corresponds to the definition of an interval in R^{n} in Definition 11.4 on p302 of [299]. The n-dimensional volume of such a bounded interval may be defined as the product of the lengths of these n intervals, as on p303 of [299].

Note that the empty set is considered to be a bounded interval in R^{n}, with n-dimensional volume equal to 0.

Let us say that a subset of R^{n} is a possibly unbounded interval if it is the Cartesian product of n intervals in the real line that are not necessarily bounded. The n-dimensional volume may be defined as the product of the lengths of these n intervals, as before. This is interpreted as being +∞ when each of the intervals has positive length, and at least one of the intervals has length +∞. If at least one of the intervals has length 0, then we interpret the product as being equal to 0, even if some of the other intervals have length +∞.

1.6.3 Some elementary subsets of R^{n}

Let us say that a subset of R^{n} is a bounded elementary set if it can be expressed as the union of finitely many bounded intervals in R^{n}. This corresponds to an elementary subset of R^{n} as defined on p303 of [299] again. The collection E(R^{n}) of all bounded elementary subsets of R^{n} is a ring of subsets of R^{n}, and not a σ-ring, as in (12) on p303 of [299]. This is also not an algebra of subsets of R^{n}, as before.

Let us say that a subset of R^{n} is a possibly unbounded elementary set if it can be expressed as the union of finitely many possibly unbounded intervals in R^{n}. The collection E_{1}(R^{n}) of all possibly unbounded elementary subsets of R^{n} is an algebra of subsets of R^{n} that is not a σ-algebra.

Sometimes one may be interested in elementary subsets of a subset of R^{n}, and this will be discussed more broadly in the next subsection.

1.6.4 Some more rings and algebras of sets

Let X be a set, let A be a nonempty collection of subsets of X, and let Y be a subset of X. Note that A_{Y} = {A ∩ Y : A ∈ A} (1.6.6)

is a nonempty collection of subsets of Y. If A is a ring, algebra, σ-ring, or σ-algebra of subsets of X, then one can check that A_{Y} has the same property as a collection of subsets of Y.

Suppose for the moment that there is an element A_{1} of A such that

Y ⊆ A_{1}. (1.6.7)

This means that Y = A_{1} ∩ Y ∈ A_{Y} . (1.6.8)

If A is a ring or σ-ring of subsets of X, then it follows that A_{Y} is an algebra or σ-algebra of subsets of Y, as appropriate.

Let A_{Y,0} be the collection of elements of A that are subsets of Y. Observe that A_{Y,0} ⊆ A_{Y} , (1.6.9) because A ∩ Y = A when A ⊆ Y, and that ∅ ∈ A_{Y,0} when ∅ ∈ A. If A is a ring or σ-ring of subsets of X, then A_{Y,0} has the same property, as a collection of subsets of Y or X. We also have that

A_{Y,0} = A_{Y} (1.6.10)

when Y ∈ A and A is a ring of subsets of X.

1.7 Finitely and countably additive measures

Let X be a set, and let A be a ring of subsets of X. Also let μ be a function on A with values in the set of nonnegative extended real numbers. We say that μ is a finitely additive measure on X with respect to A if

μ(∅) = 0 (1.7.1)

and μ(A ∪ B) = μ(A) + μ(B) (1.7.2)

for all A, B ∈ A with A∩B = ∅. If A_{1}, …, A_{n} are finitely many pairwise-disjoint elements of A, then it follows that

μ( ⋃_{j=1}^{n} A_{j}) = ∑_{j=1}^{n} μ(A_{j}). (1.7.3)

This corresponds to part of 11.1 on p118 of [158], and to part of Definition 10.3 on p126 of [159] when A is an algebra of subsets of X.

This basically corresponds to part of Definition 11.2 on p301 of [299] as well. Nonnegativity of μ is not part of this definition, although that is mostly what is considered, as in Definition 11.12 on p310 of [299]. Other types of measures are very interesting and important, but they are not our main focus here. More precisely, in [299], μ is allowed to take the value +∞ on A, or the value −∞, but not both, so that the sum on the right side of (1.7.2) is defined. One also asks that μ(A) be finite for some A ∈ A, and this is used to obtain (1.7.1), as in (5) on p301 of [299].

Suppose that μ is a finitely additive measure on X with respect to A, and that A, B are elements of A with A ⊆ B. Observe that

μ(A) ≤ μ(A) + μ(B \ A) = μ(B), (1.7.4)

as in part (c) of Theorem 1.19 on p17 of [298], and (8) on p302 of [299].

1.7.1 Countable additivity

Similarly, we say that μ is a countably additive measure on X with respect to A if (1.7.1) holds, and if

μ( ⋃_{j=1}^{∞} A_{j}) = ∑_{j=1}^{∞} μ(A_{j}) (1.7.5)

for every sequence A_{1}, A_{2}, A_{3}, … of pairwise-disjoint elements of A such that

⋃_{j=1}^{∞} A_{j} ∈ A. (1.7.6)

This corresponds to parts of 11.1 on p118 of [158], Definition 10.3 on p126 of [159], and Definition 11.2 on p301 of [299], as before. If A is an algebra of subsets of X, then μ may be called a premeasure, as on p36 of [20], p30 of [118], and p270 of [335].

Of course, (1.7.6) holds automatically when A is a σ-ring of subsets of X. This also corresponds to part (a) of Definition 1.18 on p17 of [298] when A is a σ-algebra of subsets of X. One may refer to μ as a positive or nonnegative measure, for emphasis, as in [298]. This corresponds to the definition of a measure on p41 of [14], p17 of [20], p24 of [118], and p263 of [335] when A is a σ-algebra of subsets of X as well.

1.7.2 Measurable and measure spaces

If A is a σ-algebra of subsets of X, then (X, A) may be called a measurable space, as in Definition 2.26 on p27 of [14], on p11 of [20], p25 of [118], p149 of [159], and in part (b) of Definition 1.3 on p9 of [298]. In this case, the elements of A may be called measurable subsets of X with respect to A.

If (X, A) is a measurable space and μ is a nonnegative countably additive measure on (X, A), then (X, A, μ) is called a measure space. This corresponds to Definition 2.56 on p42 of [14], remarks on p18 of [20] and p25 of [118], part of Definition 10.3 on p126 of [159], and to part (b) of Definition 1.18 on p17 of [298], and it is the same as on p263 of [335]. These terms are used a bit differently on p310 of [299].

If (X, A, μ) is a measure space and

μ(X) < +∞, (1.7.7)

then (X, A, μ) is said to be a finite measure space, and μ is said to be a finite measure, as in Definition 5.18 on p123 of [14], Definition 3.7 on p19 of [20], a remark on p25 of [118], and Definition 10.3 on p126 of [159]. If

μ(X) = 1, (1.7.8)

then (X, A, μ) may be called a probability space, and μ a probability measure, as in Definition 12.1 on p381 of [14], on p20 of [20], and on p313 of [118].

1.7.3 Some basic examples of measures

Let X be any set, and let A be the collection of all subsets of X. If A is any subset of X, then let μ(A) be the number of elements of A, interpreted as +∞ when A has infinitely many elements. One can check that this is a countably additive measure, called counting measure on X. This corresponds to the first example in 2.55 on p41 of [14], Example 3.2 on p18 of [20], an example on p25 of [118], Example 10.4 (a) on p127 of [159], and Example 1.20 (a) on p18 of [298], and it is mentioned on p310 of [299] with X = Z_{+}. This is also mentioned in (i) on p263 of [335] when X is countable.

Let x_{0} ∈ X be given. If A is any subset of X, then put

(1.7.9)  δ_{x_{0}}(A) = 1 when x_{0} ∈ A
= 0 when x_{0} ∉ A.

It is easy to see that this is a probability measure, as in the second example in 2.55 on p41 of [14], Example 3.4 of p18 of [20], an example on p25 of [118], and Example 1.20 (b) on p18 of [298].

1.7.4 Some more examples of measures

Let X be an uncountable set, and let A_{1} be the collection of subsets A of X such that either A has only finitely or countably many elements, or X \ A has only finitely or countably many elements. Put

μ_{1}(A) = 0 (1.7.10)

in the first case, and

μ_{1}(A) = 1 (1.7.11)

in the second case. One can check that A_{1} is a σ-algebra of subsets of X, and that μ_{1} defines a probability measure. This corresponds to Exercise 3.3 on p20 of [20], some remarks on p21, 25 of [118], and Exercise 6 on p33 of [298], and it is mentioned in a slightly different way on p42 of [14].

Now let X be any infinite set, and let A_{0} be the collection of subsets A of X such that either A has only finitely many elements, or X \ A has only finitely many elements. Put

μ_{0}(A) = 0 (1.7.12)

in the first case, and

μ_{0}(A) = 1 (1.7.13)

in the second case. One can verify that A_{0} is an algebra of subsets of X, and that μ_{0} is finitely additive. If X is uncountable, then A_{0} ⊆ A_{1}, and μ_{0} is the same as the restriction of μ_{1} to A_{0}. However, if X is countably infinite, then μ_{0} is not countably additive on A_{0}.

Alternatively, one can get a finitely additive measure μ on the σ-algebra of all subsets of X by taking the measure of finite subsets of X to be 0 and of infinite subsets of X to be equal to +∞. This is not countably additive, as mentioned on p25 of [118].

1.8 Measures and nonnegative sums

Let X be a nonempty set, and let A_{0} be the collection of all finite subsets of X. This is a ring of subsets of X, as mentioned in Subsection 1.5.4. Also let w be a function on X with values in the set of nonnegative extended real numbers. If E ∈ A_{0}, then put

μ_{w}(E) = ∑_{x∈E} w(x). (1.8.1)

This should be interpreted as being equal to 0 when E = ∅.

It is easy to see that μ_{w} is a finitely additive measure on X with respect to A_{0}. Conversely, if μ is any finitely additive measure on X with respect to A_{0}, then μ is of this form in a unique way. In fact, μ = μ_{w} exactly when

w(x) = μ({x}) (1.8.2)

for every x ∈ X.

1.8.1 Nonnegative sums over arbitrary sets

Let f be a nonnegative real-valued function on X. The sum

∑_{x∈X} f(x) (1.8.3)

may be defined as a nonnegative extended real number as the supremum of the finite subsums

∑_{x∈A} f(x) (1.8.4)

over all nonempty finite subsets A of X. Of course, if X has only finitely many elements, then this is the same as the usual sum over X.

We may also consider functions f on X with values in the set of nonnegative extended real numbers, by interpreting (1.8.4) as being +∞ when f(x) = +∞ for some x ∈ A. This means that (1.8.3) is equal to +∞ when f(x) = +∞ for some x ∈ X.

1.8.2 Comparison with infinite series of nonnegative numbers

Let {x_{j}}_{j=1}^{∞} be a sequence of distinct elements of X, and suppose for the moment that f(x) = 0 when x ∈ X is not one of the terms of this sequence. One can check that

∑_{x∈X} f(x) = ∑_{j=1}^{∞} f(x_{j}) (1.8.5)

under these conditions. More precisely, the sum on the right should also be interpreted as being equal to +∞ when any of its terms is +∞. Note that the partial sums

∑_{j=1}^{n} f(x_{j}) (1.8.6)

of the sum on the right side of (1.8.5) are of the form (1.8.4). It is easy to see that any finite subsum of the form (1.8.4) is less than or equal to (1.8.6) when n is sufficiently large.

1.8.3 Summable functions on X

Let us say that f is summable on X when

∑_{x∈X} f(x) < +∞. (1.8.7)

If there is an ϵ > 0 such that f(x) ≥ ϵ (1.8.8)

for infinitely many x ∈ X, then one can check that f is not summable on X. If f is summable on X, then it follows that (1.8.8) holds for only finitely many x ∈ X.

One can use this to get that

f(x) > 0 (1.8.9)

for only finitely or countably many x ∈ X when f is summable on X. More precisely, if f is summable on X, then for each positive integer l there are only finitely many x ∈ X such that (1.8.8) holds with ϵ = 1/l, and the set of x ∈ X such that (1.8.9) holds is the same as the union of this sequence of finite sets.

1.8.4 Some properties of the sum

If t is a positive real number, then one can check that

∑_{x∈X} t f(x) = t ∑_{x∈X} f(x), (1.8.10)

where the right side is interpreted as being +∞ when (1.8.3) is +∞, as usual. This uses the fact that

∑_{x∈A} t f(x) = t ∑_{x∈A} f(x) (1.8.11)

for every finite set A ⊆ X. This also works with t = 0, with the right side of (1.8.10) interpreted as being 0 even when (1.8.3) is +∞, as in a remark in Section 1.1.

If g is another nonnegative extended real-valued function on X, then

∑_{x∈X}(f(x) + g(x)) = ∑_{x∈X} f(x) + ∑_{x∈X} g(x). (1.8.12)

1.8. MEASURES AND NONNEGATIVE SUMS 21 To see this, let A be a finite subset of X, and observe that

∑_{x∈A}(f(x) + g(x)) = ∑_{x∈A} f(x) + ∑_{x∈A} g(x) ≤ ∑_{x∈X} f(x) + ∑_{x∈X} g(x). (1.8.13)

This implies that

∑_{x∈X}(f(x) + g(x)) ≤ ∑_{x∈X} f(x) + ∑_{x∈X} g(x). (1.8.14)

If B is another finite subset of X, then

∑_{x∈A} f(x) + ∑_{x∈B} g(x) ≤ ∑_{x∈A∪B} f(x) + ∑_{x∈A∪B} g(x)
(1.8.15)  = ∑_{x∈A∪B}(f(x) + g(x)) ≤ ∑_{x∈X}(f(x) + g(x)).

One can use this to get that

∑_{x∈X} f(x) + ∑_{x∈X} g(x) ≤ ∑_{x∈X}(f(x) + g(x)). (1.8.16)

If E is any subset of X, then ∑_{x∈E} f(x) may be defined as before, as the supremum of the sums (1.8.4) over all finite subsets A of E. If E_{1} and E_{2} are disjoint subsets of X, then we get that

∑_{x∈E_{1}∪E_{2}} f(x) = ∑_{x∈E_{1}} f(x) + ∑_{x∈E_{2}} f(x). (1.8.17)

This can be obtained from the remarks in the preceding paragraph, by considering the functions on X that are equal to f on E_{1} and E_{2}, and to 0 elsewhere.

1.8.5 Monotone convergence for nonnegative sums

Let {f_{j}}_{j=1}^{∞} be a sequence of functions on X with values in the nonnegative extended real numbers, and suppose that f_{j}(x) ≤ f_{j+1}(x) (1.8.18) for each j ≥ 1 and x ∈ X. Put

f(x) = sup_{l≥1} f_{l}(x) (1.8.19)

for all x ∈ X so that f_{j}(x) → f(x) as j → ∞ for every x ∈ X, as mentioned near the beginning of Subsection 1.2.2. Note that

∑_{x∈X} f_{j}(x) ≤ ∑_{x∈X} f_{j+1}(x) ≤ ∑_{x∈X} f(x) (1.8.20)

for each j. Under these conditions, a version of the monotone convergence theorem states that

∑_{x∈X} f_{j}(x) → ∑_{x∈X} f(x) as j → ∞. (1.8.21)

More precisely,

∑_{x∈X} f_{j}(x) → sup_{l≥1} f_{l}(x) as j → ∞ (1.8.22)

and

sup_{l≥1} ∑_{x∈X} f_{l}(x) ≤ ∑_{x∈X} f(x), (1.8.23)

because of (1.8.20). Thus (1.8.21) is the same as saying that

sup_{l≥1} ∑_{x∈X} f_{l}(x) = ∑_{x∈X} f(x), (1.8.24)

and it suffices to show that

∑_{x∈X} f(x) ≤ sup_{l≥1} ∑_{x∈X} f_{l}(x). (1.8.25)

To see this, let A be a finite subset of X, and observe that

∑_{x∈A} f_{j}(x) → ∑_{x∈A} f(x) as j → ∞. (1.8.26)

This means that

∑_{x∈A} f(x) = sup_{l≥1} ∑_{x∈A} f_{l}(x) ≤ sup_{l≥1} ∑_{x∈X} f_{l}(x). (1.8.27)

It follows that

∑_{x∈X} f(x) ≤ sup_{j≥1} ∑_{x∈X} f_{j}(x), (1.8.28)

by the definition of the sum on the left.

1.8.6 Sums over countable unions of subsets of X

Let f be any nonnegative extended real-valued function on X again, and let E_{1}, E_{2}, E_{3}, … be an infinite sequence of pairwise-disjoint subsets of X. One can use this version of the monotone convergence theorem to get that

∑_{j=1}^{n} ∑_{x∈E_{j}} f(x) = ∑ ; x∈⋃_{j=1}^{n} E_{j} f(x) → ∑ ; x∈⋃_{j=1}^{∞} E_{j} f(x) as n → ∞. (1.8.29)

More precisely, one can take f_{n}(x) = f(x) when x ∈ ⋃_{j=1}^{n} E_{j}, and f_{n}(x) = 0 otherwise, to get a monotonically increasing sequence of functions that converges to f pointwise on ⋃_{j=1}^{∞} E_{j}. Equivalently,

∑_{j=1}^{∞} ∑_{x∈E_{j}} f(x) = ∑ ; x∈⋃_{j=1}^{∞} E_{j} f(x). (1.8.30)

1.8.7 Measures defined using these sums

Let w be a nonnegative extended real-valued function on X again, and now let μ_{w}(E) be as in (1.8.1) for every subset E of X. This defines a nonnegative countably additive measure on the collection of all subsets of X. This is mentioned on p25 of [118], and it basically corresponds to an example mentioned on p42 of [14], Example 3.3 on p18 of [20], and to the example (i) on p263 of [335] when X is countably infinite.

Let A_{1} be the collection of all subsets of X with only finitely or countably many elements, and remember that this is a σ-ring of subsets of X, as in Subsection 1.5.4. If μ is any countably additive measure on X with respect to A_{1}, then it is easy to see that μ = μ_{w} on A_{1} exactly when w is as in (1.8.2).

1.9 Some basic properties of measures

Let X be a set, let A be a ring of subsets of X, and let μ be a countably additive nonnegative measure on X with respect to A. Also let B_{1}, B_{2}, B_{3}, … be a sequence of elements of A such that

B_{j} ⊆ B_{j+1} (1.9.1)

for each j, and put

B = ⋃_{j=1}^{∞} B_{j}. (1.9.2)

If B ∈ A, then it is well known that

μ(B_{j}) → μ(B) as j → ∞. (1.9.3)

This corresponds to 2.59 on p43 of [14], part (3) of Proposition 3.5 on p18 of [20], part c of Theorem 1.8 on p25 of [118], Theorem 10.13 on p130 of [159], part (d) of Theorem 1.19 on p17 of [298], and Theorem 11.3 on p302 of [299].

To see this, put A_{1} = B_{1}, and

A_{j} = B_{j} \ B_{j−1} (1.9.4)

for j ≥ 2. This defines a sequence of pairwise-disjoint elements of A. Observe that

B_{n} = ⋃_{j=1}^{n} A_{j} (1.9.5)

for each n, and that

B = ⋃_{j=1}^{∞} A_{j}. (1.9.6)

This implies that

μ(B_{n}) = ∑_{j=1}^{n} μ(A_{j}) (1.9.7)

for each n, and

μ(B) = ∑_{j=1}^{∞} μ(A_{j}). (1.9.8)

Clearly (1.9.8) is the same as (1.9.3), because of (1.9.7).

1.9.1 A partial converse

Now let A_{1}, A_{2}, A_{3}, … be any sequence of pairwise-disjoint elements of A, let B_{n} be as in (1.9.5) for each n, and let B be as in (1.9.6). Note that B_{1}, B_{2}, B_{3}, … is a sequence of elements of A that satisfies (1.9.1) and (1.9.2). Of course, (1.9.7) only uses finite additivity of μ on A. In this case, if B ∈ A, then (1.9.3) implies the countable additivity property (1.9.8). This corresponds to Exercise 3.1 on p20 of [20], and to the first part of Exercise 11 on p27 of [118].

1.9.2 Decreasing sequences of measurable sets

Let C_{1}, C_{2}, C_{3}, … be a sequence of elements of A such that

C_{j+1} ⊆ C_{j} (1.9.9)

for each j, and put

C = ⋂_{j=1}^{∞} C_{j}. (1.9.10)

If C ∈ A and μ(C_{1}) < +∞, (1.9.11)

then it is well known that

μ(C_{j}) → μ(C) as j → ∞. (1.9.12)

This corresponds to 2.60 on p44 of [14], part (4) of Proposition 3.5 on p18 of [20], part d of Theorem 1.8 on p25 of [118], Theorem 10.15 on p131 of [159], and part (e) of Theorem 1.19 on p17 of [298].

This can be reduced to the previous statement, by taking

B_{j} = C_{1} \ C_{j} (1.9.13)

for each j. This defines a sequence of elements of A that satisfies (1.9.1), because of (1.9.9). If B is as in (1.9.6), then

B = ⋃_{j=1}^{∞}(C_{1} \ C_{j}) = C_{1} \ ( ⋂_{j=1}^{∞} C_{j}) = C_{1} \ C (1.9.14)

under these conditions. If C ∈ A, then B ∈ A, and (1.9.3) implies that

μ(C_{1} \ C_{j}) → μ(C_{1} \ C) as j → ∞. (1.9.15)

Of course, μ(C_{1}) = μ(C_{1} \ C_{j}) + μ(C_{j}) (1.9.16)

for each j, and μ(C_{1}) = μ(C_{1} \ C) + μ(C), (1.9.17)

using finite additivity. If (1.9.11) holds, then (1.9.12) may be obtained from (1.9.15).

1.9.3 Another partial converse

Let B_{1}, B_{2}, B_{3}, … be a sequence of elements of A that satisfies (1.9.1) again, and let B be as in (1.9.2). Suppose that B ∈ A, and put

C_{j} = B \ B_{j} (1.9.18)

for each j. This implies that (1.9.9) holds, and if C is as in (1.9.10), then

C = ⋂_{j=1}^{∞}(B \ B_{j}) = B \ ( ⋃_{j=1}^{∞} B_{j}) = ∅. (1.9.19)

In this case, (1.9.12) says that

μ(B \ B_{j}) = μ(C_{j}) → 0 as j → ∞. (1.9.20)

We also have that μ(B) = μ(B_{j}) + μ(B \ B_{j}) (1.9.21)

for each j, using finite additivity. Thus (1.9.20) implies (1.9.3). This corresponds to Exercise 3.2 on p20 of [20], and to the second part of Exercise 11 on p27 of [118].

1.9.4 A counterexample with μ(C_{1}) = +∞

Let X be the set Z_{+} of positive integers, and let μ be counting measure on X. If C_{j} is the set of positive integers greater than or equal to j for each j, then (1.9.9) holds, and (1.9.10) is the empty set. However, (1.9.12) does not hold. This is Example 3.6 on p19 of [20], an example on p26 of [118], and Example 1.20 (c) on p18 of [298].

1.10 Subadditivity and outer measures

Let X be a set, let A be a ring of subsets of X, and let μ be a nonnegative extended real-valued function on A.

1.10.1 Finite subadditivity

We say that μ is finitely subadditive on A if for any finite sequence B_{1}, …, B_{n} of elements of A, we have that

μ( ⋃_{j=1}^{n} B_{j}) ≤ ∑_{j=1}^{n} μ(B_{j}). (1.10.1)

If μ is finitely additive, then it is well known that μ is finitely subadditive. To see this, put A_{1} = B_{1}, and

A_{j} = B_{j} \ ( ⋃_{l=1}^{j−1} B_{l}) (1.10.2)

for j = 2, …, n. It is easy to see that A_{1}, A_{2}, …, A_{n} are pairwise-disjoint elements of A. We also have that

⋃_{j=1}^{n} A_{j} = ⋃_{j=1}^{n} B_{j}. (1.10.3)

More precisely, if x ∈ B_{l} for some l, then x ∈ A_{j} for the first j such that x ∈ B_{j}. Note that A_{j} ⊆ B_{j} for each j, so that

μ(A_{j}) ≤ μ(B_{j}), (1.10.4)

as in (1.7.4). Thus

μ( ⋃_{j=1}^{n} B_{j}) = μ( ⋃_{j=1}^{n} A_{j}) = ∑_{j=1}^{n} μ(A_{j}) ≤ ∑_{j=1}^{n} μ(B_{j}). (1.10.5)

1.10.2 Countable subadditivity

Similarly, we say that μ is countably subadditive if for any infinite sequence B_{1}, B_{2}, B_{3}, … of elements of A with

⋃_{j=1}^{∞} B_{j} ∈ A, (1.10.6)

we have that

μ( ⋃_{j=1}^{∞} B_{j}) ≤ ∑_{j=1}^{∞} μ(B_{j}). (1.10.7)

If μ is countably additive, then it is well known that μ is countably subbaditive, for essentially the same reasons as before. Indeed, if we put A_{1} = B_{1} and let A_{j} be as in (1.10.2) for j ≥ 2, then A_{1}, A_{2}, A_{3}, … is a sequence of pairwise-disjoint elements of A such that

⋃_{j=1}^{∞} A_{j} = ⋃_{j=1}^{∞} B_{j}. (1.10.8)

It follows that

μ( ⋃_{j=1}^{∞} B_{j}) = μ( ⋃_{j=1}^{∞} A_{j}) = ∑_{j=1}^{∞} μ(A_{j}) ≤ ∑_{j=1}^{∞} μ(B_{j}), (1.10.9)

because of (1.10.4). This corresponds to 2.58 on p43 of [14], to part (2) of Proposition 3.5 on p18 of [20], and to part b of Theorem 1.8 on p25 of [118].

1.10.3 A criterion for countable additivity

Suppose that μ is finitely additive on A, and let A_{1}, A_{2}, A_{3}, … be a sequence of pairwise-disjoint elements of A whose union is an element of A. If n is any positive integer, then

∑_{j=1}^{n} μ(A_{j}) = μ( ⋃_{j=1}^{n} A_{j}) ≤ μ( ⋃_{j=1}^{∞} A_{j}). (1.10.10)

This implies that

∑_{j=1}^{∞} μ(A_{j}) ≤ μ( ⋃_{j=1}^{∞} A_{j}). (1.10.11)

If μ is also countably subadditive, then it follows that μ is countably additive.

1.10.4 Outer measures

Now let μ be a nonnegative extended real-valued function defined on the collection of all subsets of X. We say that μ is a (Carathéodory) outer measure on X if it satisfies the following three conditions, as in Definition 4.1 on p24 of [20], on p28 of [118], and Definition 10.2 on p126 of [159]. First,

μ(∅) = 0. (1.10.12)

Second, if A ⊆ B ⊆ X, then μ(A) ≤ μ(B). (1.10.13)

Third, μ should be countably subadditive on the collection of all subsets of X.

Outer measures are also known as exterior measures, as on p264 of [335]. Sometimes outer measures are simply called measures, as on p1 of [110], p53 of [113], and p8 of [247].

1.11 More on outer measures

Let X be a set, and let μ be an outer measure on X.

1.11.1 Measurability with respect to μ

A subset A of X is said to be measurable with respect to μ if

μ(Y) = μ(Y ∩ A) + μ(Y \ A) (1.11.1)

for every subset Y of X. This is due to Carathéodory, and corresponds to Definition 4.5 on p26 of [20], Definition 10.5 on p127 of [159] and Definition 1.3 on p8 of [247], and it is also mentioned on p2 of [110], p54 of [113], p29 of [118], and p264 of [335].

Of course, μ(Y) ≤ μ(Y ∩ A) + μ(Y \ A) (1.11.2)

automatically, by subadditivity. Thus, in order to get (1.11.1), it suffices to check that μ(Y) ≥ μ(Y ∩ A) + μ(Y \ A). (1.11.3)

In particular, this implies that A is measurable when μ(A) = 0.

Note that A is measurable with respect to μ if and only if X \A is measurable with respect to μ, by the definition of measurability, and because Y \ A = Y ∩ (X \ A). It is well known that

(1.11.4) the collection of subsets of X that are measurable with respect to μ is a σ-algebra,

and that

(1.11.5) the restriction of μ to this σ-algebra is countably additive.

This corresponds to Theorem 4.6 on p26 of [20], parts (i) and (ii) of Theorem 1 on p2 of [110], parts (2) and (3) of Theorem 2.1.3 on p54 of [113], Theorem 1.11 on p29 of [118], Theorem 10.11 on p129 of [159], and Theorem 1.1 on p265 of [335], and it is mentioned in parts (1) and (3) of Theorem 1.4 on p8 of [247].

1.11.2 Metric outer measures

Suppose for the moment that (X, d) be a metric space. Let A, B be nonempty subsets of X, and suppose that there is a positive real number η such that

d(x, y) ≥ η (1.11.6)

for all x ∈ A and y ∈ B. If

μ(A ∪ B) = μ(A) + μ(B) (1.11.7)

under these conditions, then μ is said to be a metric outer measure or metric exterior measure on X.

In this case, it is well known that

(1.11.8) open and closed sets in X are measurable with respect to μ, as in Theorem 5 on p9 of [110], Exercise 10.48 on p144 of [159], and Theorem 1.2 on p267 of [335]. This is part of Carathéodory’s criterion, as in (9) on p75 of [113], and as mentioned in Theorem 1.7 on p10 of [247].

In order to verify (1.11.7), it suffices to check that

μ(A ∪ B) ≥ μ(A) + μ(B), (1.11.9)

because of subadditivity, as before. Metric outer measures can often be defined using suitable coverings of subsets of X. More precisely, one may be able to use covering by subsets of X with arbitrarily small diameter. If A and B satisfy (1.11.6), then any subset of X with diameter less than η can intersect at most one of A and B. This means that a covering of A ∪ B by sets of diameter less than η can be split into coverings of A and B separately, and this can often be used to get (1.11.9).

1.11.3 Some constructions of outer measures

Let X be a set, and let C be a collection of subsets of X such that ∅ ∈ C. Also let ζ be a nonnegative extended real-valued function on C such that ζ(∅) = 0. If E is any subset of X, then consider sequences A_{1}, A_{2}, A_{3}, … of elements of C such that

E ⊆ ⋃_{j=1}^{∞} A_{j}. (1.11.10)

Using such a sequence, we get a sum

∑_{j=1}^{∞} ζ(A_{j}). (1.11.11)

Let μ^{∗}(E) be the infiumum of these sums over all such coverings of E.

Under these conditions, μ^{∗} is an outer measure on X, as in Proposition 4.2 on p24 of [20], and Proposition 1.10 on p29 of [118]. More precisely, in [20] one asks that there be a sequence of elements of C whose union is equal to X, and in [118] one asks that X ∈ C. These conditions imply that there are coverings of subsets of X as before, and otherwise one can interpret the infimum as being equal to +∞.

As a variant of this, one can consider coverings of E by finite or countable families {A_{j}}_{j} of elements of C, and take μ^{∗}(E) to be the infimum of the corresponding finite or countable sums

∑_{j} ζ(A_{j}). (1.11.12)

One can get μ^{∗}(∅) = 0 by considering the empty set to be covered by the empty family of sets, and the corresponding empty sum to be equal to 0. In a related construction on p54 of [247], one asks that ζ takes arbitrarily small values at some elements of C, without asking that ∅ ∈ C.

Suppose that X is a metric space again, and that 0 < δ ≤ ∞. Let C(δ) be the collection of elements of C with diameter less than or equal to δ, where the diameter is defined as in Section A.5. One can define μ_{δ}^{∗} in the same way as before, using C(δ) instead of C, as on p170 of [113], and p54 of [247]. This is an outer measure on X, as mentioned in [113, 247], for the same reasons as before. More precisely, in [247] one asks that there be a sequence of elements of C(δ) whose union is equal to X, and that ζ take arbitrarily small values on C(δ).

Note that C = C(+∞), so that

μ^{∗} = μ_{+∞}^{∗}. (1.11.13)

If 0 < δ_{1} ≤ δ_{2} ≤ +∞, then C(δ_{1}) ⊆ C(δ_{2}), (1.11.14)

and thus μ_{δ_{2}}^{∗}(E) ≤ μ_{δ_{1}}^{∗}(E) (1.11.15)

for every E ⊆ X. One can check that

sup_{δ>0} μ_{δ}^{∗}(E) (1.11.16)

is a metric outer measure on X, as in [113, 247]. This is known as Carathéodory’s construction associated to C and ζ.

1.11.4 A very nice extension theorem

Let X be any set again, let A be an algebra of subsets of X, and let μ be a countably additive measure on X with respect to A. Also let \overline{μ} be the outer measure μ^{∗} defined in the previous subsection with C = A and ζ = μ. It is well known that \overline{μ}(A) = μ(A) (1.11.17)

for every A ∈ A, and that

(1.11.18) the elements of A are measurable with respect to \overline{μ}.

This corresponds to parts (1), (2), and (3) of Theorem 4.17 on p37 of [20], Proposition 1.13 on p31 of [118], parts (a), (b), and (c) of Exercise 10.36 on p141 of [159], and to Lemma 1.4 on p271 of [335]. This implies the extension theorem, that μ can be extended to a countably additive measure on a σ-algebra of subsets of X that contains A, as in the first part of Theorem 1.14 on p31 of [118], part (d) of Exercise 10.36 on p142 of [159], and Theorem 1.5 on p272 of [335].

As a variant of this, let μ be a finitely additive measure on X with respect to A, and suppose that for every sequence C_{1}, C_{2}, C_{3}, … of elements of A such that C_{j+1} ⊆ C_{j} for each j and

⋂_{j=1}^{∞} C_{j} = ∅, (1.11.19)

we have that μ(C_{j}) → 0 as j → ∞. (1.11.20)

This implies that μ satisfies the condition mentioned at the beginning of Section 1.9, as in Subsection 1.9.3. It follows that μ is countably additive on X with respect to A, as in Subsection 1.9.1. Thus the remarks in the previous paragraph hold in this case as well, as in Exercise 10.37 on p142 of [159].

1.12 Some regularity conditions for measures

Let X be a metric space, or at least a topological space. Also let A be a ring of subsets of X, and let μ be a finitely additive measure on X with respect to A. Sometimes μ may satisfy some additional conditions related to the metric or topology on X.

1.12.1 An outer regularity condition

Let us say that μ is outer regular if for every A ∈ A we have that

(1.12.1) μ(A) = inf{μ(U) : U ∈ A, A ⊆ U, and U is an open set in X}.

If X is equipped with the discrete metric or topology, then every subset of X is an open set, and this outer regularity condition is trivial.

1.12.2 Two inner regularity conditions

Let us say that μ is inner regular with respect to closed sets if for every A ∈ A we have that

(1.12.2) μ(A) = sup{μ(E) : E ∈ A, E ⊆ A, and E is a closed set in X}.

If X is equipped with the discrete metric or topology, then every subset of X is a closed set, and this is trivial, as before.

Similarly, let us say that μ is inner regular with respect to compact sets if for every A ∈ A we have that

(1.12.3) μ(A) = sup{μ(K) : K ∈ A, K ⊆ A, and K is a compact set in X}.

It is well known that compact sets are closed sets in metric spaces, and in Hausdorff topological spaces. Thus inner regularity with respect to compact sets implies inner regularity with respect to closed sets in these cases.

If X is compact, then it is well known that closed sets in X are compact too. This means that inner regularity with respect to closed sets implies inner regularity with respect to compact sets in this case.

Suppose for the moment that A is an algebra of subsets of X, and that μ(X) < +∞. One can check that inner regularity with respect to closed sets is equivalent to outer regularity under these conditions. More precisely, inner regularity with respect to closed sets for A ∈ A corresponds to outer regularity for X \ A.

1.12.3 Some examples concerning inner regularity

Suppose for the moment that A contains all one-element subsets of X, and thus all finite subsets of X. Of course, finite subsets of X are automatically compact. If w is a nonnegative extended real-valued function on X and μ_{w} is defined on A as in Subsection 1.8.7, then μ_{w} is inner regular with respect to compact sets, by construction.

Suppose for the moment as well that X is equipped with discrete metric or topology. It is well known that the finite subsets of X are the only compact sets in this case. If μ is inner regular with respect to compact sets, then one can check that μ = μ_{w} on A exactly when w is obtained from μ as at the beginning of Section 1.8.

Let us continue to suppose for the moment that X is equipped with the discrete metric or topology. If X has infinitely many elements, then the finitely additive measure defined on the algebra A_{0} of subsets of X considered in Subsection 1.7.4 is not inner regular with respect to compact sets. Similarly, if X is uncountable, then the countably additive measure defined on the σ-algebra A_{1} of subsets of X considered in Subsection 1.7.4 is not inner regular with respect to compact sets.

1.12.4 Regularity and elementary sets

Let n be a positive integer, and remember that E(R^{n}) is the ring of bounded elementary subsets of R^{n}, as in Subsection 1.6.3. Also let μ be a finitely additive measure on R^{n} with respect to E(R^{n}) such that μ(A) < +∞ for every A in E(R^{n}). One can check that μ is both outer regular and inner regular with respect to closed sets if and only if μ is regular in the sense of Definition 11.5 on p303 of [299]. Of course, this uses the standard Euclidean metric on R^{n}.

If μ is any finitely additive measure on E(R^{n}), then inner regularity with respect to closed sets implies inner regularity with respect to compact sets. This uses the well-known fact that closed and bounded sets in R^{n} are compact. This also uses the fact that the elements of E(R^{n}) are bounded subsets of R^{n}, by construction.

1.13 Regularity conditions and countable additivity

Let X be a metric or a topological space, let A be a ring of subsets of X, and let μ be a finitely additive measure on X with respect to A. Suppose that μ is outer regular, as well as inner regular with respect to compact sets, as in the previous section. We would like to show that μ is countably additive on A. It suffices to show that μ is countably subadditive on A, as in Subsection 1.10.3.

Let A_{1}, A_{2}, A_{3}, … be a sequence of elements of A such that ⋃_{j=1}^{∞} A_{j} ∈ A, 1.13. REGULARITY CONDITIONS AND COUNTABLE ADDITIVITY 33 and let us show that

μ( ⋃_{j=1}^{∞} A_{j}) ≤ ∑_{j=1}^{∞} μ(A_{j}). (1.13.1)

We may as well suppose that the sum on the right is finite, since otherwise there is nothing to do. In particular, this means that μ(A_{j}) < +∞ for each j.

Let ϵ > 0 be given, and let us use outer regularity to get an open set U_{j} in X for each j such that A_{j} ⊆ U_{j}, U_{j} ∈ A, and μ(U_{j}) < μ(A_{j}) + 2^{−j} ϵ. (1.13.2) This implies that

∑_{j=1}^{∞} μ(U_{j}) < ∑_{j=1}^{∞} μ(A_{j}) + ϵ. (1.13.3)

Let K be a compact subset of X such that K ∈ A and

K ⊆ ⋃_{j=1}^{∞} A_{j}. (1.13.4)

It follows that

K ⊆ ⋃_{j=1}^{∞} U_{j}, (1.13.5)

so that

K ⊆ ⋃_{j=1}^{n} U_{j} (1.13.6)

for some positive integer n. This means that

μ(K) ≤ ∑_{j=1}^{n} μ(U_{j}), (1.13.7)

by finite subadditivity, as in Subsection 1.10.1. Thus

μ(K) ≤ ∑_{j=1}^{∞} μ(U_{j}) < ∑_{j=1}^{∞} μ(A_{j}) + ϵ. (1.13.8)

We can now use inner regularity with respect to compact sets to get that

μ( ⋃_{j=1}^{∞} A_{j}) ≤ ∑_{j=1}^{∞} μ(A_{j}) + ϵ. (1.13.9)

This implies (1.13.1), because ϵ > 0 is arbitrary.

1.13.1 Volumes of elementary sets

Let n be a positive integer, and remember that the ring E(R^{n}) of bounded elementary subsets of R^{n} and the algebra E_{1}(R^{n}) of possibly unbounded elementary subsets of R^{n} may be defined as in Subsection 1.6.3. More precisely, one can check that the elements of E(R^{n}) may be expressed as the union of finitely many pairwise-disjoint bounded intervals in R^{n}, as in (13) on p303 of [299]. Similarly, the elements of E_{1}(R^{n}) may be expressed as the union of finitely many pairwise-disjoint possibly unbounded intervals in R^{n}. This is related to Proposition 1.7 on p23 of [118], as mentioned later in Section 5.7. Remember also that the n-dimensional volume of bounded as well as possibly unbounded intervals in R^{n} may be defined as in Subsection 1.6.2.

The n-dimensional volume of a bounded elementary set in R^{n} may be defined as in (11) on p303 of [299]. That is to say, if the bounded elementary set is expressed as the union of finitely many pairwise-disjoint bounded intervals, then the volume of the elementary set is equal to the sum of the volumes of these intervals. This does not depend on the way that the elementary set is expressed as the union of finitely many pairwise-disjoint intervals, as in (14) on p303 of [299]. This defines a finitely additive measure on R^{n} with respect to E(R^{n}), as in (15) on p303 of [299]. There are analogous remarks for the n-dimensional volume of a possibly unbounded elementary set in R^{n}.

1.13.2 Regularity of the volume

One can check that n-dimensional volume is outer regular and inner regular with respect to compact sets on E(R^{n}). This corresponds to Example 11.6 (a) on p303 of [299]. More precisely, it suffices to verify these regularity properties for bounded intervals in R^{n}, and this is easy to do. It follows that n-dimensional volume is countably additive on E(R^{n}), as before.

Similarly, n-dimensional volume is inner regular with respect to compact sets on E_{1}(R^{n}). As before, it is enough to verify this for possibly unbounded intervals in R^{n}, and this is also fairly simple. One can check that 1-dimensional volume is outer regular on E_{1}(R) as well.

However, if n ≥ 2, then n-dimensional volume is not outer regular on E_{1}(R^{n}). This is because of unbounded intervals with n-dimensional volume equal to 0. Outer regularity does not work for these sets, because unbounded open sets in E_{1}(R^{n}) have n-dimensional volume +∞. We shall consider another version of outer regularity in the next section that works in this case.

1.14 Another outer regularity condition

Let X be a metric or topological space again, let A be a ring of subsets of X, and let μ be a finitely additive measure on A. Let us say that μ is outer regular with respect to subsets of compact sets if the outer regularity property (1.12.1) holds when A ∈ A and there is a compact subset K of X such that A ⊆ K and K ∈ A. Of course, this is equivalent to outer regularity when X is compact.

1.14.1 Using this condition to get countable additivity

Suppose that μ is outer regular with respect to subsets of compact sets, and inner regular with respect to compact sets. We would like to show that μ is countably additive on A under these conditions as well. It is enough to show that μ is countably subadditive on A, as before.

Let A_{1}, A_{2}, A_{3}, … be a sequence of elements of A such that ⋃_{j=1}^{∞} A_{j} ∈ A, so that we would like to show that (1.13.1) holds. To do this, let K be a compact subset of X such that K ∈ A and (1.13.4) holds. It suffices to show that

μ(K) ≤ ∑_{j=1}^{∞} μ(A_{j}), (1.14.1)

because of inner regularity with respect to compact sets.

In fact, we have that

μ(K) ≤ ∑_{j=1}^{∞} μ(A_{j} ∩ K). (1.14.2)

To see this, we may as well suppose that the sum on the right is finite, as before. Let ϵ > 0 be given, and let us use outer reguarity with respect to subsets of compact sets to get an open set V_{j} in X for each j such that A_{j} ∩ K ⊆ V_{j}, V_{j} ∈ A, and μ(V_{j}) < μ(A_{j} ∩ K) + 2^{−j} ϵ. (1.14.3)

Note that

∑_{j=1}^{∞} μ(V_{j}) < ∑_{j=1}^{∞} μ(A_{j} ∩ K) + ϵ. (1.14.4)

Using (1.13.4), we get that

K = ⋃_{j=1}^{∞}(A_{j} ∩ K) ⊆ ⋃_{j=1}^{∞} V_{j}. (1.14.5)

This implies that

K ⊆ ⋃_{j=1}^{n} V_{j} (1.14.6)

for some positive integer n, by compactness, so that

μ(K) ≤ ∑_{j=1}^{n} μ(V_{j}), (1.14.7)

because of finite subadditivity. It follows that

μ(K) ≤ ∑_{j=1}^{∞} μ(V_{j}) < ∑_{j=1}^{∞} μ(A_{j}) + ϵ. (1.14.8)

This means that (1.14.1) holds, because ϵ > 0 is arbitrary.

1.14.2 More on the n-dimensional volume on E_{1}(R^{n})

Let n be a positive integer, and let us continue with the discussion of the ndimensional volume on E_{1}(R^{n}) from the previous section. It is easy to see that the n-dimensional volume is outer regular with respect to subsets of compact sets on E_{1}(R^{n}), because of outer regularity of n-dimensional volume on E(R^{n}).

It follows that n-dimensional volume is countably additive on E_{1}(R^{n}), using the remarks in the previous subsection. This also uses the inner regularity of ndimensional volume on E_{1}(R^{n}) with respect to compact sets, as in the previous section.

1.15 Lebesgue outer measure on R^{n}

The remarks in the previous two sections suggest a way to define n-dimensional Lebesgue outer measure, using the construction discussed in Subsection 1.11.4. Often one defines Lebesgue outer measure on R^{n} a bit more directly, and shows that it has the usual properties using related arguments.

If E is any subset of R^{n}, then in the construction in Subsection 1.11.4, one considers coverings of E by sequences of elements of E_{1}(R^{n}). As a variant of this, one can consider coverings of E by sequences of possibly unbounded intervals in R^{n}. It is easy to see that this leads to an equivalent definition of the Lebesgue outer measure of E, because of the way that the volume of an element of E_{1}(R^{n}) is defined.

As another variant, one can consider coverings of E by sequences of bounded intervals. It is not too difficult to show that this also leads to an equivalent definition of Lebesgue outer measure. This is simpler when n = 1, because the 1-dimensional volume of an unbounded interval in R is +∞. If n ≥ 2, then the n-dimensional volume of an unbounded interval is either 0 or +∞, as in Subsection 1.6.2. If it is 0, then one should check that its n-dimensional Lebesgue outer measure is equal to 0 when it is defined using coverings by sequences of bounded intervals.

One can also restrict one’s attention to coverings of E by sequences of bounded open intervals. This is because any bounded interval in R^{n} is contained in a bounded open interval whose n-dimensional volume is arbitrarily close to the volume of the initial interval. This approach to defining Lebesgue outer measure on R^{n} is essentially the same as in Definition 11.7 on p304 of [299], where coverings by sequences of bounded open elementary sets is used. More precisely, in the construction in [299], one considers any finitely additive measure on E(R^{n}) that is outer regular, inner regular with respect to closed or equivalently compact sets, and finite on the elements of E(R^{n}).

One could consider coverings of E by sequences of cubes as well. To check that this leads to the same result, one can verify that bounded intervals can be covered by finitely many cubes, with the sum of the volumes of these cubes arbitrarily close to the volume of the interval. One can restrict one’s attention to closed cubes too, as on p10 of [335].

1.15. LEBESGUE OUTER MEASURE ON R^{N} 37

One-dimensional Lebesgue outer measure is defined on p14 of [14] using coverings by sequences of possibly unbounded open intervals. It is simply called “outer measure”, and it is shown afterwards to satisfy the conditions in the definition of an outer measure used here.

One-dimensional Lebesgue outer measure is defined in Example 4.3 on p25 of [20] using coverings by sequences of intervals that are open on the left and closed on the right.

One-dimensional Lebesgue outer measure is defined a bit differently on p26 of [110], using coverings of E by sequences of arbitrary subsets of the real line, and taking the infimum of the sum of the diameters of the sets in the covering, over all such coverings. This is equivalent to using coverings by sequences of intervals as before, because the diameter of an interval in R is the same as its length, or one-dimensional volume. This also uses the fact that any subset of the real line is contained in an interval with the same diameter. Lebesgue outer measure on R^{n} is obtained another way in [110] when n ≥ 2, using the n = 1 case.

Note that one gets an outer measure when defining Lebesgue outer measure using converings in these ways, as in Subsection 1.11.3. More precisely, one can check that it is enough to use coverings by sets whose diameter is as small as one likes. This implies that Lebesgue outer measure is a metric outer measure, as in Subsection 1.11.2. This corresponds to Observation 4 on p14 of [335], as mentioned on p267 of [335]. This is related to the remarks about the Carathéodory construction in Subsection 1.11.3.

One-dimensional Lebesgue outer measure is obtained using the Riemann integral on p111 of [113], and in Definition 9.19 on p120 of [159]. This is used to get n-dimensional Lebesgue outer measure beginning on p119 of [113].

1.15.1 Lebesgue measure on R^{n}

One may define n-dimensional Lebesgue measure to be the restriction of ndimensional Lebesgue outer measure to the corresponding σ-algebra of measurable sets, as in Subsection 1.11.1. A subset of R^{n} that is measurable with respect to Lebesgue outer measure may be called Lebesgue measurable. Another definition of measurability is used on p16 of [335], and Exercise 3 on p312 of [335] asks one to show that this is equivalent to measurability with respect to Lebesgue outer measure, as in Subsection 1.11.1.

Lebesgue measure on R^{n} is defined on the σ-algebra of Lebesgue measurable sets another way on p53 of [298].

Lebesgue measurability of subsets of R is defined another way on p52 of [14]. The fact that the Lebesgue measurable subsets of R form a σ-algebra, and that Lebesgue outer measure is countably additive on this σ-algebra, is discussed on p54 of [14].

Lebesgue measure on R is defined on p37 of [118] as Lebesgue–Stieltjes measure λ_{α}, as in Subsection 2.5.1, with α(x) = x.

More on measures and σ-algebras

2.1 Getting an outer measure from a measure

Let X be a set, let A be a σ-algebra of subsets of X, and μ be a nonnegative countably additive measure defined on X with respect to A. If E is any subset of X, then put \overline{μ}(E) = inf\{μ(A) : A ∈ A, E ⊆ A\}, (2.1.1)

as in Exercise 4.3 on p39 of [20], and (1.2) of p8 of [247]. It is easy to see that this is the same as in Subsection 1.11.4 under these conditions. This is related to Exercise 22 on p32 of [118].

The properties of \overline{μ} mentioned in Subsection 1.11.4 are somewhat simpler in this case. In particular, one can check more directly that \overline{μ} is an outer measure on X, as in Exercise 1 on p22 of [247]. Clearly \overline{μ} = μ on A, by construction. One can also verify that the elements of A are measurable with respect to \overline{μ}, in the sense described in Subsection 1.11.1.

This construction is related to one mentioned at the bottom of p57 of [113], and we shall say more about that in Section 2.7.

2.1.1 The infimum in the definition of \overline{μ}(E) is attained

Let E be any subset of X, and let us check that the infimum on the right side of (2.1.1) is attained. This is clear when \overline{μ}(E) = +∞, and so we suppose that \overline{μ}(E) < ∞. In this case, for each positive integer l there is an A_{l} ∈ A such that E ⊆ A_{l} and μ(A_{l}) < \overline{μ}(E) + 1/l. (2.1.2)

If we put B = ⋂_{l=1}^{∞} A_{l}, then B ∈ A, E ⊆ B, and

μ(B) ≤ \overline{μ}(E), (2.1.3) because μ(B) ≤ μ(A_{l}) for each l. It follows that

\overline{μ}(E) = μ(B), (2.1.4)

by the definition of \overline{μ}(E).

In particular, μ(B) = 0 when \overline{μ}(E) = 0. If E is measurable with respect to \overline{μ}, as in Subsection 1.11.1, then one can use (2.1.4) to get that

\overline{μ}(B \ E) = 0. (2.1.5)

2.1.2 When E is σ-finite with respect to \overline{μ}

Suppose now that E is σ-finite with respect to \overline{μ}, in the sense that there is a sequence E_{1}, E_{2}, E_{3}, … of subsets of X such that E = ⋃_{j=1}^{∞} E_{j} and \overline{μ}(E_{j}) < +∞ for each j. Let B_{j} be an element of A such that E_{j} ⊆ B_{j} and

\overline{μ}(E_{j}) = μ(B_{j}) (2.1.6)

for each j, as before. Note that E_{j} ⊆ E ∩ B_{j} for each j, so that

\overline{μ}(E ∩ B_{j}) = μ(B_{j}). (2.1.7)

If E is measurable with respect to \overline{μ}, then E ∩B_{j} is measurable with respect to \overline{μ} for each j. This implies that

\overline{μ}(B_{j} \ E) = 0 (2.1.8)

for each j, as before. It follows that

\overline{μ}(( ⋃_{j=1}^{∞} B_{j}) \ E) = \overline{μ}( ⋃_{j=1}^{∞}(B_{j} \ E)) = 0. (2.1.9)

2.1.3 Complete measure spaces

The measure space (X, A, μ) is said to be complete if for every subset E of X such that there is a B ∈ A with E ⊆ B and μ(B) = 0, we have that E ∈ A. This is mentioned on p19 of [20], on p26 of [118], in Definition 11.20 on p155 of [159], on p29 of [298], and p266 of [335]. Equivalently, this means that E ∈ A when \overline{μ}(E) = 0. Note that any outer measure is complete with respect to the corresponding σ-algebra of measurable sets, as in Subsection 1.11.1.

It is well known that one can get a completion of A as follows. A subset of X is said to measurable with respect to the completion if it can be expressed as A ∪ E, when A ∈ A and E ⊆ X is as in the previous paragraph. In this case, the measure of A ∪ E is defined to be μ(A). One can check that this defines a complete measure space, as in Exercise 3.8 on p21 of [20], Theorem 1.9 on p26 of [118], Exercise 10.38 on p142 and Theorem 11.21 on p155 of [159], Theorem 1.36 on p29 of [298], and Exercise 2 on p312 of [335].

40 CHAPTER 2. MORE ON MEASURES AND σ-ALGEBRAS

2.1.4 More on σ-finiteness

We say that A ∈ A is σ-finite with respect to μ if A can be expressed as the union of a sequence of elements of A with finite measure, as on p25 of [118], p118 of [158], p138 of [159], and p49 of [298]. If X is σ-finite with respect to μ, then (X, A, μ) is said to be σ-finite as a measure space, as in Definition 5.18 on p123 of [14], Definition 3.7 on p19 of [20], on p127 of [159], and p263 of [335]. Note that σ-finiteness of a set with respect to an outer measure is defined on p4 of [110] to mean that the set may be expressed as the union of a sequence of measurable sets with respect to the outer measure with finite measure. This property is also known as countable measurability with respect to the outer measure, as in 2.3.4 on p76 of [113].

It is easy to see that the collection of elements of A that are σ-finite with respect to μ is a σ-ring.

If X is any set equipped with counting measure, then A ⊆ X is σ-finite if and only if A has only finitely or countably many elements. This means that X is σ-finite as a measure space if and only if it has only finitely or countably many elements.

2.2 Borel sets

Let X be a set, and let E be any collection of subsets of X. Also let A(E) be the intersection of all of the σ-algebras of subsets of X that contain E. One can check that A(E) is a σ-algebra of subsets of X as well. Of course,

(2.2.1) A(E) is the smallest σ-algebra of subsets of X that contains E,

by construction. This is mentioned on p28 of [14], p12 of [20], beginning on p59 of [113], on p22 of [118], and in Definition 10.19 on p132 of [159], and it corresponds to Theorem 1.10 on p12 of [298].

Suppose now that X is a metric space, or a topological space. The collection of Borel sets in X is

(2.2.2) the smallest σ-algebra of subsets of X that contains the open sets,

as in Definition 2.29 on p28 of [14] when X = R, Definition 5.37 on p137 of [14] when X = R^{n}, on p13 of [20], p4 of [110], p60 of [113], p22 of [118], p118 of [158], in Definition 10.19 on p132 of [159], p9 of [247], p13 of [298], p309 of [299], and p23, 267 of [335]. Note that some variants of this terminology have sometimes been used.

2.2.1 Borel measures

A nonnegative countably additive measure defined on the σ-algebra of Borel sets in X may be called a Borel measure, as in the footnote on p329 of [159], in Definition 2.15 on p49 of [298], and on p269 of [335]. However, this term is sometimes also used for an outer measure μ on X such that the Borel sets in X are measurable with respect to μ, in the sense discussed in Subsection 1.11.1, as on p4 of [110], and in (2) in Definition 1.5 on p9 of [247]. We may use the term Borel outer measure for this here.

Of course, if μ is a Borel outer measure on X, then the restriction of μ to the Borel sets is a Borel measure on μ. If X is a metric space, then a metric outer measure on X is a Borel outer measure on X, as in Subsection 1.11.2.

2.2.2 Two basic families of examples of Borel sets

A subset of X is said to be an F_{σ} set if it can be expressed as the union of a sequence of closed sets. Similarly, a subset of X is said to be a G_{δ} set if it can be expressed as an intersection of a sequence of open sets. Clearly F_{σ} sets and G_{δ} sets are Borel sets in X. It is easy to see that a subset of X is an F_{σ} set if and only if its complement in X is a G_{δ} set.

Suppose that (X, d(·, ·)) is a metric space. If x ∈ X and r is a positive real number, then let B(x, r) = {y ∈ X : d(x, y) < r} (2.2.3)

be the usual open ball in X centered at x of radius r with respect to d. It is well known and not difficult to show that open balls in X are open sets. Similarly, if r is a nonnegative real number, then let

\overline{B}(x, r) = \{y ∈ X : d(x, y) ≤ r\} (2.2.4)

be the closed ball in X centered at x of radius r with respect to d. One can check that closed balls in X are closed sets.

It is easy to see that an open ball in X may be expressed as the union of a sequence of closed balls with the same center. This implies that open balls are F_{σ} sets. Similarly, a closed ball in X may be expressed as the intersection of a sequence of open balls with the same center, and is thus a G_{δ} set.

If A is any subset of X, then put

A_{r} = ⋃_{x∈A} B(x, r). (2.2.5)

This is an open set in X, because it is a union of open sets. One can check that

\overline{A} = ⋂_{r>0} A_{r}, (2.2.6)

where \overline{A} is the closure of A in X. This implies that

\overline{A} = ⋂_{l=1}^{∞} A_{1/l}, (2.2.7)

so that closed sets in X are G_{δ} sets. It follows that open sets in X are F_{σ} sets, because their complements in X are closed sets.

2.2.3 Some examples concerning intervals

Let a and b be real numbers with a < b. One can check that [a, b) and (a, b] are each both F_{σ} sets and G_{δ} sets in the real line, with respect to the standard Euclidean metric. More precisely, [a, b) and (a, b] may each be expressed as the union of a sequence of closed intervals, and as the intersection of a sequence of open intervals. Similarly, (a, b) may be expressed as the union of a sequence of closed intervals, and [a, b] may be expressed as the intersection of a sequence of open intervals. We also have that an open half-line may be expressed as the union of a sequence of closed half-lines, and a closed half-line may be expressed as the intersection of a sequence of open half-lines.

Let n be a positive integer, and remember that a subset of R^{n} may be called an interval if it can be expressed as the Cartesian product of n intervals in the real line, as in Subsection 1.6.2. If a subset of R^{n} can be expressed as a product of n open intervals in R, then one can check that this set is an open set in R^{n}, with respect to the standard Euclidean metric. Similarly, if a subset of R^{n} can be expressed as the product of n closed intervals in R, then one can verify that this set is a closed set in R^{n}. It is well known that the product of n closed intervals in R that are also bounded is compact in R^{n}.

One can check that any interval in R^{n} is both an F_{σ} set and a G_{δ} set. More precisely, it can be expressed as the union of a sequence of closed intervals, and as the intersection of a sequence of open intervals. In particular, intervals in R^{n} are Borel sets.

2.3 Separability and countable bases

Let X be a metric space, or a topological space. A collection B of open subsets of X is said to be a base for the topology of X if every open set may be expressed as a union of elements of B. One may consider the empty set as being the union of an empty collection of sets, so that it has this property automatically.

Equivalently, B is a base for the topology of X if for every open set U ⊆ X and point x ∈ U there is a V ∈ B such that x ∈ V and V ⊆ U. If X is a metric space, then this is also equivalent to saying that for every x ∈ X and r > 0 there is a V ∈ B such that x ∈ V and V ⊆ B(x, r).

The collection of all open balls in X is a base for the topology of X when X is a metric space. In fact, the collection of all open balls in X with radii of the form 1/l for some positive integer l is a base for the topology of X.

Suppose that X is a metric space again, and let E be a dense set in X. One can check that B_{E} = {B(y, 1/l) : y ∈ E, l ∈ Z_{+}} (2.3.1)

is a base for the topology of X. Indeed, if x ∈ X amd r is a positive real number, then let l be a positive integer such that 2/l < r, and let y be an element of E such that d(x, y) < 1/l. (2.3.2)

Thus B(y, 1/l) is an element of B_{E} that contains x as an element, and one can check that B(y, 1/l) ⊆ B(x, r), (2.3.3)

using the triangle inequality.

2.3.1 Separable metric and topological spaces

A metric or topological space X is said to be separable if there is a dense set E in X such that E has only finitely or countably many elements. If X is a metric space, and E ⊆ X has only finitely or countably many elements, then one can verify that (2.3.1) has only finitely or countably many elements. More precisely, it is easy to see that

{B(y, 1/l) : y ∈ E} (2.3.4)

has only finitely or countably many elements for every l ∈ Z_{+}, because E has only finitely or countably many elements. This implies that

⋃_{l=1}^{∞}{B(y, 1/l) : y ∈ E} (2.3.5)

has only finitely or countably many elements, and this set is the same as (2.3.1).

It is well known that the set Q of rational numbers is a countable dense set in the real line with respect to the standard Euclidean metric, so that R is separable. If n is a positive integer, then it is well known and not too difficult to show that the set Q^{n} of n-tuples of rational numbers is a countable dense set in R^{n} with respect to the standard Euclidean metric, so that R^{n} is separable.

If X is any set with the discrete metric or topology, then X is the only dense set in itself. It follows that X is separable if and only if X has only finitely or countably many elements.

Let X be a metric or topological space again, and let B be a base for the topology of X. If V ∈ B and V ≠ ∅, then let x(V) be an element of V. Consider the set E of these points x(V). One can check that E is dense in X, because B is a base for the topology of X. If B has only finitely or countably many elements, then E has only finitely or countably many elements, so that X is separable.

2.3.2 Bases and Borel sets

If E is any collection of Borel sets, then the σ-algebra A(E) of subsets of X generated by E, as in the previous section, is contained in the σ-algebra of Borel sets in X.

If E is a base for the topology of X with only finitely or countable many elements, then A(E) contains all Borel sets in X. More precisely, if a subset U of X can be expressed as the union of finitely or countably many elements of E, then U ∈ A(E). If this holds for all open sets U in X, then A(E) contains the Borel sets in X.

2.3.3 Some more examples of bases using intervals

The collection of all open intervals in the real line is a base for the topology of R with respect to the standard metric. More precisely, the collection of open intervals in R with rational endpoints is a countable base for the topology of R.

If n is a positive integer, then the collection of all open intervals in R^{n} is a base for the topology of R^{n} with respect to the standard metric. In fact, the collection of all open intervals in R^{n} obtained by taking the Cartesian product of n open intervals in R with rational endpoints is a countable base for the topology of R^{n}.

Let E(R^{n}) and E_{1}(R^{n}) be the collections of bounded and possibly unbounded elementary subsets of R^{n}, as in Subsection 1.6.3. Note that elementary subsets of R^{n} are Borel sets, because intervals are Borel sets, as in the previous section.

We also have that A(E(R^{n})) contains all of the Borel sets in R^{n}, because the collection of open intervals in R^{n} obtained from products of open intervals in R with rational endpoints is a countable base for the topology of R^{n}, as before. It follows that A(E(R^{n})) and A(E_{1}(R^{n})) are both the same as the collection of all Borel sets in R^{n}.

2.4 Limits of functions

We would like consider some more Borel measures on the real line, related to monotonically increasing real-valued functions on R. In order to do this, we shall first review some facts related to one-sided limits of functions on R.

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces, let E be a subset of X, and let p be an element of X that is a limit point of E. Also let f be a function on E with values in Y, and let q be an element of Y. We say that the limit of f(x), as x ∈ E approaches p, is equal to q, if for every ϵ > 0 there is a δ > 0 such that

d_{Y} (f(x), q) < ϵ (2.4.1)

for every x ∈ E with d(p, x) < δ and x ≠ p. It is well known and not difficult to show that the limit q is unique when it exists. In this case, we put

lim_{{x∈E \atop x→p}} f(x) = q, (2.4.2)

or simply

lim_{x→p} f(x) = q (2.4.3)

when E = X.

It is well known that (2.4.2) holds if and only if for every sequence {x_{j}}_{j=1}^{∞} of elements of E that converges to p and satisfies x_{j} ≠ p for each j, we have that {f(x_{j})}_{j=1}^{∞} converges to q in Y. If E = X, then it is easy to see that f is continuous at p if and only if

lim_{x→p} f(x) = f(p). (2.4.4)

If p is not a limit point of X, then any mapping from X into Y is continuous at p.

2.4.1 One-sided limits on R

Let us now take E = X = R, with the standard metric. The limit of f(x) as x ∈ R approaches p from the right is defined by

f(p+) = lim_{x→p+} f(x) = lim_{{x∈(p,+∞) \atop x→p}} f(x), (2.4.5)

when the limit on the right side exists. Similarly, the limit of f(x) as x ∈ R approaces p from the left is defined by

f(p−) = lim_{x→p−} f(x) = lim_{{x∈(−∞,p) \atop x→p}} f(x), (2.4.6)

when the limit on the right side exists. One can check that

lim_{x→p} f(x) (2.4.7)

exists if and only if the one-sided limits (2.4.5) and (2.4.6) exist and are equal. Of course, (2.4.7) is equal to the common value of the one-sided limits in this case.

Let us say that f is continuous at p from the right if

lim_{x→p+} f(x) = f(p). (2.4.8)

Similarly, we say that f is continuous at p from the left if

lim_{x→p−} f(x) = f(p). (2.4.9)

Thus f is continuous at p if and only if f is continuous at p from both the left and the right.

2.4.2 Monotonically increasing functions on R

Let f be a monotonically increasing real-valued function on R, so that

f(x) ≤ f(y) (2.4.10)

for every x, y ∈ R with x < y. It is well known that the limits of f at p ∈ R from the left and right exist, with

f(p+) = inf_{p<x<+∞} f(x) (2.4.11)

and

f(p−) = sup_{−∞<x<p} f(x). (2.4.12)

This corresponds to Theorem 4.1 H on p102 of [135] and its proof, the first part of Theorem 8.19 on p111 of [159] and its proof, and to parts of Theorem 4.29 on p95 of [299]. Note that

f(p−) ≤ f(p) ≤ f(p+), (2.4.13)

and that f(p_{1}+) ≤ f(p_{2}−) (2.4.14)

when p_{1} < p_{2}, as in [299].

It follows from (2.4.13) that f is continuous at p exactly when

f(p−) = f(p+). (2.4.15)

Thus f is not continuous at p exactly when

f(p−) < f(p+). (2.4.16)

It is well known that f is continuous at all but finitely or countably many elements of R, as in the first part of Theorem 8.19 on p111 of [159], and Theorem 4.30 on p96 of [299].

It is convenient to put

f(+∞) = sup_{x∈R} f(x) (2.4.17)

and

f(−∞) = inf_{x∈R} f(x). (2.4.18)

These satisfy f(x) → f(+∞) as x → +∞ (2.4.19)

and f(x) → f(−∞) as x → −∞, (2.4.20)

with suitable interpretations. This is related to Definition 4.1 E on p101 of [135], Exercise 11 and 21 at the end of Section 4.1 of [135], and some remarks on p98 of [299].

2.5 Lengths of intervals with respect to α

Let α be a monotonically increasing real-valued function on the real line. If a and b are real numbers with a < b, then put

(2.5.1)  λ_{α}([a, b)) = α(b−) − α(a−),
(2.5.2)  λ_{α}([a, b]) = α(b+) − α(a−),
(2.5.3)  λ_{α}((a, b]) = α(b+) − α(a+),
(2.5.4)  λ_{α}((a, b)) = α(b−) − α(a+),

2.5. LENGTHS OF INTERVALS WITH RESPECT TO α 47

as in Example 11.6 (b) on p303 of [299]. We may also use (2.5.2) when a = b. These may be described as the α-lengths of these intervals. Of course, this is the same as the ordinary length of an interval when α(x) = x for every x ∈ R.

Similarly, put

(2.5.5)  λ_{α}([a, +∞)) = α(+∞) − α(a−),
(2.5.6)  λ_{α}((a, +∞)) = α(+∞) − α(a+),
(2.5.7)  λ_{α}((−∞, b]) = α(b+) − α(−∞),
(2.5.8)  λ_{α}((−∞, b)) = α(b−) − α(−∞),
(2.5.9)  λ_{α}((−∞, +∞)) = α(+∞) − α(−∞).

Note that (2.5.9) is finite if and only if α is bounded on R. Similarly, (2.5.5) and (2.5.6) are finite exactly when α has a finite upper bound on R, and (2.5.7) and (2.5.8) are finite exactly when α has a finite lower bound on R.

Let E(R) be the ring of bounded elementary sets in R, and let E_{1}(R) be the algebra of possibly unbounded elementary sets in R, as in Subsection 1.6.3. We can extend λ_{α} to a finitely additive measure on E(R), in the same way as in Subsection 1.13.1. Similarly, we can extend λ_{α} to a finitely additive measure on E_{1}(R), and we use λ_{α} to denote this extension as well.

One can check that λ_{α} is outer regular and inner regular with respect to compact sets on E(R), as in Example 11.6 (a) on p303 of [299]. Similarly, λ_{α} is outer regular and inner regular with respect to compact sets on E_{1}(R). As in Subsection 1.13.2, it suffices to check these regularity properties for intervals, and one can do this directly from the definitions. This implies that λ_{α} is countably additive on E_{1}(R), by the remarks at the beginning of Section 1.13. This is closely related to Proposition 1.15 on p33 of [118].

2.5.1 Lebesgue–Stieltjes outer measures

Using λ_{α}, we get an outer measure \overline{λ_{α}} on the real line as in Subsection 1.11.4. This may be called the Lebesgue–Stieltjes outer measure associated to α. Note that \overline{λ_{α}} = λ_{α} on E_{1}(R), and that the elements of E_{1}(R) are measurable with respect to \overline{λ_{α}}, as before. This means that the Borel sets in R with respect to the standard metric are measurable with respect to \overline{λ_{α}}, as in Subsection 2.3.3.

In the construction in Subsection 1.11.4, one considers coverings of a subset E of the real line by sequences of elements of E_{1}R). One can consider coverings of E by sequences of possibly unbounded intervals instead, and get an equivalent definition of \overline{λ_{α}}, as in Section 1.15. One can consider coverings by sequences of bounded intervals too. In fact, one can consider coverings by sequences of bounded open intervals, and get an equivalent definition of \overline{λ_{α}}. This is essentially the same as in Definition 11.7 on p304 of [299], where coverings by sequences of bounded open elementary sets are used.

Alternatively, one can consider coverings by sequences of bounded intervals of the form (a, b], as in Example 4.4 on p25 of [20] and discussed further in Section 4.2 beginning on p28 of [20], and on p282 of [335]. One can also show that \overline{λ_{α}} is a metric outer measure on R with respect to the standard metric, as on p283 of [335]. This is another way to see that the Borel sets in R are measurable with respect to \overline{λ_{α}}, as shown also in Proposition 4.9 on p30 of [20].

One can obtain Lebesgue–Stieltjes outer measures using Riemann–Stieltjes integrals as well, as on p111 of [113], and in Definition 9.19 on p120 of [159].

One may define the Lebesgue–Stieltjes measure associated to α to be the restriction of \overline{λ_{α}} to the corresponding σ-algebra of measurable sets, or simply to the Borel sets. The restriction of \overline{λ_{α}} to the measurable sets may be denoted λ_{α} again. This is related to Theorem 1.16 on p35 of [118], and some of the remarks that follow it. Another approach to Lebesgue–Stieltjes measures may be found in the proof of part (b) of Theorem 8.14 on p173 of [298], and we shall discuss this in the next subsections.

2.5.2 When α is continuous and strictly increasing

Let us suppose in this subsection that α is continuous and strictly increasing on R. This implies that α(R) = (α(−∞), α(+∞)), (2.5.10)

by the intermediate value theorem.

One can check that the inverse of α is continuous as a real-valued function on α(R), using (2.5.10). As a related matter, it is well known that a one-to-one continuous mapping from a compact metric space onto another metric space has a continuous inverse, as in Theorem 4.17 on p90 of [299]. In fact, a oneto-one continuous mapping from a compact topological space onto a Hausdorff topological space is continuous. This implies that the restriction of α to any closed interval in the real line has a continuous inverse. One can use this to get that α^{−1} is continuous on closed intervals contained in α(R), so that α^{−1} is continuous on α(R).

Let \overline{λ} be one-dimensional Lebesgue outer measure on the real line, as in Section 1.15. One can use \overline{λ}(α(E)) (2.5.11)

as another way to define the Lebesgue–Stieltjes outer measure of E ⊆ R associated to α. It is easy to see that this basically corresponds to the previous definitions under these conditions, depending on how \overline{λ} is defined.

2.5.3 Some properties of monotone functions

If t is a real number, then it is easy to see that α^{−1}({t}) is an interval in R, or the empty set. More precisely, if x, y, z ∈ R, x < y < z, and α(x) = α(z) = t, then α(y) = t. If α is continuous, then α^{−1}({t}) is a closed set, and thus a closed interval.

It is well known that any collection of pairwise-disjoint intervals in the real line with nonempty interiors can only have finitely or countably many elements, because each of the intervals contains a distinct rational number. This implies that there are only finitely or countably many t ∈ R such that α^{−1}({t}) has more than one element, as mentioned on p174 of [298].

Suppose that x, z, t ∈ R satisfy x < z and

α(x) < t < α(z). (2.5.12)

Consider y_{0} = sup{y ∈ [x, z] : α(y) < t}. (2.5.13)

One can check that α(w) < t (2.5.14)

when w < y_{0}, and that α(w) ≥ t (2.5.15)

when w > y_{0}. In particular, this means that

α(y_{0}−) ≤ t ≤ α(y_{0}+). (2.5.16)

If α is continuous at y_{0}, then we get that α(y_{0}) = t.

2.5.4 Another approach to Lebesgue–Stieltjes outer measures

If x ∈ R, then put Φ[x] = [α(x−), α(x+)], (2.5.17)

so that Φ[x] = {α(x)} when α is continuous at x. If E is any subset of R, then we put

Φ[E] = ⋃_{x∈E} Φ[x]. (2.5.18)

It is easy to see that Φ sends the union of any family of subsets of R to the union of the sets that Φ associates to the sets in the family.

One can check that

(α(−∞), α(+∞)) ⊆ Φ[R] ⊆ [α(−∞), α(+∞)] ∩ R, (2.5.19)

using the remarks in the previous subsection. In particular, Φ[R] is a possiblyunbounded interval in R.

If E is any subset of the real line, then Φ[R \ E] is equal to the union of Φ[R] \ Φ[E] and a subset of the set of t ∈ R such that α^{−1}({t}) has at least two elements, as on p174 of [298]. This additional set has only finitely or countably many elements, as in the previous subsection.

If λ is one-dimensional Lebesgue measure on the real line, then one can use

λ(Φ[E]) (2.5.20)

as another way to define the Lebesgue–Stieltjes measure of a Borel set E ⊆ R associated to α, as on p174 of [298]. Similarly, one can use

\overline{λ}(Φ[E]) (2.5.21) as another way to define the Lebesgue–Stieltjes outer measure of E ⊆ R associated to α. Note that this is equal to 0 when Φ[E] is contained in the set of t ∈ R such that α^{−1}({t}) has more than one element, because the Lebesgue outer measure of a set with only finitely or countably many elements is equal to 0.

2.6 Monotone classes

Let X be a set, and let M be a collection of subsets of X. We say that M is a monotone class or monotone family if it satisfies the following two conditions, as in Definition 5.15 on p122 of [14], Definition 2.9 on p14 of [20], on p380 of [159], and p145 of [298]. First, if A_{1}, A_{2}, A_{3}, … is a sequence of elements of M such that A_{j} ⊆ A_{j+1} for each j, then

⋃_{j=1}^{∞} A_{j} ∈ M. (2.6.1)

Second, if B_{1}, B_{2}, B_{3}, … is a sequence of elements of M such that B_{j+1} ⊆ B_{j} for each j, then

⋂_{j=1}^{∞} B_{j} ∈ M. (2.6.2)

Note that σ-algebras of subsets of X are monotone classes.

2.6.1 Monotone classes generated by algebras

Let E be any collection of subsets of X, and let M(E) be the intersection of all of the monotone classes of subsets of X that contains E. One can check that M(E) is also a montone class of subsets of X, as mentioned on p381 of [159], and at the beginning of the proof of Theorem 7.3 on p146 of [298]. This is the smallest monotone class of subsets of X that contains E, by construction.

Let A(E) be the smallest σ-algebra of subsets of X that contains E, as in Section 2.2. Note that M(E) ⊆ A(E), (2.6.3)

because σ-algebras of subsets of X are monotone classes.

If E is an algebra of subsets of X, then it is well known that

M(E) = A(E). (2.6.4)

This corresponds to Theorem 5.17 on p122 of [14], Theorem 2.10 on p14 of [20], and Theorem 21.6 on p380 of [159], and Theorem 7.3 on p146 of [298] is a particular case of this.

2.6.2 Monotone classes and finite measures

Let A be a σ-algebra of subsets of X, and let μ, ν be countably additive nonnegative measures on X with respect to A. Suppose for the moment that

μ(X), ν(X) < +∞. (2.6.5)

It is easy to see that the collection M of E ∈ A such that

μ(E) = ν(E) (2.6.6)

is a monotone class, using the remarks in Section 1.9.

Let E be an algebra of subsets of X contained in A, and suppose that (2.6.6) holds for every E ∈ E, so that E ⊆ M. (2.6.7)

Suppose also that A is the smallest σ-algebra of subsets of X that contains E. This implies that A is the smallest monotone class of subsets of X that contains E, as before. It follows that A ⊆ M, (2.6.8)

so that A = M, (2.6.9)

because M ⊆ A, by construction. This means that (2.6.6) holds for every E ∈ A.

We would like to consider some extensions of this in the next two subsections, with the same hypotheses aside from (2.6.5).

2.6.3 Monotone classes and some sets of finite measure

Let E_{0} be an element of A such that

μ(E_{0}), ν(E_{0}) < +∞. (2.6.10)

Consider the collection M_{E_{0}} of E ∈ A such that

μ(E ∩ E_{0}) = ν(E ∩ E_{0}). (2.6.11)

One can check that M_{E_{0}} is a monotone class, using the remarks in Section 1.9, as before.

Suppose that E_{0} ∈ E, so that if E ∈ E, then E ∩ E_{0} ∈ E, and (2.6.11) holds, by hypothesis. This means that

E ⊆ M_{E_{0}}. (2.6.12)

If A is the smallest σ-algebra of subsets of X that contains E, then A is the smallest monotone class of subsets of X that contains E, and we get that

A = M_{E_{0}}. (2.6.13)

This is the same as saying that (2.6.6) holds for all E ∈ A with E ⊆ E_{0}.

2.6.4 Using a σ-finiteness condition on μ, ν

Suppose that E_{1}, E_{2}, E_{3}, … is a sequence of elements of E such that

μ(E_{j}) = ν(E_{j}) < +∞ (2.6.14)

for each j. Thus μ(E ∩ E_{j}) = ν(E ∩ E_{j}) (2.6.15)

for each E ∈ A and j ≥ 1, as in the previous subsection. Equivalently, (2.6.6) holds when E ∈ A and E ⊆ E_{j} for some j.

One can use this to get that (2.6.6) holds for every E ∈ A such that

E ⊆ ⋃_{j=1}^{∞} E_{j}. (2.6.16)

One way to do this is to reduce to the case where the E_{j}’s are pairwise disjoint, by replacing E_{j} with E_{j} \ ( ⋃_{l=1}^{j−1} E_{l}) when j ≥ 2. Alternatively, one could

reduce to the case where E_{j} ⊆ E_{j+1} for each j, by replacing E_{j} with ⋃_{l=1}^{j} E_{l}. One could also express E as the union of a sequence of pairwise-disjoint elements of A contained in the E_{j}’s. In particular, (2.6.6) holds for every E ∈ A when

⋃_{j=1}^{∞} E_{j} = X. (2.6.17)

2.7 Regular outer measures

Let X be a set, and let μ be an outer measure on X. We say that μ is regular if for every subset A of X there is a subset B of X such that A ⊆ B, B is measurable with respect to μ, and

μ(A) = μ(B). (2.7.1)

This is mentioned on p4 of [110], on p56 of [113], in Exercise 10.40 on p143 of [159], and on p9 of [247].

Note that the outer measures constructed in Section 2.1 are regular in this sense.

If μ is any outer measure on X, then the restriction of μ to the σ-algebra of subsets of X that are measurable with respect to μ is countably additive, as in Subsection 1.11.1, and this can be used to obtain an outer measure \overline{μ} on X as in Section 2.1. This is mentioned at the bottom of p57 of [113]. It is easy to see that μ(E) ≤ \overline{μ}(E) for every E ⊆ X, and of course equality holds when E is measurable with respect to μ. We also have that \overline{μ} is regular on X, as before.

2.7.1 A nice property of regular outer measures

Let μ be a regular outer measure on X, and let A_{1}, A_{2}, A_{3}, … be a sequence of subsets of X with A_{j} ⊆ A_{j+1} for each j. Under these conditions,

μ(A_{j}) → μ( ⋃_{l=1}^{∞} A_{l}) as j → ∞, (2.7.2)

as in Theorem 2 on p5 of [110], 2.1.5 (1) on p56 of [113], and mentioned on p9 of [247]. This is related to Exercise 4.17 on p41 of [20]. Of course,

μ(A_{j}) ≤ μ( ⋃_{l=1}^{∞} A_{l}) (2.7.3)

for each j.

Let C_{1}, C_{2}, C_{3}, … be a sequence of subsets of X that are measurable with respect to μ and satisfy A_{j} ⊆ C_{j} and

μ(A_{j}) = μ(C_{j}) (2.7.4)

for each j. Put

B_{j} = ⋂_{l=j}^{∞} C_{l} (2.7.5)

for each j, and note that B_{j} is measurable with respect to μ. One can check that A_{j} ⊆ B_{j} ⊆ C_{j} (2.7.6)

for each j. This implies that

μ(A_{j}) = μ(B_{j}) = μ(C_{j}) (2.7.7)

for each j.

Observe that B_{j} ⊆ B_{j+1} for each j, by construction. It follows that

μ(B_{j}) → μ( ⋃_{l=1}^{∞} B_{l}) as j → ∞, (2.7.8)

as in Section 1.9. We also have that

⋃_{l=1}^{∞} A_{l} ⊆ ⋃_{l=1}^{∞} B_{l}, (2.7.9)

so that

μ( ⋃_{l=1}^{∞} A_{l}) ≤ μ( ⋃_{l=1}^{∞} B_{l}). (2.7.10)

More precisely, equality holds in this inequality, because of (2.7.3), (2.7.7), and (2.7.8). This implies (2.7.2).

2.7.2 Regularity and σ-finiteness

Suppose that A ⊆ X is σ-finite with respect to μ, so that there is a sequence A_{1}, A_{2}, A_{3}, … of subsets of X such that A = ⋃_{j=1}^{∞} A_{j} and μ(A_{j}) < +∞ for each j. If μ is a regular outer measure on X, then there is a sequence B_{1}, B_{2}, B_{3}, … of measurable subsets of X with respect to μ such that A_{j} ⊆ B_{j} and μ(A_{j}) = μ(B_{j}) for each j. In particular, μ(B_{j}) < +∞ for each j.

Note that

A = ⋃_{j=1}^{∞}(A ∩ B_{j}) (2.7.11)

and μ(A ∩ B_{j}) < +∞ for each j. If A is measurable with respect to μ, then A ∩ B_{j} is measurable with respect to μ for each j.

2.7.3 Borel regular outer measures

Suppose now that X is a metric or topological space, and that μ is a Borel outer measure on X, so that Borel sets in X are measurable with respect to μ, as in Subsection 2.2.1. We say that μ is Borel regular as an outer measure on X if for every subset A of X there is a Borel set B ⊆ X such that A ⊆ B and (2.7.1) holds. This is mentioned on p5 of [110] and p61 of [113], and it corresponds to (3) in Definition 1.5 on p9 of [247]. Note that this implies that μ is regular on X.

If A is measurable with respect to μ and μ(A) = μ(B) < +∞, then it follows that μ(B \ A) = 0. (2.7.12)

Suppose that A is measurable and σ-finite with respect to μ. This implies that there is a sequence C_{1}, C_{2}, C_{3}, … of subsets of X that are measurable with respect to μ such that

A = ⋃_{j=1}^{∞} C_{j} (2.7.13)

and μ(C_{j}) < +∞ for each j, as in the previous subsection. This leads to a sequence B_{1}, B_{2}, B_{3}, … of Borel sets in X such that C_{j} ⊆ B_{j} and μ(B_{j} \C_{j}) = 0 for each j, as in the preceding paragraph. It follows that B = ⋃_{j=1}^{∞} B_{j} is a Borel set in X such that (2.7.12) holds, because

B \ A = ⋃_{j=1}^{∞}(B_{j} \ A) ⊆ ⋃_{j=1}^{∞}(B_{j} \ C_{j}). (2.7.14)

2.8 Using a collection C of subsets of X

Let X be a set, let ν be an outer measure on X, and let C be a collection of subsets of X that includes X as an element. If A ⊆ X, then put

ν_{C}(A) = inf{ν(C) : C ∈ C and A ⊆ C}. (2.8.1)

2.8. USING A COLLECTION C OF SUBSETS OF X 55

Note that ν(A) ≤ ν_{C}(A), (2.8.2)

and that ν(A) = ν_{C}(A), (2.8.3)

when A ∈ C. If ∅ ∈ C, and if the union of any sequence of elements of C is an element of C as well, then one can check that ν_{C} is an outer measure on X too. This is related to some remarks at the bottom of p57 of [113] when C is the collection of subsets of X that are measurable with respect to ν.

Suppose for the moment that ν_{C}(A) < +∞, and let C_{j} be an element of C such that A ⊆ C_{j} and ν(C_{j}) < ν_{C}(A) + 1/j (2.8.4)

for each positive integer j. Let C_{δ} be the collection of subsets of X that can be expressed as the intersection of a sequence of elements of C, so that ⋂_{j=1}^{∞} C_{j} ∈ C_{δ}. Observe that

ν(A) ≤ ν( ⋂_{j=1}^{∞} C_{j}) ≤ ν_{C}(A), (2.8.5)

using the fact that A ⊆ ⋂_{j=1}^{∞} C_{j} in the first step. If (2.8.3) holds, then we get that

ν(A) = ν( ⋂_{j=1}^{∞} C_{j}). (2.8.6)

Note that (2.8.6) implies that

ν(( ⋂_{j=1}^{∞} C_{j}) \ A) = 0 (2.8.7)

when A is measurable with respect to ν, because the right side of (2.8.6) is finite.

If (2.8.3) holds for every A ⊆ X, and if the elements of C are measurable with respect to ν, then it follows that ν is a regular outer measure on X. More precisely, this uses the fact that ν(A) = ν(X) when ν(A) = +∞. If X is a metric or topological space, ν is a Borel outer measure on X, (2.8.3) holds for every A ⊆ X, and the elements of C are Borel sets in X, then we get that ν is Borel regular on X.

2.8.1 Some regular outer measures

Let A be an algebra of subsets of X, let μ be a countably additive measure on A, and let ν = \overline{μ} be the corresponding outer measure on X, as in Subsection 1.11.4. If A_{1}, A_{2}, A_{3}, … is a sequence of elements of A, then

ν( ⋃_{j=1}^{∞} A_{j}) ≤ ∑_{j=1}^{∞} μ(A_{j}), (2.8.8)

by the definition of ν.

Let C = A_{σ} be the collection of subsets of X that can be expressed as the union of a sequence of elements of A. One can check that

ν_{C}(E) ≤ ν(E) (2.8.9)

for every E ⊆ X, using (2.8.8) and the definition of ν. This corresponds to part of Proposition 1.6 on p273 of [335]. The other part of that proposition corresponds to (2.8.6) in this situation.

It follows that ν_{C} = ν, because ν ≤ ν_{C} automatically, as before. Remember that the elements of A are measurable with respect to ν, as in Subsection 1.11.4, so that the elements of C are measurable with respect to ν. This implies that \overline{μ} is a regular outer measure on X, as in part (a) of Exercise 10.40 on p143 of [159].

2.8.2 Outer regular outer measures

Now let X be a metric space, or a topological space, and let ν be an outer measure on X. Let us say that ν is outer regular if for every A ⊆ X we have that (2.8.10) ν(A) = inf{ν(U) : U ⊆ X is an open set, and A ⊆ U}.

This implies that there is a G_{δ} set B ⊆ X such that A ⊆ B and ν(A) = ν(B), by the remarks at the beginning of the section.

If ν is also a Borel outer measure on X, then it follows that ν is Borel regular on X. In this case, we get that the restriction of ν to the σ-algebra of measurable subsets of X with respect to ν is outer regular as in Subsection 1.12.1.

Conversely, suppose that ν is Borel regular on X. If the restriction of ν to the σ-algera of Borel sets in X is outer regular as in Subsection 1.12.1, then one can check that ν is outer regular as an outer measure on X.

If n is a positive integer, then Lebesgue outer measure on R^{n} is outer regular. This can be obtained using the characterization of Lebesgue outer measure in terms of coverings by sequences of bounded open intervals, as in Section 1.15.

If α is a monotonically increasing real-valued function on the real line, then the corresponding Lebesgue–Stieltjes outer measure \overline{λ_{α}} on R is outer regular. This can be obtained using the characterization of \overline{λ_{α}} in terms of coverings by sequences of bounded open intervals, as in Subsection 2.5.1.

2.9 More on the extension theorem

Let X be a set, let A be an algebra of subsets of X, and let μ be a countably additive measure on X with respect to A. Also let \overline{μ} be the corresponding outer measure on X, as in Subsection 1.11.4. Suppose that A_{1} is a σ-algebra of subsets of X that contains A and is contained in the σ-algebra of measurable subsets of X with respect to \overline{μ}. Let μ_{1} be the restriction of \overline{μ} to A_{1}, and note that μ_{1} is countably additive, as in Subsection 1.11.1. Remember that \overline{μ} = μ on A, so that μ_{1} = μ on A.

Suppose that ν_{1} is another countably additive measure on X with respect to A_{1} such that ν_{1} = μ on A, so that ν_{1} = μ_{1} on A. We would like to compare ν_{1} with μ_{1} on A_{1}, as in part (4) of Theorem 4.17 on p37 of [20], Theorem 1.14 on p31 of [118], Exercise 10.39 on p142 of [159], and some remarks beginning on p272 of [335]. More precisely, in [20, 118, 159, 335], one takes A_{1} to be the σ-algebra of subsets of X generated by A, and essentially the same arguments can be used here.

Let A_{σ} be the collection of subsets of X that may be expressed as the union of a sequence of elements of A, as before. In fact, the elements of A_{σ} may be expressed as the union of a sequence of pairwise-disjoint elements of A, as in Section 1.10. This implies that

ν_{1} = μ_{1} on A_{σ}, (2.9.1)

because of countable additivity. This corresponds to part of the hint to part (a) of Exercise 10.39 on p142 of [159]. This is also mentioned on p38 of [20], p31 of [118], and p272 of [335], using a slightly different argument.

2.9.1 Showing that ν_{1} ≤ μ_{1}

Let E be an element of A_{1}, and let A_{1}, A_{2}, A_{3}, … be a sequence of elements of A such that E ⊆ ⋃_{j=1}^{∞} A_{j}. Observe that

ν_{1}(E) ≤ ν_{1}( ⋃_{j=1}^{∞} A_{j}) ≤ ∑_{j=1}^{∞} ν_{1}(A_{j}) = ∑_{j=1}^{∞} μ(A_{j}). (2.9.2)

This implies that ν_{1}(E) ≤ \overline{μ}(E) = μ_{1}(E), (2.9.3)

by the definition of \overline{μ} in Subsection 1.11.4. This corresponds to (4.7) on p38 of [20], part of Theorem 1.14 on p31 of [118], part (a) of Exercise 10.39 on p142 of [159], and to one of the steps on p272 of [335]. In particular, if μ_{1}(E) = 0, then ν_{1}(E) = 0.

2.9.2 When μ_{1}(E) < +∞

Suppose that E ∈ A_{1} and μ_{1}(E) < ∞, and let ϵ > 0 be given. There is a sequence A_{1}, A_{2}, A_{3}, … of elements of A such that E ⊆ ⋃_{j=1}^{∞} A_{j}

∑_{j=1}^{∞} μ(A_{j}) < μ_{1}(E) + ϵ, (2.9.4)

by the definition of μ_{1}(E) = \overline{μ}(E). This implies that

μ_{1}( ⋃_{j=1}^{∞} A_{j}) ≤ ∑_{j=1}^{∞} μ_{1}(A_{j}) < μ_{1}(E) + ϵ, (2.9.5)

so that

μ_{1}(( ⋃_{j=1}^{∞} A_{j}) \ E) = μ_{1}( ⋃_{j=1}^{∞} A_{j}) − μ_{1}(E) < ϵ. (2.9.6)

It follows that

ν_{1}(( ⋃_{j=1}^{∞} A_{j}) \ E) < ϵ, (2.9.7)

as in the previous subsection.

Note that

ν_{1}( ⋃_{j=1}^{∞} A_{j}) = μ_{1}( ⋃_{j=1}^{∞} A_{j}), (2.9.8)

as in (2.9.1). Using this, we get that

μ_{1}(E) ≤ μ_{1}( ⋃_{j=1}^{∞} A_{j}) = ν_{1}( ⋃_{j=1}^{∞} A_{j})

(2.9.9)  = ν_{1}(E) + ν_{1}(( ⋃_{j=1}^{∞} A_{j}) \ E) < ν_{1}(E) + ϵ.

This implies that μ_{1}(E) ≤ ν_{1}(E), (2.9.10)

because ϵ > 0 is arbitrary. This means that

ν_{1}(E) = μ_{1}(E), (2.9.11)

because of (2.9.3). This corresponds to part (b) of Exercise 10.39 on p142 of [159], part of Theorem 1.14 on p31 of [118], and to some of the remarks on p31 of [118], and p272 of [335].

2.9.3 When E is σ-finite with respect to μ_{1}

Suppose that

E = ⋃_{j=1}^{∞} E_{j}, (2.9.12)

where E_{j} ∈ A_{1} and μ_{1}(E_{j}) < +∞ for each j. One can use the remarks in the previous subsection to get that (2.9.11) holds under these conditions as well. More precisely, one can reduce to the case where the E_{j}’s are pairwise disjoint, or to the case where E_{j} ⊆ E_{j+1} for each j.

In particular, (2.9.11) holds for all E ∈ A_{1} when X can be expressed as the union of a sequence of elements of A_{1} with finite measure with respect to μ_{1}. Note that any subset of X with finite measure with respect to \overline{μ} is contained in the union of a sequence of elements of A with finite measure with respect to μ, by definition of \overline{μ}, as in Subsection 1.11.4. If X can be expressed as the union of a sequence of elements of A_{1} with finite measure with respect to μ_{1}, then it follows that X can be expressed as the union of a sequence of elements of A with finite measure with respect to μ, because the union of a sequence of countable sets is countable too. This is the way that the σ-finiteness condition is stated in part (4) of Theorem 4.17 on p37 of [20], Theorem 1.14 on p31 of [118], part (c) of Exercise 10.39 on p142 of [159], and on p272 of [335].

2.10 Some conditions concerning finite measure

Let (X, d) be a metric space, and let μ be an outer measure on X, or a nonnegative countably additive measure defined on a σ-algebra of subsets of X that contains the Borel sets. Let us say that μ is locally finite on X if for every x ∈ X there is an r > 0 such that

μ(B(x, r)) < +∞. (2.10.1)

This is the same as saying that for every x ∈ X there is a t > 0 such that

μ(\overline{B}(x, t)) < +∞. (2.10.2)

More precisely, (2.10.2) implies (2.10.1) with r = t, and (2.10.1) implies (2.10.2) with t < r. Local finiteness for outer measures on metric spaces is defined in part (1) of Definition 1.5 on p9 of [247] in terms of closed balls, using the notation for open and closed balls on p7 of [247].

Equivalently, μ is locally finite on X if for every x ∈ X there is an open set U ⊆ X such that x ∈ U and μ(U) < +∞. (2.10.3)

This formulation works as well when X is a topological space.

We may sometimes be concerned with situations where

μ(K) < +∞ (2.10.4)

for every compact set K ⊆ X. If X is a topological space, then one may ask that X be Hausdorff, so that compact sets in X are closed sets, and thus Borel sets. If μ is locally finite on X, then it is easy to see that (2.10.4) holds for every compact set K in X. Indeed, local finiteness implies that K can be covered by a family of open sets with finite measure. It follows that K can be covered by finitely many open sets with finite measure, because K is compact, so that (2.10.4) holds.

2.10.1 Local compactness

We say that X is locally compact at a point x ∈ X if there is an open set U ⊆ X and a compact set K ⊆ X such that x ∈ U and U ⊆ K. Sometimes this is defined by taking K to be the closure \overline{U} of U in K. This version implies the previous one, and the converse holds when X is Hausdorff, so that compact sets in X are closed sets. If X is a metric space, then local compactness at x is the same as saying that there is an r > 0 such that \overline{B}(x, r) is compact.

If X is locally compact at every x ∈ X, then we say that X is locally compact as a metric or topological space. In this case, if (2.10.4) holds for every compact set K ⊆ X, then μ is locally finite on X. Locally compact Hausdorff topological spaces have a lot of nice properties related to analysis.

2.10.2 σ-Compact sets

A subset of X is said to be σ-compact if it can be expressed as the union of a sequence of compact sets. If (2.10.4) holds for every compact set K ⊆ X, then σ-compact sets are σ-finite with respect to μ.

It is easy to see that the union of finitely many compact subsets of X is compact as well. This implies that a σ-compact subset A of X may be expressed as the union of a sequence of compact subsets of X that is increasing with respect to inclusion. More precisely, if A can be expressed as the union of a sequence of compact sets, then one can replace the nth term of the sequence with the union of the first n terms of the sequence.

Suppose that X is σ-compact, so that there is a sequence K_{1}, K_{2}, K_{3}, … of compact subsets of X such that

X = ⋃_{j=1}^{∞} K_{j}. (2.10.5)

If E is a closed set in X, then it is well known that E ∩ K_{j} is compact for each j. This implies that

E = ⋃_{j=1}^{∞}(E ∩ K_{j}) (2.10.6)

is σ-compact.

It is easy to see that a countable union of σ-compact sets is σ-compact, because a countable union of countable sets is countable. If X is σ-compact, then it follows that F_{σ} sets in X are σ-compact.

If X is locally compact, and if there is a countable base for the topology of X, then one can check that X is σ-compact. In particular, locally compact separable metric spaces are σ-compact.

2.10.3 σ-Compactness and inner regularity

Suppose now that μ is a Borel measure on X, and that A ⊆ X is an F_{σ} set, so that there is a sequence E_{1}, E_{2}, E_{3}, … of closed sets in X such that A = ⋃_{j=1}^{∞} E_{j}. We may as well ask also that E_{j} ⊆ E_{j+1} for each j, by replacing E_{j} with ⋃_{l=1}^{j} E_{l}, if necessary. Under these conditions, we get that

μ(E_{j}) → μ(A) as j → ∞, (2.10.7) as in Section 1.9. This implies that A satisfies the inner regularity property with respect to closed sets mentioned in Subsection 1.12.2.

Similarly, if A is σ-compact, then A satisfies the inner regularity property with respect to compact sets mentioned in Subsection 1.12.2.

If X is σ-compact, then one can check that inner regulariy with respect to closed sets implies inner regularity with respect to compact sets. This uses the fact that closed sets in X are σ-compact in this case, as in the previous subsection, so that closed sets satisfy the inner regularity property with respect to compact sets.

2.10.4 Another basic regularity property

Suppose now that A is a G_{δ} set, so that there is a sequence U_{1}, U_{2}, U_{3}, … of open sets in X such that A = ⋂_{j=1}^{∞} U_{j}. As before, we may as well ask that U_{j+1} ⊆ U_{j} for each j, by replacing U_{j} with ⋂_{l=1}^{j} U_{l}, if necessary.

Suppose also that there is an open set V ⊆ X such that A ⊆ V and μ(V) is finite. We may as well ask that U_{j} ⊆ V for each j, by replacing U_{j} with U_{j} ∩ V, if necessary. This means that μ(U_{j}) is finite too.

It follows that μ(U_{j}) → μ(A) as j → ∞, (2.10.8)

as in Subsection 1.9.2. This implies that A satisfies the outer regularity property mentioned in Subsection 1.12.1.

2.11 Some nice families of sets

Let X be a set, and let F be a family of subsets of X that contains X and the empty set as elements. Let us say that F is σ-nice if for every sequence of elements of F, the union and intersection of the sets in the sequence is an element of F as well. This corresponds to the conditions (2) and (3) near the bottom of p59 of [113]. Of course, a σ-algebra of subsets of X is a σ-nice family, and a σ-nice family may be considered as a monotone class in particular.

If F is a σ-nice family of subsets of X, then one can check that

F_{0} = {A ∈ F : X \ A ∈ F} (2.11.1)

is a σ-algebra of subsets of X. This corresponds to a remark near the top of p60 of [113].

It is easy to see that the intersection of any nonempty collection of σ-nice families of subsets of X is also a σ-nice family of subsets of X. If E is any collection of subsets of X, then the intersection of all of the σ-nice families of subsets of X that contains E is the smallest σ-nice family of subsets of X that contains E.

Suppose that E contains the complement in X of each of its elements. If F is a σ-nice family of subsets of X that contains E, then it follows that (2.11.1) contains E too. This implies that the smallest σ-nice family of subsets of X that contains E is the same as the smallest σ-algebra of subsets of X that contains E, as in [113].

2.11.1 Another characterization of Borel sets

Suppose now that X is a metric space, and remember that

(2.11.2) open sets in X are F_{σ} sets,

as in Subsection 2.2.2. Equivalently, this means that

(2.11.3) closed sets in X are G_{δ} sets,

as before. More precisely, it would be enough to ask that X be a topological space that satisifes these conditions for the moment.

Let F be a σ-nice family of subsets of X that contains all of the open sets in X. This implies that F contains all of the closed sets in X in this case. It follows that (2.11.1) contains all of the open sets in X. Using this, we get that the collection of Borel sets in X is the same as the smallest σ-nice family of subsets of X that contains the open sets in X, as mentioned on p60 of [113].

Similarly, one can check that the collection of Borel sets in X is the same as the smallest σ-nice family of subsets of X that contains the closed sets in X, as mentioned in [113].

2.11.2 Regularity of finite Borel measures

Let us continue to ask that X be a metric space, or a topological space that satisfies the additional condition mentioned in the previous subsection, and let μ be a finite Borel measure on X. It is well known that

(2.11.4) μ is outer regular, and inner regular with respect to closed sets,

as in Section 1.12. This corresponds to part of the proof of part (1) of Theorem 1.10 on p11 of [247], and to part of Proposition 1.3 on p269 of [335]. This also corresponds to part of Lemma 1 on p6 of [110], and Theorem 2.2.2 on p60 of [113], and there are some related results in [159, 298]. We shall say more about these matters in the next two subsections, and the next section.

To see this, let A be the collection of all Borel sets A ⊆ X such that for each ϵ > 0 there are an open set U ⊆ X and a closed set E ⊆ X with E ⊆ A ⊆ U and μ(U \ E) < ϵ. (2.11.5)

One can check that open sets in X are elements of A, using the remarks at the beginning of Subsection 2.10.3, because open sets are F_{σ} sets and μ is finite, by hypothesis. Similarly, one can check that closed sets in X are elements of A, using the remarks in Subsection 2.10.4, because closed sets are G_{δ} sets and μ is finite, by hypothesis.

One can also verify that A is a σ-algebra of subsets of X, as mentioned in [247]. In particular, if A ∈ A, then it is easy to see that X \ A ∈ A, because of the way that A is defined. This implies that A contains the Borel sets in X. It follows that μ is outer regular, and inner regular with respect to closed sets.

2.11.3 An analogous argument for inner regularity of μ

Let us continue to suppose that μ is a finite Borel measure on X. A variant of the previous argument is used in the proof of part (1) in Theorem 2.2.2 on p60 of [113]. In this argument, we let F be the family of Borel sets A ⊆ X such that for each ϵ > 0 there is a closed set E ⊆ X such that E ⊆ A and

μ(A \ E) < ϵ. (2.11.6)

Clearly closed sets in X are contained in F. One can check that F is a σ-nice family of subsets of X, as on p61 of [113]. It follows that F contains the Borel sets in X, as in Subsection 2.11.1. Remember that inner regularity is equivalent to outer regularity when μ(X) < +∞, as mentioned in Subsection 1.12.2.

2.11.4 Inner regularity for Borel sets of finite measure

Now let μ be any Borel measure on X. If B ⊆ X is a Borel set, then

μ_{B}(A) = μ(A ∩ B) (2.11.7)

defines another Borel measure on X. Suppose that μ(B) < +∞, so that

μ_{B}(X) = μ(B) < +∞. (2.11.8)

Thus μ_{B} is outer regular, and inner regular with respect to closed sets, as before.

One can use the inner regularity of μ_{B} to get that B satisfies the inner regularity property with respect to closed sets and μ mentioned in Subsection 1.12.2. This corresponds to part (i) of Lemma 1 on p6 of [110], part (1) of Theorem 2.2.2 on p60 of [113], and part (1) of Theorem 1.10 on p11 of [247].

Some additional regularity properties under suitable conditions will be discussed in the next section.

2.12 More on Borel measures

Let X be a metric space, or a topological space with the property mentioned in Subsection 2.11.1, and let μ be a Borel measure on X.

2.12.1 Outer regularity and open sets of finite measure

Suppose that W is an open set in X with μ(W) < +∞, and consider the corresponding Borel measure

μ_{W} (A) = μ(A ∩ W) (2.12.1) on X. Thus μ_{W} (X) = μ(W) < +∞, so that μ_{W} is outer regular, and inner regular with respect to closed sets, as in the previous section.

If B ⊆ X is a Borel set and ϵ is a positive real number, then there is an open set U in X such that B ⊆ U and

μ_{W} (U \ B) < ϵ, (2.12.2)

because of outer regularity. This means that

μ((U ∩ W) \ B) = μ((U \ B) ∩ W) = μ_{W} (U \ B) < ϵ. (2.12.3)

If B ⊆ W, then U ∩ W is also an open set that contains B.

Suppose now that W_{1}, W_{2}, W_{3}, … is a sequence of open subsets of X such that μ(W_{j}) < +∞ (2.12.4)

for each j, and let B be a Borel set in X with

B ⊆ ⋃_{j=1}^{∞} W_{j}. (2.12.5)

If ϵ is a positive real number, then we can use the argument in the two previous paragraphs to get a sequence of open sets V_{1}, V_{2}, V_{3}, … in X such that

B ∩ W_{j} ⊆ V_{j} (2.12.6)

and μ(V_{j} \ (B ∩ W_{j})) < 2^{−j} ϵ (2.12.7)

for each j. Note that ⋃_{j=1}^{∞} V_{j} is an open set in X with

B = ⋃_{j=1}^{∞}(B ∩ W_{j}) ⊆ ⋃_{j=1}^{∞} V_{j}. (2.12.8)

We also have that

( ⋃_{j=1}^{∞} V_{j}) \ B = ⋃_{j=1}^{∞}(V_{j} \ B) ⊆ ⋃_{j=1}^{∞}(V_{j} \ (B ∩ W_{j})), (2.12.9)

so that

μ(( ⋃_{j=1}^{∞} V_{j}) \ B) ≤ μ( ⋃_{j=1}^{∞}(V_{j} \ (B ∩ W_{j})))

(2.12.10)  ≤ ∑_{j=1}^{∞} μ(V_{j} \ (B ∩ W_{j})) < ∑_{j=1}^{∞} 2^{−j} ϵ = ϵ.

This corresponds to part (ii) of Lemma 1 on p6 of [110], part (2) of Theorem 2.2.2 on p61 of [113], part (2) of Theorem 1.10 on p11 of [247], and part of Proposition 1.3 on p269 of [335].

2.12.2 More on inner regularity

Suppose that B is a Borel set in X that is σ-finite with respect to μ, so that there is a sequence B_{1}, B_{2}, B_{3}, … of Borel sets in X such that B = ⋃_{j=1}^{∞} B_{j} and μ(B_{j}) < +∞ for each j. We may as well ask that B_{j} ⊆ B_{j+1} for each j, by replacing B_{j} with ⋃_{l=1}^{j} B_{l}, if necessary, as usual. This implies that μ(B_{j}) tends to μ(B) as j → ∞, as in Section 1.9. One can check that B satisfies the inner regularity property with respect to closed sets mentioned in Subsection 1.12.2, because of the analogous properties of the B_{j}’s in Subsection 2.11.4. Note that it suffices to consider the case where μ(B) = +∞ here.

Suppose now that X is a metric space, and that open balls in X have finite measure with respect to μ. Of course, this is the same as saying that closed balls in X have finite measure. In particular, this means that X is σ-finite with respect to μ. If A is a Borel set in X and ϵ is a positive real number, then part of Proposition 1.3 on p269 of [335] states that there is a closed set E in X such that E ⊆ A and μ(A \ E) < ϵ. This is equivalent to the usual inner regularity property of A with respect to closed sets when μ(A) < +∞.

Otherwise, one can use the hypothesis on X to express A as the union of a sequence A_{1}, A_{2}, A_{3}, … of finite measure such that every element of X has a neighborhood that intersects at most two of the A_{j}’s. If ϵ is a positive real number, then there is a sequence E_{1}, E_{2}, E_{3}, … of closed sets in X such that E_{j} ⊆ A_{j} and μ(A_{j} \ E_{j}) < 2^{−j} ϵ (2.12.11)

for each j, as in Subsection 2.11.4. One can check that E = ⋃_{j=1}^{∞} E_{j} is a closed set under these conditions, because every element of X has a neighborhood that intersects at most two of the E_{j}’s. In addition,

A \ E ⊆ ⋃_{j=1}^{∞}(A_{j} \ E) ⊆ ⋃_{j=1}^{∞}(A_{j} \ E_{j}), (2.12.12)

and thus

μ(A \ E) ≤ ∑_{j=1}^{∞} μ(A_{j} \ E_{j}) < ∑_{j=1}^{∞} 2^{−j} ϵ = ϵ. (2.12.13)

2.12.3 Some more regularity results

Let X be a metric space, or a Hausdorff topological space, that is locally compact. Suppose that μ is a Borel measure on X such that μ(K) < +∞ for all compact sets K in X. A condition on X that is sometimes used to get regularity properties of μ is that every open set X be σ-compact, as in Exercise 12.55 on p185 of [159], and Theorem 2.18 on p50 of [298]. This is closely related to the regularity properties discussed in this and the previous section. Note that this condition on X is the same as saying that X is σ-compact, and that every open set in X is an F_{σ} set, as in Subsection 2.10.2.

If K is a compact set in X, then one can use local compactness to cover K by finitely many open sets that are contained in compact sets. This implies that K is contained in an open set U that is contained in a compact set. In particular, this means that μ(U) < +∞. If X is σ-compact, then one can use this to get that X may be expressed as the union of a sequence of open sets that are contained in compact sets, and thus have finite measure.

2.13 Locally finite Borel measures on the real line

Let μ be a locally finite Borel measure on the real line, as in Section 2.10. The local finiteness of μ on R is equivalent to the condition that μ(K) < +∞ for every compact set K ⊆ R, as before. This means that μ is finite on bounded Borel sets in R.

Of course, R may be expressed as the union of a sequence of bounded open intervals. One can use this to get that μ is outer regular on R, as in Subsection 2.12.1.

It follows that μ is uniquely determined as a Borel measure on R by its restriction to the collection of open sets in R. It is well known that every open set in R may be expressed as the union of finitely or countably many pairwisedisjoint open intervals. This means that the measure of such an open set is equal to the sum of the measures of the corresponding intervals. Thus μ is uniquely determined by its restriction to the collection of open intervals in R.

It is easy to see that any unbounded open interval in the real line may be expressed as the union of a sequence of bounded open intervals that are increasing with respect to inclusion. Under these conditions, the measures of the bounded open intervals tends to the measure of the unbounded open interval, as in Section 1.9. This implies that μ is uniquely determined by its restriction to the collection of bounded open intervals in R. This corresponds to Exercise 3.9 on p21 of [20], although the reader is probably expected to use a somewhat different argument there.

Let α be a monotonically increasing real-valued function on R, and let λ_{α} be the corresponding Lebesgue–Stieltjes measure on R, as in Section 2.5, considered as a Borel measure on R. Remember that the measure of an open interval in R with respect to λ_{α} may be expressed in terms of α, as before. The remarks in the previous paragraphs imply that λ_{α} is uniquely determined by this property, as a Borel measure on R. This corresponds to part of the uniqueness part of part (b) of Theorem 8.14 on p173 of [298], and to the uniqueness part of the first part of Theorem 3.5 on p282 of [335]. This also corresponds to part of Theorem 19.48 on p331 of [159], using Remark 19.49 on p332 of [159]. Note that the uniqueness of λ_{α} could be obtained as in Section 2.9 as well. This is related to part of Theorem 1.16 on p35 of [118].

2.13.1 Some normalizing conditions on α

However, α is not uniquely determined by λ_{α}, or equivalently by the α-lengths of intervals as in Section 2.5. One can add a constant to α without changing 2.13. LOCALLY FINITE BOREL MEASURES ON THE REAL LINE 67

the α-lengths of intervals or λ_{α}. In order to deal with this, one may ask that

α(0) = 0, (2.13.14)

as in Theorem 19.48 on p331 of [159], and Theorem 3.5 on p282 of [335]. If α has a finite lower bound on R, then one may ask that

α(−∞) = 0, (2.13.15)

as on p171 of [298].

If α is continuous on R, then it is easy to see that α is uniquely determined by the α-lengths of intervals and either of the additional conditions (2.13.14) or (2.13.15). However, the values of α at the points of discontinuity are not determined. To deal with this, one may ask that α be continuous from the left at each point, as in Theorem 19.48 on p331 of [159], and on p171 of [298], or one may ask that α be continuous from the right at each point, as on p28 of [20], p33 of [118], and p282 of [335].

If α is any monotonically increasing real-valued function on R, then put

α_{L}(x) = α(x−) (2.13.16)

and α_{R}(x) = α(x+) (2.13.17)

for every x ∈ R. These are monotonically increasing real-valued functions on R with α_{L}(x) ≤ α(x) ≤ α_{R}(x) (2.13.18)

for every x ∈ R, as in Subsection 2.4.2. One can check that

α_{L}(x−) = α(x−) = α_{R}(x−) (2.13.19)

and α_{L}(x+) = α(x+) = α_{R}(x+) (2.13.20)

for every x ∈ R. In particular, this means that α_{L} is continuous from the left at each point, and that α_{R} is continuous from the right at each point. We also get that the α-lengths of intervals are the same as the α_{L}-lengths and the α_{R}-lengths.

2.13.2 Getting monotonically increasing functions

Let μ be a locally finite Borel measure on R again. If x ∈ R, then put

(2.13.21)  α_{μ,0,L}(x) = μ([0, x)) when x > 0
= 0  when x = 0
= −μ([x, 0)) when x < 0,

as in Theorem 19.45 on p329 of [159]. One can check that α_{μ,0,L} is monotonically increasing on R, and continuous from the left at every point, as in [159]. Theorem 19.48 on p331 of [159] and Remark 19.49 on p332 of [159] say that

μ ↦ α_{μ,0,L} (2.13.22)

is a one-to-one mapping from the set of all locally finite Borel measures on R onto the set of monotonically increasing real-valued functions on R that are equal to 0 at 0 and are left continuous at every point. The inverse mapping sends such a monotonically increasing function on R to the corresponding Lebesgue– Stieltjes measure.

Similarly, put

(2.13.23)  α_{μ,0,R}(x) = μ((0, x]) when x > 0
= 0  when x = 0
= −μ((x, 0]) when x < 0,

as in Exercise 4.1 on p39 of [20], Theorem 1.16 on p35 of [118], and Theorem 3.5 on p282 of [335]. One can verify that α_{μ,0,R} is monotonically increasing on R, and continuous from the right at every point, as mentioned on p31 of [20], and in [118, 335].

If μ((−∞, 0)) < +∞, then put

α_{μ,−∞,L}(x) = μ((−∞, x)) (2.13.24)

for every x ∈ R, as in part (a) of Theorem 8.14 on p173 of [298]. One can check that α_{μ,−∞,L} is monotonically increasing and left continuous at every point, with α_{μ,−∞,L}(−∞) = 0, (2.13.25)

as in [298]. One can also verify that

α_{μ,−∞,L}(+∞) = μ(R). (2.13.26)

In this case, we can consider

α_{μ,−∞,R}(x) = μ((−∞, x]) (2.13.27)

on R as well. One can check that α_{μ,−∞,R} is monotonically increasing and right continuous at every point, with

α_{μ,−∞,R}(−∞) = 0 (2.13.28)

and α_{μ,−∞,R}(+∞) = μ(R). (2.13.29)

2.14. SOME MORE FAMILIES OF RINGS AND ALGEBRAS OF SETS 69

2.14 Some more families of rings and algebras of sets

Let X be a metric space, or a topological space. Consider the collection A_{op−cl} of subsets of X that are both open and closed. It is easy to see that this is an algebra of subsets of X. In particular, ∅ and X are elements of A_{op−cl}, and these are the only elements of A_{op−cl} if and only if X is connected.

Similarly, let A_{op−cm} be the collection of subsets of X that are both open and compact. If X is a metric space, or a Hausdorff topological space, then compact subsets of X are closed, and one can check that A_{op−cm} is a ring of subsets of X. This uses the fact that the intersection of a closed set and a compact set is compact in any metric space. If X is not necessarily Hausdorff, then one can take A_{op−cm} to be the collection of subsets of X that are open, closed, and compact, to get a ring of subsets of X. Using this, we get that

A_{op−cl} = A_{op−cm} (2.14.1)

when X is compact.

If μ is a finitely additive measure on A_{op−cm}, then μ is automatically countably additive. Indeed, let A_{1}, A_{2}, A_{3}, … be an infinite sequence of pairwisedisjoint elements of A_{op−cm} such that ⋃_{j=1}^{∞} A_{j} ∈ A_{op−cm}. Under these conditions, there is a positive integer n such that

⋃_{j=1}^{n} A_{j} = ⋃_{j=1}^{∞} A_{j}, (2.14.2)

because the A_{j}’s are open sets and ⋃_{j=1}^{∞} A_{j} is compact, by hypothesis. This means that A_{j} = ∅ for j > n, because the A_{j}’s are pairwise disjoint. Thus we can use finite additivity to get that

μ( ⋃_{j=1}^{∞} A_{j}) = μ( ⋃_{j=1}^{n} A_{j}) = ∑_{j=1}^{n} μ(A_{j}) = ∑_{j=1}^{∞} μ(A_{j}). (2.14.3)

2.14.1 Some remarks related to the Lindelöf property

A subset E of X is said to have the Lindelöf property if every open covering of E in X can be reduced to a subcovering with only finitely or countably many elements. It is easy to see that this happens when E is σ-compact.

If there is a base for the topology of X with only finitely or countably many elements, then Lindelöf’s theorem implies that every subset of X has the Lindelöf property.

Suppose for the moment that X is a metric space with the Lindelöf property. This implies that

(2.14.4) for every r > 0, X can be expressed as the union of finitely or countably many open balls of radius r.

If X is any metric space that satisfies (2.14.4), then one can get a base for the topology of X consisting of finitely or countably many open balls, using the union of the collections of open balls of radius r mentioned in (2.14.4) for r ∈ Z_{+}. Note that one can get a dense set in X with only finitely or countably many elements using the centers of these open balls.

2.14.2 Zero-dimensional spaces

We say that X is zero dimensional at a point x ∈ X if for every open set V ⊆ X with x ∈ V there is an open set U ⊆ X such that x ∈ U, U ⊆ V, and U is a closed set in X too. This is the same as saying that the collection of elements of A_{op−cl} that contain x is a local base for the topology of X.

If X is zero dimensional at every x ∈ X, then we say that X is zero dimensional as a metric or topological space. This means that A_{op−cl} is a base for the topology of X.

If U is an open set in X such that

(2.14.5) U can be expressed as the union of a sequence of elements of A_{op−cl},

then U is an element of any σ-algebra of subsets of X that contains A_{op−cl}. If X is zero dimensional, then this holds when U has the Lindelöf property. If X is zero dimensional, and every open set in X has the Lindelöf property, then we get that any σ-algebra of subsets of X that contains A_{op−cl} also contains the Borel sets. In particular, this works when X is zero dimensional and there is a countable base for the topology of X, by Lindelöf’s theorem.

2.14.3 Some remarks about countable bases

Suppose that B_{0} is a base for the topology of X with only finitely or countably many elements, and that B is another base for the topology of X. Under these conditions, it is well known that there is a base B_{1} for the topology of X such that

B_{1} ⊆ B (2.14.6)

and

(2.14.7) B_{1} has only finitely or countable many elements.

Indeed, Lindelöf’s theorem implies that every element of B_{0} can be expressed as the union of finitely or countably many elements of B. One can use this to get a collection B_{1} of finitely or countably many elements of B such that every element of B_{0} may be expressed as a union of elements of B_{1}. This implies that B_{1} is a base for the topology of X too.

2.15. ULTRAMETRICS 71

2.15 Ultrametrics

Let (X, d) be a metric space. We say that d is an ultrametric on X if d(x, z) ≤ max(d(x, y), d(y, z)) (2.15.1) for every x, y, z ∈ X. Note that this implies the usual version of the triangle inequality for d on X.

The discrete metric is defined on X by putting d(x, y) equal to 0 when x = y, and to 1 when x ≠ y. It is easy to see that this defines an ultrametric on X.

2.15.1 Open balls in ultrametric spaces

Suppose that d is an ultrametric on X. If x, w ∈ X satisfy d(x, w) < r (2.15.2) for some positive real number r, then one can check that B(w, r) ⊆ B(x, r). (2.15.3) Similarly, we have that B(x, r) ⊆ B(x, r), (2.15.4) so that B(w, r) = B(x, r). (2.15.5)

Suppose that B(x, r) ∩ B(x, z) ≠ ∅ (2.15.6) for some z ∈ X, and let w be an element of B(x, r) ∩ B(z, r). Observe that B(x, r) = B(w, r) = B(z, r), (2.15.7) as in the preceding paragraph.

Thus, if B(x, r) ≠ B(z, r), (2.15.8) then B(x, r) ∩ B(z, r) = ∅. (2.15.9) If z ∈ X \ B(x, r), then (2.15.8) holds, so that B(z, r) ⊆ X \ B(x, r), (2.15.10) as in (2.15.9).

This shows that X \ B(x, r) is an open set, so that (2.15.11) B(x, r) is a closed set. Alternatively, one can check that B(x, r) contains all of its limit points in X, using (2.15.3). It follows that X is zero dimensional with respect to d, as in the previous section.

2.15.2 Closed balls in ultrametric spaces

Similarly, if d(x, w) ≤ r, (2.15.12)

then one can verify that \overline{B}(w, r) ⊆ \overline{B}(x, r). (2.15.13)

We also have that \overline{B}(x, r) ⊆ \overline{B}(w, r), (2.15.14)

for the same reasons, so that

\overline{B}(w, r) = \overline{B}(x, r). (2.15.15)

In particular, (2.15.13) implies that

(2.15.16) \overline{B}(x, r) is an open set.

This can be used to get that X is zero dimensional with respect to d as well.

If \overline{B}(x, r) ∩ \overline{B}(z, r) ≠ ∅, (2.15.17)

and w ∈ \overline{B}(x, r) ∩ \overline{B}(z, r), then

\overline{B}(x, r) = \overline{B}(w, r) = \overline{B}(z, r), (2.15.18)

as before. If \overline{B}(x, r) ≠ \overline{B}(z, r), (2.15.19)

then we get that \overline{B}(x, r) ∩ \overline{B}(z, r) = ∅. (2.15.20)

Note that (2.15.20) holds when z ∈ X \ \overline{B}(x, r). One can use this as another way to get that X \ \overline{B}(x, r) contains all of its limit points, and is thus a closed set.

Some additional topics related to ultrametrics will be discussed in Appendix A.

Measurable functions and integration

3.1 Measurable functions

Let X and Y be sets, and let f be a mapping from X into Y. If E ⊆ Y, then it is easy to see that f^{−1}(Y \ E) = X \ f^{−1}(E). (3.1.1)

Let I be a nonempty set, and let E_{j} be a subset of Y for each j ∈ I. One can check that

f^{−1}( ⋃_{j∈I} E_{j}) = ⋃_{j∈I} f^{−1}(E_{j}) (3.1.2)

and

f^{−1}( ⋂_{j∈I} E_{j}) = ⋂_{j∈I} f^{−1}(E_{j}). (3.1.3)

Let A_{X} be a σ-algebra of subsets of X. One can verify that

{E ⊆ Y : f^{−1}(E) ∈ A_{X}} (3.1.4)

is a σ-algebra of subsets of Y, using the previous remarks. This corresponds to part (a) of Theorem 1.12 on p13 of [298], and it is mentioned in the proof of Proposition 2.1 on p43 of [118]. This is also part of part (a) of Exercise 11.38 on p161 of [159].

3.1.1 When Y is a metric or topological space

Suppose that Y is a metric space, or a topological space. We say that f is measurable if for every open set V ⊆ Y, we have that

f^{−1}(V) ∈ A_{X}. (3.1.5)

This corresponds to part (c) of Definition 1.3 on p9 of [298].

Suppose for the moment that X is a metric or topological space, and that A_{X} contains the σ-algebra of Borel sets in X. If f is continuous, then f is automatically measurable in this case, as mentioned on p13 of [298].

3.1.2 Measurable mappings into measurable spaces

Let A_{Y} be a σ-algebra of subsets of Y. We say that f is measurable with respect to A_{Y} on Y if (3.1.5) holds for every V ∈ A_{Y}, as on p43 of [118].

Suppose that Y is a metric space or a topological space again, and that A_{Y} is the corresponding σ-algebra of Borel sets in Y. If f is a measurable function in the previous sense, so that (3.1.5) holds when V is an open set in Y, then f is a measurable mapping with respect to A_{Y}. This is because (3.1.4) is a σ-algebra of subsets of Y that contains the open sets, by hypothesis. This corresponds to part (b) of Theorem 1.12 on p13 of [298]. This is the way that measurability of real-valued functions is defined on p31 of [14].

In particular, if X is a metric or topological space, f is continuous, and A_{X} contains the Borel sets in X, then f is measurable with respect to A_{Y}. This corresponds to Corollary 2.2 on p44 of [118], and part (a) of Exercise 10.42 on p143 of [159].

3.1.3 Compositions of measurable mappings

Let A_{Y} be any σ-algebra of subsets of Y again, and let Z be another set with a σ-algebra A_{Z} of its subsets. If f is measurable with respect to A_{Y}, and if g is a mapping from Y into Z that is measurable with respect to A_{Z}, then it is easy to see that the composition g ∘ f of f and g is measurable with respect to A_{Z}, as a mapping from X into Z, as mentioned on p43 of [118].

Suppose that Y and Z are metric or topological spaces, and that A_{Y} and A_{Z} are the corresponding σ-algebras of Borel sets. In this case, the previous statement about measurability of compositions corresponds to part (d) of Theorem 1.12 on p13 of [298]. In particular, we get that g ∘ f is measurable when f is measurable and g is continuous, as in part (b) of Theorem 1.7 on p11 of [298], and mentioned immediately after the statement of Theorem 1.12 on p13 of [298].

3.1.4 Measurability and bases for topologies

Suppose that Y is a metric or topological space, B is a base for the topology of Y, as in Section 2.3, and that every open set in Y can be expressed as the union of finitely or countably many elements of B. Of course, this holds automatically when B has only finitely or countably many elements. If (3.1.5) holds for every V ∈ B, then it is easy to see that it also holds for every open set V in Y, so that f is measurable.

3.2. INCLUSION MAPPINGS AND INDUCED σ-ALGEBRAS 75

3.2 Inclusion mappings and induced σ-algebras

Let X be a set, and let X_{0} be a subset of X. Also let ι = ι_{X_{0},X} be the obvious inclusion mapping of X_{0} into X, so that ι sends each element of X_{0} to itself, as an element of X. If A ⊆ X, then

ι^{−1}(A) = A ∩ X_{0}. (3.2.1)

Let A_{X} be a σ-algebra of subsets of X, and put

A_{X,X_{0}} = {A ∩ X_{0} : A ∈ A_{X}}, (3.2.2)

as in Subsection 1.6.4. This is a σ-algebra of subsets of X_{0}, as before, and we may call this the σ-algebra of subsets of X_{0} induced by A_{X}. Clearly ι is measurable as a mapping from X_{0} into X with respect to these σ-algebras. In fact, A_{X,X_{0}} is the smallest σ-algebra of subsets of X_{0} with this property. If X_{0} ∈ A_{X}, then A_{X,X_{0}} = {A ∈ A_{X} : A ⊆ X_{0}}, (3.2.3)

as in Subsection 1.6.4.

If A_{X_{0}} is any σ-algebra of subsets of X_{0}, then one can check that

A_{X_{0},X} = {A ⊆ X : A ∩ X_{0} ∈ A_{X_{0}}} (3.2.4)

is a σ-algebra of subsets of X. This is the largest σ-algebra of subsets of X such that ι is measurable with respect to A_{X_{0}} on X_{0}.

3.2.1 More on induced σ-algebras

Suppose now that X is a metric space, or a topological space. A subset of X_{0} is said to be relatively open if it can be expressed as U ∩ X_{0} for some open set U ⊆ X. It is well known and not difficult to show that this defines a topology on X_{0}, called the induced topology associated to the topology on X. If X is a metric space, then the restriction of the metric on X to X_{0} is a metric on X_{0}. In this case, it is well known that the relatively open sets in X_{0} are the same as the open sets in X_{0} with respect to the restriction of the metric on X to X_{0}.

Let A_{X} and A_{X_{0}} be the σ-algebras of Borel sets in X and X_{0}, respectively. Thus A_{X_{0}} is the smallest σ-algebra of subsets of X_{0} that contains the relatively open sets in X_{0}. If A_{X,X_{0}} is as in (3.2.2), then A_{X,X_{0}} is a σ-algebra of subsets of X_{0} that contains the relatively open sets, so that

A_{X_{0}} ⊆ A_{X,X_{0}}. (3.2.5)

If A_{X_{0},X} is as in (3.2.4), then A_{X_{0},X} is a σ-algebra of subsets of X that contains the open sets in X, so that

A_{X} ⊆ A_{X_{0},X}. (3.2.6)

Equivalently, ι is continuous as a mapping from X_{0} into X, by construction, and this implies that ι is measurable with respect to A_{X} and A_{X_{0}}, as in the previous section. It follows that

A_{X,X_{0}} ⊆ A_{X_{0}}. (3.2.7)

Thus A_{X_{0}} = A_{X,X_{0}} (3.2.8)

because of (3.2.5) and (3.2.7). If X_{0} is a Borel set in X, then we get that the Borel sets in X_{0} are the same as the Borel sets in X that are contained in X_{0}, as in (3.2.3).

This is related to the definition of Borel measurability of a real-valued function on a Borel set in the real line on p33 of [14]. This is also related to 2.44 on p34 of [14], concerning the composition of a measurable real-valued function with a real-valued Borel measurable function on a subset of the real line.

3.2.2 Borel sets in the set of extended real numbers

A subset E of the set of extended real numbers is defined to be a Borel set on p36 of [14] if (3.2.9) E ∩ R is a Borel set in R

in the previous sense. It is not too difficult to show that this is equivalent to defining Borel sets in the set of extended real numbers using the previous definition and the topology on the set of extended real numbers mentioned in Subsection 1.1.2.

More precisely, this topology on the set of extended real numbers may be defined by saying that open intervals in R are open sets, as well as intervals of the form (a, +∞] and [−∞, b) with a, b ∈ R, and arbitrary unions of these basic open sets. In particular, R is an open set in the set of extended real numbers with respect to this topology, and the standard topology on the real line is the same as the topology induced on R by this topology on the set of extended real numbers. One can use this and the remarks in the previous subsection to get that the definition of a Borel set of extended real numbers mentioned in the preceding paragraph is equivalent to the previous definition using this topology on the set of extended real numbers.

3.3 Extended real-valued measurable functions

Let (X, A_{X}) be a measurable space, as in Subsection 1.7.2, and let f be an extended real-valued function on X. It is well known and not too difficult to show that the following four conditions are equivalent. First,

f^{−1}((a, +∞]) ∈ A_{X} (3.3.1)

for every a ∈ R. Second, f^{−1}([a, +∞]) ∈ A_{X} (3.3.2) for every a ∈ R. Third, f^{−1}([−∞, b)) ∈ A_{X} (3.3.3)

for every b ∈ R. Fourth, f^{−1}([−∞, b]) ∈ A_{X} (3.3.4)

for every b ∈ R. This corresponds to part of Theorem 11.3 on p149 of [159], Theorem 11.15 on p311 of [299], and to some remarks on p28 of [335]. This also corresponds to Proposition 5.5 on p44 of [20] and part of Proposition 2.3 on p44 of [118] for real-valued functions.

In fact, measurability of f is defined using (3.3.1) in Definition 11.2 on p149 of [159], and Definition 11.13 on p310 of [299], as well as Definition 5.1 on p43 of [20] for real-valued functions. Similarly, (3.3.3) is used to define measurability on p28, 273 of [335]. If a, b ∈ R and a < b, then we can use (3.3.1) and (3.3.3) to get that

f^{−1}((a, b)) = f^{−1}((a, +∞]) ∩ f^{−1}([−∞, b)) ∈ A_{X}. (3.3.5)

One can also define measurability of f as in Subsection 3.1.1, using the standard topology on the set of extended real numbers, as in Subsection 1.1.2 and the previous section. This topology may be described by saying that (a, +∞] and [−∞, b) are open sets for all a, b ∈ R, so that (a, b) is an open set too when a < b, and that these open sets form a base for the topology. In particular, measurability with respect to this topology implies that (3.3.1) and (3.3.3) hold for all a, b ∈ R. To get the converse, one can use the fact that every open set of extended real numbers may be expressed as the union of finitely or countably many of these basic open sets. This corresponds to part (c) of Theorem 1.12 on p13 of [298].

Measurability of a real-valued function f on X is defined to p44 of [118] to mean that f is measurable as a mapping into R with respect to the σ-algebra of Borel sets in R, as in Subsection 3.1.2. The equivalence of this with the conditions mentioned at the beginning of the section is also part of Proposition 2.3 on p44 of [118].

3.3.1 Some additional remarks about measurability

In order to check that f is measurable, it suffices to verify any of the previous conditions with a or b in a dense set in R, as appropriate, as in Theorem 11.3 on p149 of [159]. This is related to Exercise 5.1 on p50 of [20], and Exercise 3 on p32 of [298].

Of course, if f is real-valued on X, then measurability of f may be defined as in Subsection 3.1.1 using the standard Euclidean metric on R, and the equivalence of this with the measurability conditions discussed in this section is a bit simpler. This corresponds to 2.39 on p32 of [14], and it is related to Property 1 on p28 of [335].

Measurability of extended real-valued functions is defined on p37 of [14] to mean measurability as a mapping with respect to the σ-algebra of Borel sets in the extended real numbers, using the definition of these Borel sets mentioned in the previous section. The equivalence of this with (3.3.1) corresponds to 2.52 on p37 of [14]. This is related to Proposition 5.11 on p46 of [20], and Theorem 11.4 on p149 of [159].

Suppose for the moment that X is a metric or topological space, and that A_{X} contains the Borel sets in X. The measurability of continuous real-valued functions on X corresponds to Proposition 5.6 on p44 of [20], and is part of Corollary 11.5 on p150 of [159], and of Property 2 on p29 of [335].

Measurability of compositions of measurable real or extended real-valued functions with continuous or measurable functions is mentioned in the second part of Property 2 on p29 of [335], and in Theorem 11.7 on p151 of [159]. This is related to Exercise 5.9 on p51 of [20].

3.4 Some mappings into finite products

Let X be a set, let Y_{1}, …, Y_{n} be finitely many sets, and let

Y = Y_{1} × Y_{2} × ⋯ × Y_{n} = ∏_{j=1}^{n} Y_{j} (3.4.1)

be their Cartesian product, as in Subsection 1.6.2. If f_{j} is a mapping from X into Y_{j} for each j = 1, …, n, then

f(x) = (f_{1}(x), …, f_{n}(x)) (3.4.2)

defines a mapping from X into Y, and every mapping from X into Y is of this form.

3.4.1 Continuity of mappings into Y

If Y_{1}, …, Y_{n} are metric spaces, then suitable metrics on Y may be defined as in Section A.4. In particular, if Y_{j} = R for each j, then Y = R^{n}, and one can use the standard Euclidean metric on R^{n}.

If Y_{1}, …, Y_{n} are topological spaces, then Y may be considered as a topological space with respect to the corresponding product topology. This is compatible with the metrics on Y just mentioned when the Y_{j}’s are metric spaces.

Suppose for the moment that X is a metric or topological space. It is well known and not difficult to show that f is continuous as a mapping from X into Y if and only if f_{j} is continuous as a mapping from X into Y_{j} for each j. This may be more familiar when Y_{j} = R for each j.

3.4.2 Measurability of mappings into Y

Now let A_{X} be a σ-algebra of subsets of X. If f is measurable as a mapping from X into Y, then one can check that f_{l} is measurable as a mapping from X into Y_{l} for each l. One way to do this is to observe that f_{l} is the same as the composition of f with the mapping p_{l}(y) = y_{l} from Y into Y_{l}, and that p_{l} is continuous. Alternatively, if V_{l} is an open set in Y_{l}, then f_{l}^{−1}(V_{l}) may be expressed as in (3.4.4), with V_{j} = Y_{j} when j ≠ l.

Suppose now that f_{j} is measurable as a mapping from X into Y_{j} for each j. Let V_{j} be an open set in Y_{j} for each j, and note that

V = ∏_{j=1}^{n} V_{j} (3.4.3)

is an open set in Y, as in Subsection A.4.2, or by definition of the product topology on Y if one uses that. In this case, we have that

f^{−1}(V) = ⋂_{j=1}^{n} f_{j}^{−1}(V_{j}) ∈ A_{X}. (3.4.4)

One might like to use this to get that f is measurable as a mapping from X into Y. This works when every open set in Y can be expressed as the union of finitely or countably many open sets of the form (3.4.3).

Suppose that B_{j} is a base for the topology of Y_{j} with only finitely or countably many elements for each j. Let B be the collection of open subsets of Y of the form (3.4.3), when V_{j} ∈ B_{j} for each j. It is well known and not difficult to show that B is a base for the topology of Y with only finitely or countably many elements. It follows that f is measurable as a mapping from X into Y under these conditions.

This is related to Propositions 1.5, 2.4 on p23, 44 of [118]. If Y_{j} = R for each j, then this corresponds to part of the proof of Theorem 1.8 on p11 of [298] when n = 2, and it is related to Exercise 2 on p32 of [298]. This also basically corresponds to part of the proof of Theorem 11.18 on p312 of [299] when n = 2 and Y_{1} = Y_{2} = R.

3.4.3 Measurability of complex-valued functions

In particular, if one defines measurability of complex-valued functions as in Subsection 3.1.1, then it follows that this is equivalent to the measurability of the real and imaginary parts of the function, as in parts (a) and (b) of 1.9 on p12 of [298]. Similarly, measurability of a complex-valued function on X is defined on p44 of [118] to mean measurability as a mapping with respect to the σ-algebra of Borel sets in the complex plane C, and Corollary 2.5 on p45 of [118] says that this is equivalent to the measurability of the real and imaginary parts of the function.

Measurability of complex-valued functions is defined in terms of the measurability of the real and imaginary parts of the functions on p156 of [14], on p43 of [20], on p154 of [159], on p325 of [299], and on p68 of [335]. The equivalence of this with the definition in Subsection 3.1.1 corresponds to Exercise 14 on p334 of [299].

3.5 More on measurable functions

Let (X, A_{X}) be a measurable space, and let f and g be measurable complexvalued functions on X. This leads to a measurable mapping from X into C^{2} = C × C, as in the previous section. It is well known that addition and multiplication of complex numbers correspond to continuous functions from C^{2} into C. One can use this to get that

(3.5.1) f + g and f g are measurable on X,

as in Subsection 3.1.3.

This corresponds to Proposition 2.6 on p45 of [118], and to part (c) of 1.9 on p12 of [298]. More precisely, in [298], this argument is used initially for realvalued functions, and this is used to get the analogous statement for complexvalued functions. The statement for real-valued functions is mentioned in Theorem 11.18 on p312 of [299] too, using essentially the same argument.

3.5.1 More on real-valued functions

If f and g are real-valued, then the measurability of f + g is part of part (a) of 2.46 on p35 of [14], and part of Proposition 5.7 on p44 of [20]. This is also part of part (ii) of Property 5 on p29 of [335] when f and g are defined on a Lebesgue measurable subset of R^{n}, and is mentioned on p273 of [335] for functions on abstract measure spaces. In the proofs in [14, 335], one uses the fact that

{x ∈ X : f(x) + g(x) > a} (3.5.2)

= ⋃_{r∈Q}({x ∈ X : f(x) > r} ∪ {x ∈ X : g(x) > a − r})

for every a ∈ R. The argument in [20] is essentially the same, using < instead of >.

One can check directly that f^{2} is measurable. In order to get that f g is measurable, one can use the measurability of f^{2}, g^{2}, and (f + g)^{2}. This is another part of part (a) of 2.46 on p35 of [14], and Proposition 5.7 on p44 of [20]. This is another part of part (ii) of Property 5 on p29 of [335] for functions on R^{n} as well, and it is mentioned on p273 of [335] for functions on abstract measure spaces.

One may want to consider extended real-valued functions too, and then one should be a bit careful about the points at which the sum or product is not defined. One often uses an arbitrary constant value on that set, as in Theorems 11.10 and 11.11 beginning on p152 of [159]. The product is defined if one uses the convention that 0 · (±∞) = 0, although one should be a bit more careful about the measurability of the sum in this case, as mentioned on p45 of [118].

3.5.2 The set where two measurable functions are equal

If f and g are measurable extended real-valued functions on X, then Lemma 11.9 on p152 of [159] states that

{x ∈ X : f(x) > g(x)}, (3.5.3) {x ∈ X : f(x) ≥ g(x)}, (3.5.4) {x ∈ X : f(x) = g(x)} (3.5.5)

are measurable sets in X. Indeed, one can check that (3.5.3) is the same as

⋃_{r∈Q}({x ∈ X : f(x) > r} ∩ {x ∈ X : g(x) < r}). (3.5.6)

One use this to get the measurability of (3.5.3), and to obtain the measurability of (3.5.4) and (3.5.5) from that.

If f and g are real-valued on X, then one can also get the measurability of these sets using the measurability of f − g. One can reduce to this case as well, by considering the sets where these functions are ±∞ separately. Of course, one can get the measurability of (3.5.5) when f and g are complex-valued on X using the measurability of f − g again, or by considering the real and imaginary parts of f and g.

Alternatively, if Y_{1} = Y_{2} is the set of extended real numbers, then one can look at the measurability of these sets in terms of the measurability of

x ↦ (f(x), g(x)) (3.5.7)

as a mapping from X into Y = Y_{1} × Y_{2}, as in the previous section.

3.6 Some more measurability properties

Let (X, A_{X}) be a measurable space again, and let {f_{j}}_{j=1}^{∞} be a sequence of extended real-valued measurable functions on X. It is well known that

(3.6.1)  sup_{j≥1} f_{j}(x) is measurable on X.

Indeed, if a ∈ R, then it is easy to see that

{x ∈ X : sup_{j≥1} f_{j}(x) > a} = ⋃_{j=1}^{∞}{x ∈ X : f_{j}(x) > a}. (3.6.2)

The analogous statement for the infimum may be obtained using a similar argument, or by reducing to the previous case. This corresponds to 2.53 on p37 of [14], parts of Proposition 5.8 on p45 of [20], parts of Proposition 2.7 on p45 of [118], parts of Theorem 11.12 on p153 of [159], part of Theorem 1.14 on p15 of [298], part of Theorem 11.17 on p311 of [299], and parts of Property 3 on p29 of [335].

In particular, the maximum or minimum of finitely many extended realvalued measurable functions on X is measurable on X. This is a bit simpler, and indeed the case of two such functions is mentioned in Proposition 5.7 on p45 of [20].

If f is an extended real-valued measurable function on X, then we get that

(3.6.3) f^{+}(x) = max(f(x), 0) and f^{−}(x) = − min(f(x), 0)

are measurable as well. Of course, this can also be verified more directly.

3.6.1 Measurability of pointwise limits

Let {f_{j}}_{j=1}^{∞} be a sequence of extended real-valued measurable functions on X again. Observe that

(3.6.4)  lim sup_{j→∞} f_{j}(x) and lim inf_{j→∞} f_{j}(x) are measurable on X,

using the characterizations of the upper and lower limits discussed in Subsection 1.3.4. This corresponds to part (a) of Exercise 17 on p86 of [14], parts of Proposition 5.8 on p45 of [20], parts of Proposition 2.7 on p45 of [118], parts of Theorem 11.12 on p153 of [159], part of Theorem 1.14 on p15 of [298], part of Theorem 11.17 on p311 of [299], and parts of Property 3 on p29 of [335].

In particular, if {f_{j}}_{j=1}^{∞} converges pointwise on X, then one can use (3.6.4) to get that

(3.6.5)  lim_{j→∞} f_{j}(x) is measurable on X.

This is shown directly for sequences of real-valued functions on X in 2.48 on p36 of [14]. This is also part of Proposition 2.7 on p45 of [118], Corollary 11.14 on p154 of [159], Corollary (a) on p15 of [298], Corollary (b) on p312 of [299], and Property 4 on p29 of [335].

One can use (3.6.4) to get that

(3.6.6) {x ∈ X : lim inf_{j→∞} f_{j}(x) = lim sup_{j→∞} f_{j}(x)}

is a measurable set in X, as in the previous section. This is the same as the set of x ∈ X such that f_{j}(x) tends to some extended real number as j → ∞, as in Subsections 1.3.1 and 1.3.3. The measurability of this set corresponds to Exercise 5.4 on p50 of [20], Exercise 3 on p48 of [118], Exercise 5 on p32 of [298], and Exercise 3 on p332 of [299]. If {f_{j}}_{j=1}^{∞} is a sequence of real-valued functions on X, then the measurability of the set of x ∈ X such that {f_{j}(x)}_{j=1}^{∞} converges corresponds to Exercise 14 on p39 of [14].

3.6.2 Measurability of monotone functions

If f is a real-valued function on the real line that is monotonically increasing or decreasing, then f is Borel measurable on R, i.e., it is measurable with respect to the σ-algebra of Borel sets in R. This is Proposition 5.10 on p46 of [20], and Exercise 8 on p48 of [118]. This is also stated for monotone functions on Borel sets in R in 2.43 on p34 of [14].

Exercise 17 on p39 of [14] asks one to show that if f is a real valued function on a Borel set in R that is continuous at all but finitely or countably many points in its domain, then f is Borel measurable. Exercise 22 on p39 of [14] asks one to show that if f is a monotonically increasing function on any subset of R, then f is continuous at all but finitely or countably many elements of its domain.

3.7 Simple functions

Let X be a set. A real or complex-valued function ϕ on X is said to be a simple function on X if it takes only finitely many values on X, as in Definition 2.88 on p65 of [14], Definition 5.13 on p47 of [20], on p46 of [118], Definition 11.34 on p159 of [159], Definition 1.16 on p16 of [298], and Definition 11.19 on p313 of [298]. Note that functions taking the values ±∞ are not considered here.

Suppose that a_{1}, …, a_{n} is a list of the values of ϕ on X, without repetitions, and put A_{j} = ϕ^{−1}({a_{j}}) for j = 1, …, n. Observe that

ϕ(x) = ∑_{j=1}^{n} a_{j} 1_{A_{j}}(x) (3.7.1)

on X, where 1_{A_{j}} is the indicator function of A_{j} on X, as in Section 1.4. This may be called the standard representation of ϕ, as on p79 of [14], and p46 of [118]. It can be convenient to include a term with coefficient 0, as mentioned in [118], particularly when working with additional simple functions on X.

Suppose now that (X, A_{X}) is a measurable space. It is easy to see that a simple function f on X is measurable if and only if the sets A_{j} mentioned in the preceding paragraph are elements of A_{X} for each j. Note that measurability is included in the definition of a simple function in [20, 118].

If A is any subset of X, then the corresponding indicator function 1_{A} is measurable if and only if A ∈ A_{X}. Any linear combination of indicator functions on X is a simple function, and any linear combination of indicator functions of measurable subsets of X is a measurable simple function on X. Simple functions on σ-finite measure spaces are defined on p27, 274 of [335] to be linear combinations of indicator functions of measurable sets of finite measure. The canonical form is defined on p50 of [335] to be like the standard representation, except that one does not include a term with coefficient 0.

One can check that the sum and product of two simple functions on X is a simple function, as in [118]. The measurability of the sum or product of two measurable simple functions is a bit easier to get than for arbitrary functions.

3.7.1 Approximation by simple functions

Let f be a nonnegative extended real-valued measurable function on X. It is well known and not too difficult to show that there is a monotonically increasing sequence of nonnegative measurable simple functions on X that tends to f(x) at every x ∈ X. This is Proposition 5.14 on p47 of [20], part of part (a) of Theorem 2.10 on p47 of [118], part of Theorem 11.35 on p159 of [159], Theorem 1.17 on p16 of [298], and part of Theorem 11.20 on p313 of [299]. This is also mentioned on p274 of [335] for σ-finite measure spaces, to get the additional condition on simple functions used in [335], as before.

If f is bounded on X, then this sequence of simple functions converges to f uniformly on X, as in [118, 159, 298, 299]. In fact, this sequence of simple functions converges to f uniformly on any set on which f is bounded, as in [118].

Suppose now that f is an extended real-valued or complex-valued measurable function on X. In these two cases, one can get a sequence of real or complexvalued measurable simple functions on X, as appropriate, that tends to f(x) at every x ∈ X, and whose absolute values are monotonically increasing to |f(x)|. This corresponds to 2.89 on p65 of [14], part (b) of Theorem 2.10 on p47 of [118], and another part of Theorem 11.35 on p159 of [159]. This is mentioned on p274 of [335] for σ-finite measure spaces as well, with the additional condition on simple functions as before.

This sequence converges to f uniformly on X when f is bounded on X, and more precisely it converges to f uniformly on any set on which f is bounded, as in [14, 118, 159].

3.8 Integrating nonnegative functions

Let (X, A, μ) be a measure space, as in Subsection 1.7.2. Also let ϕ be a nonnegative real-valued measurable simple function on X, as in (3.7.1), and let E be an element of A. Under these conditions, we put

∫_{E} ϕ dμ = ∑_{j=1}^{n} a_{j} μ(A_{j} ∩ E), (3.8.1)

as in (2) on p20 of [298], and (52) on p314 of [299]. This uses the convention 0 · (+∞) = 0 when a_{j} = 0 and μ(A_{j} ∩ E) = +∞. More precisely, (3.8.1) is defined as a nonnegative extended real number.

3.8.1 Defining the Lebesgue integral

Now let f be a nonnegative extended real-valued measurable function on X. The Lebesgue integral of f over E with respect to μ may be defined as a nonnegative extended real number by

∫_{E} f dμ = sup { ∫_{E} ϕ dμ : 0 ≤ ϕ ≤ f}, (3.8.2) where more precisely the supremum is taken over all nonnegative real-valued measurable simple functions ϕ on X such that ϕ ≤ f on X. This corresponds to (3) on p20 of [298], and (53) on p314 of [299]. It is sometimes convenient to express this integral as

∫_{E} f(x) dμ(x), (3.8.3)

in order to indicate the variable of integration x. Sometimes the notation

∫_{E} f(x) μ(dx) (3.8.4)

is used, as mentioned on p54 of [20], and on p49 of [118].

More precisely, (3.8.1) is denoted I_{E}(ϕ) in [299], to avoid confusion with (3.8.2) when f is a simple function. It is not difficult to check that these two definitions of the integral of a simple function over E are the same, as indicated in [298]. This corresponds to the second remark on p54 of [20], and to part (c) of Proposition 2.13 on p49 of [118] when E = X, as mentioned on p50 of [118].

If E = X, then (3.8.1) and (3.8.2) correspond to (6.1) and (6.2) on p53 of [20], and they are also defined in this way on p49, 50 of [118]. One can define integrals over E in terms of integrals over X by

∫_{E} ϕ dμ = ∫_{X} ϕ 1_{E} dμ (3.8.5)

and ∫_{E} f dμ = ∫_{X} f 1_{E} dμ, (3.8.6)

as on p54 of [20], p49 of [118], Definition 12.31 on p175 of [159], and p50, 55, 59 of [335]. This is mentioned on p21 of [298] as well.

3.8.2 Conditions when the integral is equal to 0

Note that ∫_{E} ϕ dμ = 0 (3.8.7)

automatically when μ(E) = 0. This implies that

∫_{E} f dμ = 0, (3.8.8)

as in part (4) of Proposition 6.3 on p54 of [20], 1.24 (e) on p21 of [298], and Remark 11.23 (e) on p315 of [299].

Of course, (3.8.8) also holds when f(x) = 0 for every x ∈ E, even if μ(E) = +∞, as in 1.24 (d) on p21 of [298]. In fact, (3.8.7) holds when ϕ(x) = 0 for almost every x ∈ E with respect to μ, in the sense that the measure of the set where ϕ(x) ≠ 0 is equal to 0. This implies that (3.8.8) holds when f(x) = 0 for almost every x ∈ E with respect to μ, as in the “only if” part of Exercise 3 on p84 of [14], the “if” part of Proposition 2.16 on p51 of [118], and the “if” part of Theorem 12.13 on p169 of [159].

3.8.3 Using finite partitions of X

Let P = {A_{1}, …, A_{n}} be a collection of finitely many elements of A that form a partition of X, in the sense that these sets are pairwise disjoint and their union is equal to X. The lower Lebesgue sum of f with respect to P may be defined by

L(f, P) = ∑_{j=1}^{n} μ(A_{j}) ( inf_{A_{j}} f), (3.8.9)

as in Definition 3.2 on p74 of [14]. The infimum of f over A_{j} is interpreted as being equal to 0 when A_{j} = ∅ here, as on p164 of [159].

Using this, one can define the Lebesgue integral of f over X with respect to μ by

∫_{X} f dμ = sup_{P} L(f, P), (3.8.10)

where the supremum is taken over all such partitions P of X. This corresponds to Definition 3.3 on p74 of [14], and Definition 12.2 on p164 of [159]. Note that arbitrary nonnegative extended real-valued functions f on X are considered in [159], without asking f to be measurable.

One can check that this definition of the Lebesgue integral is equivalent to the previous one, as in 3.9 on p77 of [14]. As before, one can also verify that this definition agrees with (3.8.1) for simple functions. This corresponds to 3.4, 3.7 on p75, 76 of [14], and to Theorem 12.4 on p165 of [159].

If μ(X) < +∞ and f is also bounded on X, then the Lebesgue of f over X with respect to μ may be obtained using suitable upper Lebesgue sums, as in part (a) of Exercise 4 on p99 of [14].

3.9 More on integrating nonnegative functions

Let (X, A, μ) be a measure space again, and let ϕ be a nonnegative real-valued measurable simple function on X. If c is a nonnegative real number, then it is easy to see that

∫_{X} c ϕ dμ = c ∫_{X} ϕ dμ, (3.9.1)

as in part (a) of Proposition 2.13 on p49 of [118]. If ψ is another nonnegative real-valued measurable simple function on X, then one can verify that

∫_{X}(ϕ + ψ) dμ = ∫_{X} ϕ dμ + ∫_{X} ψ dμ, (3.9.2)

as in part (b) of Proposition 2.13 on p49 of [118], Theorem 12.7 on p165 of [159], and the second part of Proposition 1.25 on p21 of [298].

It follows from (3.9.1) and (3.9.2) that if a simple function is expressed as a linear combination with nonnegative coefficients of indicator functions of measurable sets, then its integral is equal to the corresponding linear combination of the measures of the sets, without necessarily using the standard representation of the simple function. This corresponds to 3.13, 3.15 on p79, 80 of [14], and a remark on p54 of [20] and Exercise 6.2 on p55 of [20]. This also appears to be implicit in Definition 11.21 on p3.14 of [299], as indicated in the beginning of the proof of Theorem 11.29 on p3.19 of [298]. Conversely, this property can be used to get (3.9.2).

3.9.1 Integrating some bounded measurable functions

The Lebesgue integral is defined initially on R^{n} in Section 1 of Chapter 2 of [335], and it is mentioned on p275 of [335] that the same approach works on σ-finite measure spaces. Remember that simple functions are defined on p27, 274 of [335] to be linear combinations of indicator functions of measurable sets of finite measure. The integral of such a simple function is defined using its canonical form in the usual way in “Stage one” beginning on p50 of [335]. Note that the simple function is not asked to be nonnegative here, and its integral is automatically finite. The facts that the integral of any linear combination of measurable sets of finite measure is equal to the corresponding linear combination of the measures of the sets, and that the integral of simple functions is linear, are parts (i) and (ii) of Proposition 1.1 on p51 of [335].

Let E be a measurable set of finite measure, and let f be a bounded measurable function that is equal to 0 on the complement of E, as in “Stage two” beginning on p53 of [335]. Also let {ϕ_{j}}_{j=1}^{∞} be a sequence of uniformly bounded measurable simple functions that are equal to 0 on the complement of E and converge to f pointwise almost everywhere on E. Under these conditions, Lemma 1.2 on p53 of [335] says that the corresponding sequence of integrals of the ϕ_{j}’s converges, and that it converges to 0 when f = 0 almost everywhere. Note that this would be significantly simpler if one asked that {ϕ_{j}}_{j=1}^{∞} converges to f uniformly, as mentioned at the beginning of the proof in [335].

Remember that one can always get such a sequence of simple functions, as in Subsection 3.7.1. This is used to define the Lebesgue integral of f, as the limit of the sequence of Lebesgue integrals of the ϕ_{j}’s. It is easy to see that this does not depend on the particular sequence of ϕ_{j}’s, using the second part of the result mentioned in the preceding paragraph, as in [335].

Now let f be a nonnegative extended real-valued measurable function. The Lebesgue integral of f is defined in “Stage three” beginning on p58 of [335] as the supremum of the Lebesgue integrals of bounded nonnegative real-valued measurable functions that are less than or equal to f and equal to 0 on the complement of a measurable set of finite measure. One can check that this is equivalent to taking the supremum of the integrals of the nonnegative realvalued measurable simple functions that are less than or equal to f and equal to 0 on the complement of a measurable set of finite measure, by approximating bounded measurable functions by measurable simple functions as in Subsection 3.7.1.

3.9.2 Two basic classes of examples

Suppose for the moment that X is any set, and that every subset of X is considered to be measurable. Of course, this means that every function on X is measurable. Let x_{0} ∈ X be given, and let δ_{x_{0}} be the probability measure associated to x_{0} as in Subsection 1.7.3. If f is a nonnegative extended realvalued function on X, then

∫_{X} f dδ_{x_{0}} = f(x_{0}). (3.9.3)

This corresponds to parts of Exercise 2 on p84 of [14], and Exercise 6.4 on p55 of [20].

Suppose now that X is equipped with counting measure, as in Subsection 1.7.3. In this case, one can check that the integral of a nonnegative extended real-valued function f on X is the same as the sum of f(x) over x ∈ X, as in Subsection 1.8.1. This corresponds to Example 3.6 and Exercise 5 on p76, 85 of [14], Exercise 6.5 on p55 of [20], and to part of a remark in (i) on p263 of [335].

3.10 Some properties of these integrals

Let (X, A, μ) be any measure space again, and let f be a nonnegative extended real-valued measurable function on X. If c is a nonnegative extended real number, then one can check that

∫_{X} c f dμ = c ∫_{X} f dμ. (3.10.1)

This uses the convention that

0 · (+∞) = (+∞) · 0 = 0. (3.10.2)

This corresponds to part of 3.20 on p82 of [14], part (3) of Proposition 6.3 on p54 of [20], a remark on p50 of [118], Theorem 12.8 on p166 of [159], 1.24 (c) on p21 of [298], Remark 11.23 (d) on p315 of [299], and part of part (i) of Proposition 1.6 on p59 of [335].

Let g be another nonnegative extended real-valued measurable function on X. If f ≤ g on X, then it is easy to see that

∫_{X} f dμ ≤ ∫_{X} g dμ, (3.10.3)

as in 3.8 on p77 of [14], part (2) of Proposition 6.3 on p54 of [20], a remark on p50 of [118], Theorem 12.5 on p165 of [159], 1.24 (a) on p21 of [298], Remark 11.23 (c) on p315 of [299], and part (iii) of Proposition 1.6 on p59 of [335]. Of course, these verifications depend on how the integral is defined, as in the previous two sections.

3.10.1 When the integral is 0

If ∫_{X} f dμ = 0, (3.10.4)

then one can check that f(x) = 0 for almost every x ∈ X with respect to μ. This corresponds to the “if” part of Exercise 3 on p84 of [14], Proposition 8.1 on p71 of [20], the “only if” part of Proposition 2.16 on p51 of [118], Theorem 12.6 and the “only if” part of Theorem 12.13 on p165, 169 of [159], part (a) of Theorem 1.39 on p31 of [298], Exercise 1 on p332 of [299], and part (vi) of Proposition 1.6 on p60 of [335].

3.10.2 When the integral is finite

If ∫_{X} f dμ < +∞, (3.10.5)

then it is easy to see that f(x) < +∞ for almost every x ∈ X with respect to μ, as in Exercise 6.1 on p55 of [20], part of Proposition 2.20 on p52 of [118], part of Theorem 12.15 on p169 of [159], and part (v) of Proposition 1.6 on p60 of [335].

We also have that the set of x ∈ X such that f(x) > 0 is σ-finite with respect to μ, as in Exercise 11 on p100 of [14], Exercise 8.7 on p75 of [20], Proposition 2.20 on p52 of [118], and Exercise 19 on p61 of [298]. More precisely, one can check that for each positive integer n, the set of x ∈ X such that f(x) ≥ 1/n has finite measure.

3.10.3 The monotone convergence theorem

Let {f_{j}}_{j=1}^{∞} be a monotonically increasing sequence of nonnegative extended real-valued measurable functions on X. This means that

f_{j}(x) ≤ f_{j+1}(x) (3.10.6)

for each j and x ∈ X. Put

f(x) = sup_{l≥1} f_{l}(x) (3.10.7)

for every x ∈ X, so that f_{j}(x) → f(x) as j → ∞. Under these conditions, the monotone convergence theorem states that

∫_{X} f_{j} dμ → ∫_{X} f dμ as j → ∞. (3.10.8)

This corresponds to 3.11 on p78 of [14], Theorem 7.1 on p57 of [20], Theorem 2.14 on p50 of [118], part of B. Levi’s theorem 12.22 on p172 of [159], Lebesgue’s monotone convergence theorem 1.26 on p22 of [298] and 11.28 on p318 of [299], and Corollary 1.9 on p62 and (ii) on p275 of [335].

Note that Theorems 12.10 and 12.11 beginning on p167 of [159] are related results for sequences of simple functions. A version for Riemann integrals is discussed in [352].

Remember that the monotone convergence theorem for sums was discussed in Subsection 1.8.5. The case where the functions are indicator functions of measurable subsets of X corresponds to the remarks at the beginning of Section 1.9.

The initial steps in the proof of the monotone convergence theorem are essentially the same as in Subsection 1.8.5. Using (3.10.6), we get that

∫_{X} f_{j}dμ ≤ ∫_{X} f_{j+1} dμ (3.10.9)

for each j, so that

∫_{X} f_{j} dμ → sup_{l≥1} ∫_{X} f_{l} dμ as j → ∞. (3.10.10)

Thus it suffices to show that

sup_{l≥1} ∫_{X} f_{l} dμ = ∫_{X} f dμ. (3.10.11)

We also clearly have that

sup_{l≥1} ∫_{X} f_{l} dμ ≤ ∫_{X} f dμ. (3.10.12)

In order to complete the proof, one should show that

∫_{X} f dμ ≤ sup_{l≥1} ∫_{X} f_{l} dμ. (3.10.13)

3.11 Using the monotone convergence theorem

Let (X, A, μ) be a measure space. If f and g are any nonnegative extended real-valued measurable functions on X, then it is well known that

∫_{X}(f + g) dμ = ∫_{X} f dμ + ∫_{X} g dμ, (3.11.1)

as in 3.16 on p80 of [14], part of Theorem 7.4 on p58 of [20], part of Theorem 2.15 on p51 of [118], Theorem 12.12 on p169 of [159], the first part of the proof of Theorem 1.27 on p23 of [298], part of Theorem 11.29 on p319 of [299], and part (i) of Proposition 1.6 on p59 of [335].

This is often obtained using the monotone convergence theorem to reduce to the analogous statement (3.9.2) for simple functions. More precisely, there are monotonically increasing sequences {ϕ_{j}}_{j=1}^{∞} and {ψ_{j}}_{j=1}^{∞} of nonnegative measurable simple functions on X that tend to f and g, respectively, pointwise on X, as in Subsection 3.7.1. We also have that

∫_{X}(ϕ_{j} + ψ_{j}) dμ = ∫_{X} ϕ_{j} dμ + ∫_{X} ψ_{j} dμ (3.11.2)

for each j, as in (3.9.2). Note that {ϕ_{j} +ψ_{j}}_{j=1}^{∞} increases monotonically to f +g pointwise on X. Thus one can use the monotone convergence theorem to get (3.11.1) from (3.11.2).

3.11.1 Some integrals of sums

If {f_{j}}_{j=1}^{∞} is any sequence of nonnegative extended real-valued measurable functions on X, then

∫_{X} ( ∑_{j=1}^{∞} f_{j}) dμ = ∑_{j=1}^{∞} ∫_{X} f_{j} dμ, (3.11.3)

as in Exercise 10 on p85 of [14], Proposition 7.6 on p60 of [20], Theorem 2.15 on p51 of [118], Lebesgue’s theorem 12.21 on p171 of [159], Theorem 1.27 on p23 of [298], Theorem 11.30 on p3.20 of [299], and Corollary 1.10 on p62 of [335].

Indeed,

∫_{X} ( ∑_{j=1}^{n} f_{j}) dμ = ∑_{j=1}^{n} ∫_{X} f_{j} dμ (3.11.4)

for every positive integer n, because of (3.11.1). One can use this and the monotone convergence theorem to get (3.11.3).

3.11.2 Another result about monotone convergence

Let {f_{j}}_{j=1}^{∞} be a monotonically decreasing sequence of nonnegative extended real-valued measurable functions on X, so that

f_{j+1}(x) ≤ f_{j}(x) (3.11.5)

for each j and x ∈ X. Put

f(x) = inf_{j≥1} f_{j}(x) (3.11.6)

for every x ∈ X, so that f_{j}(x) → f(x) as j → ∞. If

∫_{X} f_{1} dμ < +∞, (3.11.7)

then it is well known that

lim_{j→∞} ∫_{X} f_{j} dμ = ∫_{X} f dμ. (3.11.8)

This corresponds to Exercise 7.7 on p64 of [20], Exercise 15 on p52 of [118], and Exercise 7 on p33 of [298]. This is stated more broadly for extended real-valued functions on X in Exercise 20 on p87 of [14].

Note that ∫_{X}(f_{1} − f_{j}) dμ + ∫_{X} f_{j} dμ = ∫_{X} f_{1} dμ (3.11.9)

for each j, and similarly

∫_{X}(f_{1} − f) dμ + ∫_{X} f dμ = ∫_{X} f_{1} dμ, (3.11.10)

as in (3.11.1). We also have that

∫_{X}(f_{1} − f_{j}) dμ → ∫_{X}(f_{1} − f) dμ as j → ∞, (3.11.11)

by the monotone convergence theorem. One can get (3.11.8) from these statements when (3.11.7) holds.

3.11.3 Fatou’s lemma

If {f_{j}}_{j=1}^{∞} is any sequence of nonnegative extended real-valued measurable functions on X again, then Fatou’s lemma states that

(3.11.12) ∫_{X} ( lim inf_{j→∞} f_{j}) dμ ≤ lim inf_{j→∞} ∫_{X} f_{j} dμ,

as in part (b) of Exercise 17 on p86 of [14], Theorem 7.8 on p62 of [20], 2.18 on p52 of [118], 12.23 on p172 of [159], 1.28 on p24 of [298], 11.31 on p320 of [299], and (i) on p275 of [335].

This can be obtained from the monotone convergence theorem, by considering the sequence of functions

inf_{j≥l} f_{j}(x), (3.11.13)

that increase monotonically to lim inf_{j→∞} f_{j}(x), as in Subsection 1.3.4. This implies that

(3.11.14) ∫_{X} ( inf_{j≥l} f_{j}) dμ → ∫_{X} ( lim inf_{j→∞} f_{j}) dμ as l → ∞.

If r is a positive integer with r ≥ l, then (3.11.13) is less than or equal to f_{r}, so that

∫_{X} ( inf_{j≥l} f_{j}) dμ ≤ ∫_{X} f_{r} dμ. (3.11.15)

This means that

∫_{X} ( inf_{j≥l} f_{j}) dμ ≤ inf_{r≥l} ∫_{X} f_{r} dμ (3.11.16)

for each l. The right side of (3.11.16) tends to the right side of (3.11.12) as l → ∞, as in Subsection 1.3.4 again.

In the formulation of Fatou’s lemma on p61 of [335], {f_{j}(x)}_{j=1}^{∞} is asked to converge for almost every x ∈ X with respect to μ. This is often the case in practice, and one can reduce to this case by considering the sequence of functions (3.11.13), as in the previous paragraph. In fact, the monotone convergence theorem is obtained from this version of Fatou’s lemma on p62 of [335]. This is related to a remark at the end of Exercise 17 on p86 of [14], and to Exercise 17 on p52 of [118], where one is asked to obtain the monotone convergence theorem from Fatou’s lemma.

3.12 Real and complex-valued functions

Let (X, A, μ) be a measure space, and let f be a measurable extended real-valued function on X. Remember that f^{+} and f^{−} are defined on X as in (3.6.3), and are nonnegative extended real-valued measurable functions on X. Note that

f(x) = f^{+}(x) − f^{−}(x) (3.12.1)

and |f(x)| = f^{+}(x) + f^{−}(x) (3.12.2)

for every x ∈ X. The integrals of f^{+} and f^{−} over X with respect to μ may be defined as nonnegative extended real numbers as before. If either of these integrals is finite, then the Lebesgue integral of f over X with respect to μ is defined as an extended real number by

∫_{X} f dμ = ∫_{X} f^{+} dμ − ∫_{X} f^{−} dμ, (3.12.3)

as in Definition 3.18 on p81 of [14], (6.3) on p53 of [20], on p53 of [118], Definition 12.2 on p164 of [159], (2) on p26 of [298], (56) on p315 of [299], and the beginning of “Stage four” on p64 of [335].

If the integrals of f^{+} and f^{−} over X with respect to μ are both finite, then f is said to be integrable on X with respect to μ. This is the same as saying that ∫_{X} |f| dμ < +∞, (3.12.4)

because of (3.12.2). This implies that (3.12.3) is finite, and more precisely we have that |∫_{X} f dμ| ≤ ∫_{X} |f| dμ. (3.12.5)

3.12.1 Some remarks about complex numbers

If a is a complex number, then let Re a, Im a ∈ R be the real and imaginary parts of a, respectively, so that a = Re a + i Im a. The complex conjugate of a is the complex number \overline{a} = Re a − i Im a. (3.12.6)

If b is another complex number, then it is easy to see that

\overline{(a + b)} = \overline{a} + \overline{b} (3.12.7) and \overline{(a b)} = \overline{a} \overline{b}. (3.12.8)

The absolute value or modulus of a is the nonnegative real number defined by |a| = ((Re a)^{2} + (Im a)^{2})^{1/2}. (3.12.9)

This is the same as the usual absolute value of a real number when a ∈ R. This is also the same as the standard Euclidean norm of (Re a, Im a), as an element of R^{2}, for any a ∈ C. Note that

| Re a|, | Im a| ≤ |a|. (3.12.10)

One can check that a \overline{a} = |a|^{2}. (3.12.11)

One can use this and (3.12.8) to get that

|a b| = |a| |b|. (3.12.12)

If a ≠ 0, then 1/a = \overline{a}/|a|^{2}. (3.12.13)

It is well known that |a + b| ≤ |a| + |b|. (3.12.14)

This is equivalent to the triangle inequality for the standard Euclidean norm on R^{2}. This can be verified using (3.12.11) as well.

3.12.2 Integrating complex-valued functions

Now let f be a measurable complex-valued function on X. It is not too difficult to see that |f(x)| is measurable as a nonnegative real-valued function on X, although the details of this may depend on how measurability of complex-valued functions is defined. If (3.12.4) holds, then f is said to be integrable on X with respect to μ. This is equivalent to the integrability of the real and imaginary parts of f, as real-valued functions on X.

In this case, the Lebesgue integral of f over X with respect to μ is defined as a complex number by

∫_{X} f dμ = ∫_{X}(Re f) dμ + i ∫_{X}(Im f) dμ, (3.12.15)

as in Definition 6.21 on p156 of [14], (6.4) on p54 of [20], on p53 of [118], Definition 12.26 on p173 of [159], Definition 1.31 on p25 of [298], on p325 of [299], and on p68 of [335].

If g is another complex-valued function on X that is integrable with respect to μ, then f + g is integrable on X with respect to μ, and

∫_{X}(f + g) dμ = ∫_{X} f dμ + ∫_{X} g dμ. (3.12.16)

Similarly, if c is a complex number, then one can check c f is integrable on X with respect to μ as well, and that

∫_{X} c f dμ = c ∫_{X} f dμ. (3.12.17)

This corresponds to some remarks on p157 of [14], Theorem 7.4 on p58 and part (3) of Proposition 7.5 on p60 of [20], some remarks on p53 of [118], Theorem 12.27 on p174 of [159], Theorem 1.32 on p26 of [299], some remarks on p325 of [299], and part of Proposition 1.11 on p65 of [335].

It is well known that (3.12.5) also holds for complex-valued integrable functions, as in 6.22 on p157 of [14], Proposition 7.7 on p61 of [20], Proposition 2.22 on p53 of [118], Theorem 12.28 on p174 of [159], Theorem 1.33 on p27 of [298], and some remarks on p325 of [299]. More precisely, it is easy to see that

(3.12.18) |∫_{X} f dμ| ≤ |∫_{X}(Re f) dμ| + |∫_{X}(Im f) dμ|
≤ ∫_{X} | Re f| dμ + ∫_{X} | Im f| dμ ≤ 2 ∫_{X} |f| dμ.

If f is a simple function, then one can verify (3.12.5) directly. Otherwise, one can approximate f by simple functions, using (3.12.18). Alternatively, if a is any complex number with |a| = 1, then

Re (a ∫_{X} f dμ) = ∫_{X} Re(a f) dμ ≤ ∫_{X} |f| dμ, (3.12.19)

and one can choose a so that the left side is the same as the left side of (3.12.5).

3.12.3 More on integrals over measurable subsets of X

If f is integrable on X with respect to μ and E ∈ A, then the integral of f over E with respect to μ can be defined by reducing to the corresponding definition for nonnegative real-valued functions in Section 3.8, or equivalently as the integral of f 1_{E} over X with respect to μ. If

∫_{E} f dμ = 0 (3.12.20)

for every E ∈ A, then it is well known that f(x) = 0 for almost every x ∈ X with respect to μ. This corresponds to Proposition 8.2 on p71 of [20], part (b) of Proposition 2.23 on p54 of [118], part (b) of Theorem 1.39 on p31 of [299], and Exercise 2 on p332 of [299]. It is easy to reduce to the case where f is real-valued on X. In this case, one can check that

∫_{X} f^{+} dμ = ∫_{X} f^{−} dμ = 0, (3.12.21)

and use the analogous statement for nonnegative functions in Subsection 3.10.1.

Suppose now that f is real-valued on X, and that

∫_{E} f dμ ≥ 0 (3.12.22)

for every E ∈ A. This implies that the integral of f^{−} over X with respect to μ is equal to 0, so that f^{−}(x) = 0 for almost every x ∈ X with respect to μ, and thus f(x) ≥ 0 for almost every x ∈ X with respect to μ.

3.13 The dominated convergence theorem

Let (X, A, μ) be a measure space, and let {f_{j}}_{j=1}^{∞} be a sequence of complexvalued measurable functions on X that converges pointwise to a complex-valued function f on X. Suppose that g is a nonnegative real-valued measurable function on X that is integrable with respect to μ such that

|f_{j}(x)| ≤ g(x) (3.13.1)

for each j and x ∈ X. Note that this implies that

|f(x)| ≤ g(x) (3.13.2)

for every x ∈ X, and that

(3.13.3) the f_{j}’s and f are integrable with respect to μ on X

as well. Under these conditions, Lebesgue’s dominated convergence theorem states that

lim_{j→∞} ∫_{X} |f_{j} − f| dμ = 0, (3.13.4)

and in particular

lim_{j→∞} ∫_{X} f_{j} dμ = ∫_{X} f dμ. (3.13.5)

This corresponds to 3.31 on p92 and some remarks on p157 of [14], Theorem 7.9 and Exercise 7.1 on p62, 63 of [20], 2.24 on p54 of [118], 12.24 and 12.30 on p172, 174 of [159], 1.34 on p27 of [298], 11.32 on p321 and some remarks on p325 of [299], and Theorem 1.13 on p67 and (iii) on p275 of [335].

More precisely,

|∫_{X} f_{j} dμ − ∫_{X} f dμ| = |∫_{X}(f_{j} − f) dμ| ≤ ∫_{X} |f_{j} − f| dμ (3.13.6)

for each j, so that (3.13.4) implies (3.13.5). Sometimes the conclusion of the dominated convergence theorem is only stated as (3.13.5). However, (3.13.4) may be obtained using that version for the sequence of functions |f_{j} −f|, because

|f_{j}(x) − f(x)| ≤ |f_{j}(x)| + |f(x)| ≤ 2 g(x) (3.13.7) for each j and x ∈ X. This is related to Exercise 7.2 on p63 of [20].

If the f_{j}’s and thus f are real-valued, then (3.13.5) may be obtained from Fatou’s lemma, by considering the sequences of functions

{g + f_{j}}_{j=1}^{∞} and {g − f_{j}}_{j=1}^{∞}, (3.13.8)

as in [20, 118, 159, 299]. Of course, one can reduce to this case, by considering the real and imaginary parts of the f_{j}’s and f. Similarly, (3.13.4) may be obtained using Fatou’s lemma, by considering the sequence of functions

{2 g − |f_{j} − f|}_{j=1}^{∞}, (3.13.9)

as in [298]. Another type of approach is used in [14, 335].

3.13.1 Convergence almost everywhere

Sometimes one only asks that {f_{j}}_{j=1}^{∞} converges to f pointwise almost everywhere with respect to μ on X, and that (3.13.1) hold almost everywhere on X with respect to μ for each j. In this case, one should be a bit careful about the measurability of f, and this can be easily arranged by adjusting f on a set of measure 0, if necessary. This is related to some remarks on p90 of [14], p63 of [20], p54 of [118], p170 of [159], beginning on p28 of [298], on p317 of [299], and p30 of [335]. This is a bit simpler when (X, A, μ) is complete as a measure space, as in Subsection 2.1.3.

3.13.2 The bounded convergence theorem

If μ(X) < +∞, then constant functions are integrable on X, so that one can take g to be a constant function in the dominated convergence theorem. This is known as the bounded convergence theorem, as in 3.26 on p89 of [14], Exercise 7.8 on p64 of [20], the corollary on p322 of [299], and Theorem 1.4 on p56 of [335].

See [99, 138, 193, 227, 237, 353] concerning related results for Riemann integrals.

3.13.3 Some more integrals of sums

Suppose now that {f_{j}}_{j=1}^{∞} is a sequence of complex-valued measurable functions on X that are integrable with respect to μ such that

∫_{X} ( ∑_{j=1}^{∞} |f_{j}|) dμ = ∑_{j=1}^{∞} ∫_{X} |f_{j}| dμ < +∞, (3.13.10)

where the first step is as in Subsection 3.11.1. In particular, this implies that

∑_{j=1}^{∞} |f_{j}(x)| < +∞ (3.13.11)

for almost every x ∈ X with respect to μ, so that ∑_{j=1}^{∞} f_{j}(x) converges absolutely for almost every x ∈ X with respect to μ. Let us simply interpret the sum as being equal to 0 at points where it does not converge. We also have that

∫_{X} |∑_{j=1}^{∞} f_{j}| dμ ≤ ∫_{X} ( ∑_{j=1}^{∞} |f_{j}|) dμ < +∞, (3.13.12)

so that ∑_{j=1}^{∞} f_{j} is integrable on X with respect to μ.

Note that

|∑_{j=1}^{n} f_{j}| ≤ ∑_{j=1}^{n} |f_{j}| ≤ ∑_{j=1}^{∞} |f_{j}| (3.13.13)

for each positive integer n. One can use the dominated convergence theorem to get that

∑_{j=1}^{∞} ∫_{X} f_{j} dμ = ∫_{X} ( ∑_{j=1}^{∞} f_{j}) dμ (3.13.14)

under these conditions, as in Exercise 7.4 on p64 of [20], Theorem 2.25 on p55 of [118], Corollary 12.33 on p175 of [159], and Theorem 1.38 on p30 of [298].

Alternatively,

|∑_{j=1}^{∞} f_{j} − ∑_{j=1}^{n} f_{j}| = | ∑_{j=n+1}^{∞} f_{j}| ≤ ∑_{j=n+1}^{∞} |f_{j}| (3.13.15)

for each positive integer n, so that

∫_{X} |∑_{j=1}^{∞} f_{j} − ∑_{j=1}^{n} f_{j}| dμ ≤ ∫_{X} ( ∑_{j=n+1}^{∞} |f_{j}|) dμ = ∑_{j=n+1}^{∞} ∫_{X} |f_{j}| dμ, (3.13.16)

where the second step is as in Subsection 3.11.1 again. The sum on the right tends to 0 as n → ∞, because of the convergence of the corresponding infinite series, by hypothesis. This implies that the integral on the left tends to 0 as n → ∞, and one can use this to get (3.13.14).

3.14 Connection with Riemann integrals

Let a and b be real numbers, with a < b, and let f be a bounded real-valued function on [a, b]. It is well known that f is Riemann integrable on [a, b] if and only if the set of points in [a, b] at which f is not continuous has Lebesgue measure 0, as in the first part of 3.34 on p93 of [14], the first part of Theorem 9.1 on p78 of [20], part b of Theorem 2.28 on p57 of [118], Theorem 7.3 A on p163 of [135], part (e) of Exercise 12.51 on p184 of [159], part (b) of Theorem 11.33 on p323 of [299], and Theorem 1.7 on p287 of [334].

If f is Riemann integrable on [a, b], then it is well known that f is Lebesgue integrable on [a, b], with Lebesgue integral equal to the Riemann integral, as in the second part of 3.34 on p93 of [14], the second part of Theorem 9.1 on p78 of [20], part a of Theorem 2.28 on p57 of [118], Theorem 11.5 G on p292 of [135], part (f) of Exercise 12.51 of [159], on p54 of [298], part (a) of Theorem 11.33 on p323 of [299], and Theorem 1.5 on p57 of [335].

3.14.1 Lebesgue integrals on R

Suppose now that f is an extended real-valued function on the real line that is Lebesgue measurable, meaning that it is measurable with respect to the σalgebra of Lebesgue measurable subsets of R. The Lebesgue integral of f over [a, b] with respect to Lebesgue measure may be denoted

∫_{a}^{b} f(x) dx, (3.14.1)

when it exists, for convenience. This is the same as the integrals over (a, b), [a, b), and (a, b], because finite subsets of R have Lebesgue measure 0. Similarly, we may use this notation when f is a complex-valued Lebesgue measurable function that is integrable on [a, b]. We may also denote integrals over half-lines in R, or over R, as in (3.14.1), with a = −∞ or b = +∞, as appropriate, under suitable conditions.

3.14.2 Invariance under translations

If t is any real number, then (3.14.1) is equal to

∫_{a+t}^{b+t} f(x − t) dx, (3.14.2)

under the conditions mentioned in the preceding paragraph. This uses the fact that Lebesgue measure on the real line is invariant under translations, as in 2.7 on p16 and Exercise 8 on p60 of [14], Theorem 1.21 of [118], part (c) of Theorem 2.20 on p52 of [298], and on p22, 73 of [335].

3.15 Some measures from integrals

Let (X, A, μ) be a measure space, and let f be a nonnegative extended realvalued measurable function on X. Under these conditions, one can check that

ν(E) = ∫_{E} fdμ (3.15.1)

defines another countably additive measure on X with respect to A, as in Subsection 1.7.1. This corresponds to Exercise 9 on p85 of [14], part (1) of Exercise 7.27 on p67 of [20], part of Exercise 14 on p52 of [118], the first part of Theorem 1.29 on p24 of [298], and part (a) of Theorem 11.24 on p315 of [299]. More precisely, countable additivity can be obtained from the monotone convergence theorem, as in Subsection 3.11.1, although a more direct argument is used in [299]. This is related to 9.4 on p258 of [14], Example 19.9 on p307 of [159], and some remarks on p286 of [335].

3.15.1 Integrals with respect to ν

If g is a nonnegative extended real-valued measurable function on X, then

∫_{X} g dν = ∫_{X} f g dμ, (3.15.2)

as in Exercise 5 on p266 of [14], Exercise 13.8 on p117 of [20], Exercise 14 on p52 of [118], and the second part of Theorem 1.29 on p24 of [298]. This follows from the definition of ν when g is the indicator function of a measurable set in X, and one can use this to get (3.15.2) when g is a nonnegative measurable simple function on X. One can obtain (3.15.2) for any nonnegative measurable function on X by approximating by simple functions, as in Subsection 3.7.1, and using the monotone convergence theorem.

If g is a real or complex-valued measurable function on X, then it follows that ∫_{X} |g| dν = ∫_{X} f |g| dμ. (3.15.3)

This means that g is integrable with respect to ν if and only if f g is integrable with respect to μ. In this case, one can check that (3.15.2) holds, as in part (2) of Exercise 7.27 on p67 of [20].

If A is the collection of all subsets of X and μ is counting measure on X, then (3.15.1) corresponds to a measure of the type mentioned in Subsection 1.8.7, and (3.15.2) corresponds to Exercise 7 on p85 of [14].

3.15.2 Some additional properties of ν

Note that

{x ∈ X : f(x) < +∞} = ⋃_{n=1}^{∞}{x ∈ X : f(x) ≤ n}. (3.15.4)

If f is finite almost everywhere on X with respect to μ, and if μ is σ-finite on X, then one can check that (3.15.1) is σ-finite on X. More precisely, it is enough to ask that {x ∈ X : f(x) > 0} (3.15.5)

be σ-finite with respect to μ, instead of the σ-finiteness of μ on X.

Of course,

ν(X) = ∫_{X} f dμ (3.15.6)

is finite exactly when f is integrable with respect to μ on X. In this case, f is finite almost everywhere on X with respect to μ, and (3.15.5) is σ-finite with respect to μ, as in Subsection 3.10.2.

Similarly, if E ∈ A and (3.15.1) is finite, then f is finite almost everywhere on E with respect to μ, and the set of x ∈ E such that f(x) > 0 is σ-finite with respect to μ. If X is σ-finite with respect to ν, then it follows that f is finite almost everywhere on X with respect to μ, and that (3.15.5) is σ-finite with respect to μ.

Suppose that f is integrable on X with respect to μ. It is well known that for every ϵ > 0 there is a δ > 0 such that if E ∈ A and

μ(E) < δ, (3.15.7)

then ν(E) < ϵ. (3.15.8)

This corresponds to 3.28 on p90 of [14], Exercise 6.8 on p56 of [20], Theorem 12.34 on p176 of [159], Exercise 12 on p33 of [298], and part (ii) of Proposition 1.12 of [335].

More on functions and measures

4.1 Absolutely continuous measures

Let (X, A, μ) be a measure space, and let λ be another countably additive measure on X with respect to A. We say that λ is absolutely continuous with respect to μ if (4.1.1) μ(E) = 0 implies that λ(E) = 0

for every E ∈ A. This may be expressed by

λ ≪ μ, (4.1.2)

as in Definition 9.32 on p270 of [14], Definition 13.1 on p112 of [20], on p88 of [118], Definition 6.7 on p128 of [299], and (12) on p288 of [335]. Definition 19.19 on p312 of [159] considers absolute continuity with respect to a signed or complex measure, and reduces to (4.1.1) when μ is nonnegative.

Note that (3.15.1) is absolutely continuous with respect to μ, by construction.

4.1.1 The Radon–Nikodym theorem

Suppose that λ(X) < +∞, and that μ is σ-finite on X. If λ is absolutely continuous with respect to μ, then the Radon–Nikodym theorem states that λ is of the form (3.15.1), where f is also integrable on X with respect to μ. This corresponds to 9.36 on p272 (≈ 100e) of [14], Theorem 13.4 and Exercise 19.8 on p113, 215 of [20], part of 3.8 on p90 of [118], Theorems 19.23 and 19.24 on p315 of [159], part (b) of 6.9 on p129 of [298], and part of Theorem 4.3 on p290 of [335].

This also works when λ is σ-finite on X, except that f may not be integrable on X with respect to μ, as in [118, 335], Theorem 19.24 on p315 of [159], and some remarks in 6.10 on p132 of [298]. However, f is finite almost everywhere with respect to μ in this case, and the set where f is positive is σ-finite with respect to μ, as in Subsection 3.15.2.

Note that f is essentially unique, in the sense that any other such function is equal to f almost everywhere on X with respect to μ, as in Subsection 3.12.3. This is called the Radon–Nikodym derivative of λ with respect to μ, and may be denoted

\frac{dλ}{dμ}. (4.1.3)

4.1.2 More on absolute continuity

Suppose that for every ϵ > 0 there is a δ > 0 such that

(4.1.4) μ(E) < δ implies that λ(E) < ϵ

for every E ∈ A. It is easy to see that (4.1.1) holds, so that (4.1.2) holds.

Conversely, if λ(X) < +∞ and (4.1.2) holds, then it is well known that the property described in the preceding paragraph holds, as in Exercise 14 on p279 of [14], Proposition 13.2 on p112 of [20], Theorem 3.5 on p89 of [118], Exercise 19.67 on p339 of [159], Theorem 6.11 on p132 of [298], and Proposition 4.2 on p289 of [335].

4.2 Mutually singular measures

Let (X, A) be a measurable space, and let λ and μ be countably additive measures on X with respect to A. We say that λ and μ are mutually singular on X if there is an A ∈ A such that

λ(X \ A) = μ(A) = 0, (4.2.1)

as in Definition 9.28 on p268 of [14], on p106 of [20], on p87 of [118], Definition 19.39 on p326 of [159], part of Definitions 6.7 on p128 of [298], and on p288 of [335]. We may also say that each of λ and μ is singular with respect to the other in this case, and this may be expressed by

λ ⊥ μ. (4.2.2)

It is easy to see that λ = 0 when (4.1.2) and (4.2.2) hold.

4.2.1 The Lebesgue decomposition theorem

If μ is σ-finite on X and λ(X) < +∞, then the Lebesgue decomposition theorem states that are are unique measures λ_{a} and λ_{s} on X with respect to A such that

λ = λ_{a} + λ_{s}, λ_{a} ≪ μ, and λ_{s} ⊥ μ, (4.2.3) as in 9.35 on p271 of [14], Theorem 13.5 and Exercise 13.13 on p115, 118 of [20], the first part of 3.8 on p90 of [118], 19.42 on p326 of [159], part (a) of Theorem 6.9 and part of 6.10 on p129, 130 of [298], and the first part of Theorem 4.3 on p290 of [335].

More precisely, this works without asking that μ be σ-finite, as in [14]. This works when λ is σ-finite on X too, as in [118, 159, 335], and some remarks in 6.10 on p132 of [298].

4.2.2 More on the Lebesgue decomposition

The fact that λ_{s} is singular with respect to μ means that there is an A_{s} ∈ A such that λ_{s}(X \ A_{s}) = μ(A_{s}) = 0, (4.2.4)

as before. It follows that λ_{a}(A_{s}) = 0, (4.2.5)

because λ_{a} is absolutely continuous with respect to μ. In particular, λ_{a} and λ_{s} are mutually singular on X.

If E ∈ A, then

λ(E \ A_{s}) = λ_{a}(E \ A_{s}) + λ_{s}(E \ A_{s}) = λ_{a}(E \ A_{s}) = λ_{a}(E). (4.2.6)

Similarly,

λ(E ∩ A_{s}) = λ_{a}(E ∩ A_{s}) + λ_{s}(E ∩ A_{s}) = λ_{s}(E ∩ A_{s}) = λ_{s}(E). (4.2.7)

If A_{s} is any element of A, then it is easy to see that (4.2.6) and (4.2.7) define measures on X with respect to A. In order to get a Lebesgue decomposition, one needs to have that μ(A_{s}) = 0, and that λ_{a} be absolutely continuous with respect to μ on X. The second part means that

(4.2.8) λ(E) = 0 for every E ∈ A with E ⊆ X \ A_{s} and μ(E) = 0.

4.3 Differentiating monotone functions

Let α be a monotonically increasing real-valued function on the real line. A version of Lebesgue’s differentiation theorem states that α is differentiable at almost every point in R with respect to Lebesgue measure. This corresponds to Theorem 14.8 on p131 of [20], part (b) of Theorem 3.23 on p101 of [118], Theorem 17.12 on p264 of [159], Theorem 8.19 on p177 of [298], and Theorem 3.4 on p121 of [335].

4.3.1 A simpler version of α′

Remember that α is Borel measurable on R, as in Subsection 3.6.2. Of course,

j (α(x + (1/j)) − α(x)) ≥ 0 (4.3.1) for each positive integer j and x ∈ R, because α is monotonically increasing on R, by hypothesis. It follows that

lim inf_{j→∞} j (α(x + (1/j)) − α(x)) (4.3.2)

is a Borel measurable nonnegative extended real-valued function on R, as in Subsection 3.6.1. This function is equal to the derivative α′(x) of α at x when α is differentiable at x.

4.3.2 Integrating α′ on an interval

If a, b ∈ R and a < b, then

∫_{a}^{b} α′(x) dx ≤ lim inf_{j→∞} ∫_{a}^{b} j (α(x + (1/j)) − α(x)) dx, (4.3.3)

because of Fatou’s lemma, as in Subsection 3.11.3. This uses the notation for Lebesgue integrals over intervals in the real line with respect to Lebesgue measure mentioned in Subsection 3.14.1, although the integral on the right may be considered as a Riemann integral, as before.

Observe that

∫_{a}^{b} j (α(x + (1/j)) − α(x)) dx = j ∫_{a}^{b} α(x + (1/j)) dx − j ∫_{a}^{b} α(x) dx
(4.3.4)  = j ∫_{a+(1/j)}^{b+(1/j)} α(x) dx − j ∫_{a}^{b} α(x) dx

= j ∫_{b}^{b+(1/j)} α(x) dx − j ∫_{a}^{a+(1/j)} α(x) dx

for each j. Remember that the one-sided limits of α exist at every point in R, as in Subsection 2.4.2. One can check that

α(p+) = lim_{j→∞} j ∫_{p}^{p+(1/j)} α(x) dx (4.3.5)

for every p ∈ R. More precisely, this is equivalent to saying that

lim_{j→∞} j ∫_{p}^{p+(1/j)}(α(x) − α(p+)) dx = 0, (4.3.6)

and this can be obtained from the definition of α(p+), as in Subsection 2.4.1. It follows that

lim_{j→∞} ∫_{a}^{b} j (α(x + (1/j)) − α(x)) dx = α(b+) − α(a+). (4.3.7)

Combining this with (4.3.3), we obtain that

∫_{a}^{b} α′(x) dx ≤ α(b+) − α(a+). (4.3.8)

Similarly, if l is a positive integer such that a < b − (1/l), then

∫_{a}^{b−(1/l)} α′(x) dx ≤ α((b − (1/l))+) − α(a+) ≤ α(b−) − α(a+), (4.3.9)

using a property of one-sided limits of monotonically increasing functions mentioned in Subsection 2.4.2. This implies that

∫_{a}^{b} α′(x) dx = lim_{l→∞} ∫_{a}^{b−(1/l)} α′(x) dx ≤ α(b−) − α(a+), (4.3.10)

using the monotone convergence theorem in the first step. In particular,

∫_{a}^{b} α′(x) dx ≤ α(b) − α(a), (4.3.11)

using properties of one-sided limits of monotonically increasing functions mentioned in Subsection 2.4.2 again. This corresponds to another part of Theorem 14.8 on p131 of [20], part of Theorem 18.14 on p284 of [159], and to Corollary 3.7 on p125 of [335].

4.3.3 Equality in the estimate for the integral of α′

Of course, we might like to have that

∫_{a}^{b} α′(x) dx = α(b) − α(a). (4.3.12)

It is well known that this holds when α is continuously differentiable on [a, b], as in Theorem 7.8 E of [135]. In fact, this holds when α is differentiable at every point in [a, b], and the derivative of α is Riemann integrable on [a, b], as in Exercise 8 on p189 of [135], and 6.21 on p134 of [299]. A broader condition under which this holds will be discussed in the next section.

It is easy to see that the inequality in (4.3.11) can be strict when α is not continuous. It is well known that there are continuous monotonically increasing real-valued functions on [0, 1] that are not constant, and whose derivative is equal to 0 almost everywhere with respect to Lebesgue measure. This is discussed beginning on p58 of [14], in Example 4.11 on p32 of [20], on p39 of [118], in Exercise 8.28 on p113 of [159], Example 8.20 (b) on p179 of [298], and beginning on p125 of [335]. See also Subsection A.12.4.

4.4 Absolutely continuous functions

Let a and b be real numbers with a < b again, and let F be a real or complexvalued function on [a, b]. We say that F is absolutely continuous on [a, b] if for every ϵ > 0 there is a δ > 0 with the following property. If (a_{1}, b_{1}), …, (a_{l}, b_{l}) are finitely many pairwise-disjoint open intervals contained in [a, b] such that

∑_{j=1}^{l}(b_{j} − a_{j}) < δ, (4.4.1)

then

∑_{j=1}^{l} |F (b_{j}) − F (a_{j})| < ϵ. (4.4.2)

This corresponds to the definition at the beginning of Section 14.5 on p135 of [20], on p105 of [118], Definition 18.10 on p282 of [159], 8.15 on p175 of [298], and on p127 of [335].

Note that there is no restriction on l in this definition. If we take l = 1, then we get that F is uniformly continuous on [a, b].

It is easy to see that F is absolutely continuous on [a, b] when F is Lipschitz of order 1 on [a, b], with respect to the restriction to [a, b] of the standard Euclidean metric on [a, b], as in Section A.11.

4.4.1 Absolute continuity and differentiability

If F is absolutely continuous on [a, b], then it is well known that F is differentiable almost everywhere on [a, b]. More precisely, one can show that F has bounded variation on [a, b]. If F is real-valued, then this implies that F can be expressed as a difference of monotonically increasing functions. Otherwise, if F is complex-valued, then the real and imaginary parts have this property. One can use this to reduce to the case of monotonically increasing real-valued functions on [a, b]. This argument also implies that F′ is integrable on [a, b] with respect to Lebesgue measure, as in the previous section. This corresponds to Lemmas 14.10 and 14.14 on p133, 135 of [20], Lemmas 3.26 and 3.24 on p102, 106 of [118], Theorems 18.12 and 18.13 on p283 of [159], a remark on p175 and Theorem 8.19 on p177 of [298], and Theorem 3.3 on p119 and a remark on p127 of [335].

In fact, if F is absolutely continuous on [a, b], then

∫_{a}^{x} F′(t) dt = F (x) − F (a) (4.4.3)

for every x ∈ [a, b], as in Theorem 14.16 on p137 of [20], part of 3.35 on p106 of [118], Theorem 18.16 on p285 of [159], Theorem 8.18 on p177 of [298], and part of Theorem 3.11 on p130 of [335].

4.4.2 Integrating integrable functions on [a, b]

If f is a real or complex-valued function on [a, b] that is integrable with respect to Lebesgue measure, then it is well known that

∫_{a}^{x} f(t) dt (4.4.4) is absolutely continuous as a function of x on [a, b]. This corresponds to Exercise 14.2 on p138 of [20], part of 3.35 on p106 of [118], Example 18.11 (a) on p282 of [159], part of Theorem 8.17 on p176 of [298], and a remark on p128 of [335]. More precisely, if f is real-valued and nonnegative on [a, b], then this follows from a remark in Subsection 3.15.2, and otherwise one can reduce to this case.

Another version of Lebesgue’s differentiation theorem implies that the derivative of (4.4.4) is equal to f(x) for almost every x ∈ [a, b] with respect to Lebesgue measure, as in 4.19 on p111 of [14], Theorem 14.5 on p127 of [20], part of Corollary 3.33 on p105 of [118], Theorem 18.3 on p275 of [159], part of Theorem 8.17 on p176 of [298], and part of Theorem 3.11 on p130 of [335].

4.4.3 A criterion for absolute continuity

Let F be a real or complex-valued function on [a, b] that is differentiable at every point in [a, b]. If F′ is Lebesgue integrable on [a, b], then Theorem 8.21 on p179 of [298] states that (4.4.3) holds for every x ∈ [a, b]. This corresponds to Problem 5 on p153 of [335], where F is asked to be continuous on [a, b] and differentiable at every point in (a, b). This also corresponds to part (d) of Exercise 18.41 on p299 of [159], where F is asked to be continuous on [a, b] and differentiable at all but finitely or countably many points in (a, b). This is related to Exercise 14.8 on p139 of [20] as well.

4.5 More on absolutely continuous functions

Let f be a nonnegative extended real-valued Lebesgue measurable function on the real line, and suppose that f is locally integrable with respect to Lebesgue measure on R, in the sense that

∫_{a}^{b} f(x) dx < +∞ (4.5.1)

for all real numbers a, b with a < b. Let ν = ν_{f} be the measure defined on R by

ν_{f}(E) = ∫_{E} f(x) dx (4.5.2)

for all Lebesgue measurable subsets E of R, as in Section 3.15, where the integral on the right is taken with respect to Lebesgue measure on R. Remember that this measure is absolutely continuous with respect to Lebesgue measure on R, by construction, as mentioned near the beginning of Section 4.1. This measure is also locally finite on R, as in Section 2.10, by hypothesis.

Let F be the real-valued function defined on the real line by

(4.5.3)  F (x) = ∫_{0}^{x} f(t) dt  when x ≥ 0
= − ∫_{x}^{0} f(t) dt when x ≤ 0.

One can check that F is continuous and monotonically increasing on R, and in fact F is absolutely continuous on every closed interval in R, as in the previous section. Note that the continuity of F on R corresponds to Exercise 7.11 on p65 of [20]. We also have that F is differentiable at almost every point in R with respect to Lebesgue measure, with

F′(x) = f(x) (4.5.4)

for almost every x ∈ R, as in the previous section.

Let λ_{F} be the Lebesgue–Stieltjes measure on the real line associated to F as in Subsection 2.5.1. If a, b ∈ R and a < b again, then

λ_{F} ((a, b)) = F (b) − F (a), (4.5.5)

as before. Similarly,

ν_{f}((a, b)) = ∫_{a}^{b} f(t) dt = F (b) − F (a), (4.5.6)

by construction. It follows that

λ_{F} = ν_{f} (4.5.7)

as Borel measures on R, as in Section 2.13.

4.5.1 More on differentiating monotone functions

Let α be a monotonically increasing real-valued function on the real line, and suppose now that α′ = f almost everywhere on R with respect to Lebesgue measure. If a, b ∈ R and a < b, then

F (b) − F (a) = ∫_{a}^{b} α′(t) dt ≤ α(b) − α(a), (4.5.8)

where the second step is as in Subsection 4.3.2. This means that

(4.5.9) α − F is monotonically increasing on R.

Note that (α − F)′ = 0 (4.5.10)

almost everywhere on R with respect to Lebesgue measure. If α is absolutely continuous on every closed interval in the real line, then equality holds in the second step in (4.5.8), and α − F is constant on R.

Let β be a monotonically increasing real-valued on the real line, and let λ_{β} be the corresponding Lebesgue–Stieltjes measure, considered as a Borel measure on R, as in Subsection 2.5.1. If

(4.5.11) λ_{β} is mutually singular with respect to Lebesgue measure on R, as Borel measures on R, then it is well known that

(4.5.12) β′ = 0 almost everywhere with respect to Lebesgue measure on R.

This corresponds to Lemma 14.6 on p128 of [20], part of Theorem 3.22 on p99 and some remarks on p106 of [118], part of Theorem 19.60 on p336 of [159], and part of the corollary on p166 of [298].

Conversely, it is also well known that (4.5.12) implies (4.5.11), as in [118, 159, 298] and part of Exercise 11 on p314 of [335].

4.5.2 One way to organize results like these

Results like these may be organized in various ways. Sometimes one obtains properties of measures that are mutually singular with respect to Lebesgue measure directly, to get that (4.5.11) implies (4.5.12), and also shows the Lebesgue differentiation theorem as in Subsection 4.4.2.

If α is a monotonically increasing real-valued function on the real line, and λ_{α} is the corresponding Lebesgue–Stieltjes measure on R, then one can use the Radon–Nikodym and Lebesgue decomposition theorems to get that λ_{α} is the sum of Borel measures on R that are absolutely continuous and mutually singular with respect to Lebesgue measure. This can be used to get that α is differentiable almost everywhere with respect to Lebesgue measure, with derivative equal almost everywhere to the Radon–Nikodym derivative of the absolutely continuous part of λ_{α}.

In particular, if α′ = 0 almost everywhere with respect to Lebesgue measure on R, then the absolutely continuous part of λ_{α} is equal to 0, so that (4.5.12) implies (4.5.11).

4.5.3 Another way to organize results like these

Alternatively, one might start with both versions of Lebesgue’s differentation theorem, for the differentiability almost everywhere of monotonically increasing functions on R, as well as the version in Subsection 4.4.2. If α is monotonically increasing on R, then one can use the Radon–Nikodym and Lebesgue decomposition theorems to get that α′ is greater than or equal to the Radon– Nikodym derivative of the absolutely continuous part of λ_{α} almost everywhere with respect to Lebesgue measure.

In particular, if λ_{α} is not mutually singular with respect to Lebesgue measure, so that the absolutely continuous part is positive, then α′ > 0 on a set of positive Lebesgue measure, so that (4.5.12) implies (4.5.11).

If f and F are as at the beginning of this subsection, then it follows that the Lebesgue–Stieltjes measure associated to α − F is mutually singular with respect to Lebesgue measure on R, so that λ_{F} is the absolutely continuous part of λ_{α}.

If λ_{α} is mutually singular with respect to Lebesgue measure, then λ_{F} = 0, so that α′ = f = 0 almost everywhere with respect to Lebesgue measure, and (4.5.11) implies (4.5.12).

4.5.4 The derivative as a limit in an integral sense

Now let f be a real or complex-valued Lebesgue measurable function on the real line, and suppose that f is locally integrable with respect to Lebesgue measure, in the sense that the restriction of f to any bounded interval in R is integrable, as before. One can define F as a real or complex-valued function on R, as appropriate, as in (4.5.3) again. As before, F is continuous on R, and absolutely continuous on every closed interval in R. Note that

F (w) − F (v) = ∫_{v}^{w} f(t) dt (4.5.13)

for every v, w ∈ R with v ≤ w. Thus

\frac{F (x + h) − F (x)}{h} = h^{−1} ∫_{x}^{x+h} f(t) dt (4.5.14)

and

\frac{F (x) − F (x − h)}{h} = h^{−1} ∫_{x−h}^{x} f(t) dt (4.5.15)

for every x ∈ R and h > 0.

If a and b are real numbers with a < b, then it is well known and not too difficult to show that

lim_{h→0} ∫_{a}^{b} |\frac{F (x + h) − F (x)}{h} − f(x)| dx = 0 (4.5.16)

under these conditions. More precisely, if f is continuous on R, then one can check that (4.5.14) and (4.5.15) converge to f(x) as h → 0 uniformly on bounded subsets of R, because f is uniformly continuous on bounded subsets of R. Otherwise, one can approximate f by continuous functions on bounded intervals in R in a suitable integral sense. Similarly, if f is integrable on R, then one can show that

lim_{h→0} ∫_{−∞}^{+∞} |\frac{F (x + h) − F (x)}{h} − f(x)| dx = 0, (4.5.17)

by approximating f by continuous functions on R with compact support. This corresponds to differentiability of F in the L^{1} norm, as on p4 of [337].

If F is any continuous real or complex-valued function on the real line and v, w ∈ R, v < w, and 0 < h < w − v, then

∫_{v}^{w} \frac{F (x + h) − F (x)}{h} dx = h^{−1} ∫_{v}^{w} F (x + h) dx − h^{−1} ∫_{v}^{w} F (x) dx
(4.5.18)  = h^{−1} ∫_{v+h}^{w+h} F (x) dx − h^{−1} ∫_{v}^{w} F (x) dx
= h^{−1} ∫_{w}^{w+h} F (x) dx − h^{−1} ∫_{v}^{v+h} F (x) dx.

Similarly,

∫_{v}^{w} \frac{F (x) − F (x − h)}{h} dx = h^{−1} ∫_{v}^{w} F (x) dx − h^{−1} ∫_{v}^{w} F (x − h) dx
(4.5.19)  = h^{−1} ∫_{v}^{w} F (x) dx − h^{−1} ∫_{v−h}^{w−h} F (x) dx
= h^{−1} ∫_{w−h}^{w} F (x) dx − h^{−1} ∫_{v−h}^{v} F (x) dx.

In both cases, the right side tends to F (w) − F (v) as h → 0, because F is continuous at v and w, by hypothesis. If (4.5.16) holds with a = v and b = w, then it follows that (4.5.13) holds. In fact, this works when the analogous limit along a sequence of nonzero real numbers converging to 0 is equal to 0.

Of course, (4.5.13) implies (4.5.3) when F (0) = 0. One can also consider differentiability in the L^{p} norm, as on p4 of [337], and in 4.4 beginning on p31 of [337]. This is discussed in Section 6 of Chapter IV of [45] when p = 2, as indicated in [337]. The p = 1 case is related to part (ii) of Theorem 3 on p8 of [45].

4.6 The L^{1} norm

Let (X, A, μ) be a measure space. If f is a real or complex-valued measurable function on X that is integrable with respect to μ, then put

∥f∥_{1} = ∥f∥_{L^{1}} = ∥f∥_{L^{1}(X)} = ∥f∥_{L^{1}(X,μ)} = ∫_{X} |f| dμ. (4.6.1)

It is sometimes convenient to interpret this as being equal to +∞ when f is not integrable on X with respect to μ.

If t is a real or complex number, as appropriate, then it is easy to see that

∥t f∥_{1} = |t| ∥f∥_{1}. (4.6.2)

If g is another real or complex-valued measurable function on X that is integrable with respect to μ, then f + g is integrable with respect to μ as well, and ∥f + g∥_{1} = ∫_{X} |f + g| dμ ≤ ∫_{X}(|f| + |g|) dμ = ∥f∥_{1} + ∥g∥_{1}. (4.6.3)

The space of all real or complex-valued measurable functions on X is a linear subspace of the space of all real or complex-valued functions on X, as appropriate. However, (4.6.1) is not quite a norm on this space, because ∥f∥_{1} = 0 if and only if f = 0 almost everywhere on X with respect to μ, and this does not necessarily mean that f = 0 everywhere on X. In order to deal with this, one can identify two integrable functions on X that are equal to each other almost everywhere with respect to μ, as on p202 of [14], p144 of [20], p54 of [118], in Theorem 13.10 on p192 of [159], in the remarks 3.10 beginning on p68 of [298], and on p69, 276 of [335]. Let us not be too concerned about this right now, although in effect we shall do something similar in practice.

4.6. THE L^{1} NORM 113

4.6.1 Convergence in L^{1}

Let us say that a sequence {f_{j}}_{j=1}^{∞} of real or complex-valued integrable functions on X converges in L^{1} to a real or complex-valued integrable function f on X, as appropriate, if ∥f_{j} − f∥_{1} → 0 as j → ∞, (4.6.4)

as in Definition 10.1 on p83 of [20]. In this case, one can check that if {f_{j}}_{j=1}^{∞} converges in L^{1} to another real or complex-valued integrable function f̃ on X, as appropriate, then f = f̃ almost everywhere on X with respect to μ.

Similarly, let us say that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{1} if

∥f_{j} − f_{l}∥_{1} → 0 as j, l → ∞. (4.6.5)

If {f_{j}}_{j=1}^{∞} converges in L^{1}, then one can check that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{1}, using a standard argument.

4.6.2 Absolutely convergent series in L^{1}

Let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that are integrable with respect to μ. Let us say that the corresponding infinite series ∑_{j=1}^{∞} a_{j} converges absolutely in L^{1} if

∑_{j=1}^{∞} ∥a_{j}∥_{1} < +∞. (4.6.6)

This implies that

∑_{j=1}^{∞} |a_{j}(x)| < +∞ (4.6.7)

for almost every x ∈ X with respect to μ, as in Subsection 3.13.3. This means that

∑_{j=1}^{∞} a_{j}(x) (4.6.8)

converges absolutely for almost every x ∈ X. Let us interpret this sum as being equal to 0 at points where it does not converge, as before. This defines an integrable function on X with respect to μ, as before. We also have that the sequence of partial sums

∑_{j=1}^{n} a_{j}(x) (4.6.9)

converges to (4.6.8) in L^{1}, as before.

4.6.3 Convergence in L^{1} of Cauchy sequences in L^{1}

If {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{1}, then it is well known that {f_{j}}_{j=1}^{∞} converges in L^{1}, as in 7.20 on p204 of [14], Theorem 15.4 on p145 of [20], Theorem 6.6 on p183 of [118], Theorem 13.11 on p192 of [159], Theorem 3.11 on p69 of [298], and Theorem 2.2 on p70 and a remark on p276 of [335]. One can start by getting a subsequence {f_{j_{r}}}_{r=1}^{∞} of {f_{j}}_{j=1}^{∞} such that

∑_{r=1}^{∞} ∥f_{j_{r+1}} − f_{j_{r}}∥_{1} < +∞. (4.6.10)

More precisely, one can use the Cauchy condition to get a subsequence such that ∥f_{j_{r+1}} − f_{j_{r}}∥_{1} ≤ 2^{−r} (4.6.11)

for each r, for instance.

Put

f = f_{j_{1}} + ∑_{r=1}^{∞}(f_{j_{r+1}} − f_{j_{r}}), (4.6.12)

where the sum on the right is interpreted as in the previous subsection. If n is a positive integer, then

f_{j_{1}} + ∑_{r=1}^{n}(f_{j_{r+1}} − f_{j_{r}}) = f_{j_{n+1}} (4.6.13)

on X. It follows that {f_{j_{n}}}_{n=1}^{∞} converges to f pointwise almost everywhere on X with respect to μ and in L^{1}, by construction. One can use this and the Cauchy condition for {f_{j}}_{j=1}^{∞} in L^{1} to get that {f_{j}}_{j=1}^{∞} converges to f in L^{1}, by a standard argument.

4.6.4 Pointwise convergent subsequences

If {f_{j}}_{j=1}^{∞} converges in L^{1} to an integrable function f̃ on X, then {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{1}, as mentioned previously. We also have that f̃ is equal to the function f in the preceding subsection almost everywhere with respect to μ on X. This means that the subsequence {f_{j_{r}}}_{r=1}^{∞} converges to f̃ pointwise almost everywhere on X with respect to μ. This corresponds to 7.23 on p205 of [14], Corollary 2.32 on p62 of [118], Theorem 3.12 on p70 of [298], and Corollary 2.3 on p71 of [335].

4.7 Convergence in measure

Let (X, A, μ) be a measure space. A sequence of real or complex-valued measurable functions {f_{j}}_{j=1}^{∞} on X is said to converge to a real or complex-valued measurable function f on X in measure if for every ϵ > 0,

μ({x ∈ X : |f_{j}(x) − f(x)| > ϵ}) → 0 as j → ∞, (4.7.1) as in Definition 10.1 on p83 of [20], on p61 of [118], Definition 11.25 on p156 of [159], Exercise 18 on p76 of [298], and Exercise 3 on p95 of [335]. Convergence in measure is also known as convergence in probability when μ is a probability measure on X.

Equivalently, {f_{j}}_{j=1}^{∞} converges to f in measure if for every ϵ, ϵ′ > 0 there is a positive integer L such that

μ({x ∈ X : |f_{j}(x) − f(x)| > ϵ}) < ϵ′ (4.7.2)

for all j ≥ L. Convergence in measure is defined in [298] to mean that this holds when ϵ = ϵ′. One can check that this implies the previous version, by replacing ϵ and ϵ′ with their minimum. This corresponds to Exercise 35 on p63 of [118].

If {f_{j}}_{j=1}^{∞} converges to f in measure, and if {f_{j}}_{j=1}^{∞} converges in measure to another real or complex-valued measurable function f̃ on X, then one can verify that f = f̃ almost everywhere on X with respect to μ.

4.7.1 Convergence in L^{1} implies convergence in measure

If f is an integrable function on X and a is a positive real number, then

a μ({x ∈ X : |f(x)| ≥ a}) ≤ ∫_{{x∈X:|f(x)|≥a}} |f| dμ ≤ ∫_{X} |f| dμ. (4.7.3)

This implies that

μ({x ∈ X : |f(x)| ≥ a}) ≤ a^{−1} ∫_{X} |f| dμ, (4.7.4)

as in 4.1 on p102 of [14], Lemma 10.4 on p85 of [20], 6.17 on p193 of [118], and Exercise 9 on p91 of [335]. This is known as Chebyshev’s (or Tchebychev’s) or Markov’s inequality.

If {f_{j}}_{j=1}^{∞} is a sequence of integrable functions on X that converges to an integrable function f in L^{1}, then it follows that {f_{j}}_{j=1}^{∞} converges to f in measure, as in Proposition 10.5 on p85 of [20], Proposition 2.29 on p61 of [118], part (b) of Exercise 18 on p77 of [298], and part of Problem 3 on p95 of [335].

4.7.2 Pointwise convergence and Egorov’s theorem

Let us suppose in this subsection that μ(X) < +∞. Let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that converges to a real or complex-valued function f on X pointwise almost everywhere with respect to μ. It is well known that {f_{j}}_{j=1}^{∞} converges to f in measure on X, as in part (1) of Proposition 10.2 on p83 of [20], Theorem 11.31 on p158 of [159], and part (a) of Exercise 18 on p77 of [298].

In fact, Egorov’s (or Egoroff’s) theorem states that for every ϵ > 0 there is a measurable subset E of X such that

μ(X \ E) < ϵ (4.7.5) and {f_{j}}_{j=1}^{∞} converges to f uniformly on E. This corresponds to 2.85 on p63 of [14], Theorem 10.8 on p86 of [20], 2.33 on p62 of [118], Theorem 11.32 on p158 of [159], Exercise 16 on p75 of [298], and Theorem 4.4 on p33 of [335].

This type of convergence is sometimes called almost uniform convergence, as in [20, 118], and in the index of [159]. However, this term is mentioned on p230 of [298] in connection with uniform convergence on compact subsets of an open set in the complex plane.

4.7.3 More on pointwise convergent subsequences

Let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that converges to a real or complex-valued measurable function f on X in measure. It is well known that there is a subsequence of {f_{j}}_{j=1}^{∞} that converges to f pointwise almost everywhere on X with respect to μ, as in part (2) of Proposition 10.2 on p83 of [20], Theorem 2.30 on p61 of [118], Theorem 11.26 on p156 of [159], and part (c) of Exercise 18 on p77 of [298].

4.7.4 Cauchy sequences in measure

Let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X again. We say that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure if for every ϵ > 0,

μ({x ∈ X : |f_{j}(x) − f_{l}(x)| > ϵ}) → 0 as j, l → ∞, (4.7.6)

as on p61 of [118], and Exercise 3 on p95 of [335]. This means that for every ϵ, ϵ′ > 0 there is a positive integer L such that

μ({x ∈ X : |f_{j}(x) − f_{l}(x)| > ϵ}) < ϵ′ (4.7.7)

for all j, l ≥ L, as in Exercise 10.1 on p86 of [20]. If {f_{j}}_{j=1}^{∞} converges in measure on X, then one can check that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure.

If {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure on X, then Theorem 2.30 on p61 of [118] states that {f_{j}}_{j=1}^{∞} converges in measure to a real or complex-valued measurable function f on X, and that there is a subsequence of {f_{j}}_{j=1}^{∞} that converges to f pointwise almost everywhere on X with respect to μ. This is also mentioned in Problem N on p68 of [189] when μ(X) is finite.

4.8 More on bounded convergence

Let (X, A, μ) be a measure space, and let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that converges to a real or complexvalued measurable function f on X in measure. Suppose that {f_{j}}_{j=1}^{∞} is uniformly bounded almost everywhere on X with respect to μ, in the sense that there is a nonnegative real number C such that

|f_{j}| ≤ C (4.8.1) almost everywhere on X with respect to μ for each j. One can check that this implies that |f| ≤ C (4.8.2)

almost everywhere on X with respect to μ.

If μ(X) < +∞, then another version of the bounded convergence theorem states that

lim_{j→∞} ∫_{X} |f_{j} − f| dμ = 0, (4.8.3)

so that

lim_{j→∞} ∫_{X} f_{j} dμ = ∫_{X} f dμ. (4.8.4)

To see this, let ϵ > 0 be given, and observe that

∫_{X} |f_{j} − f| dμ = ∫_{{x∈X:|f_{j}(x)−f(x)|≤ϵ}} |f_{j} − f| dμ

+ ∫_{{x∈X:|f_{j}(x)−f(x)|>ϵ}} |f_{j} − f| dμ (4.8.5)

≤ ϵ μ(X) + 2 C μ({x ∈ X : |f_{j}(x) − f(x)| > ϵ})

for each j. The second term on the right tends to 0 as j → ∞, by hypothesis. This is essentially the same as the proof of the bounded convergence theorem using pointwise convergence almost everywhere in 3.26 on p89 of [14], Exercise 12.49 on p183 of [159], and Theorem 1.4 on p56 of [335], except that convergence in measure can be used directly in place of Egorov’s theorem.

4.9 Some more integral distance functions

Let (X, A, μ) be a measure space such that μ(X) < +∞. If f and g are real or complex-valued measurable functions on X, then

min(|f − g|, 1) (4.9.1)

and

\frac{|f − g|}{1 + |f − g|} (4.9.2)

are measurable on X too.

Put d(f, g) = ∫_{X} min(|f − g|, 1) dμ (4.9.3)

and

d̃(f, g) = ∫_{X} \frac{|f − g|}{1 + |f − g|} dμ. (4.9.4)

These are approximately the same, as we shall see in a moment, and the second is used in Exercise 10.2 on p87 of [20], Exercise 32 on p63 of [118], Exercise 12.47 on p182 of [159], and Exercise 13 on p38 of [297]. This is also related to Problems L, N on p55, 68 of [189], respectively.

4.9.1 Some properties of d, d̃

It is easy to see that d(f, g) = 0 and d̃(f, g) = 0 each occur if and only if f = g almost everywhere on X with respect to μ, as in Subsection 3.10.1. Clearly

d(f, g) = d(g, f) and d̃(f, g) = d̃(g, f). (4.9.5)

If h is another real or complex-valued measurable function on X, then

d(f, h) ≤ d(f, g) + d(g, h) (4.9.6)

and d̃(f, h) ≤ d̃(f, g) + d̃(g, h), (4.9.7) because of analogous inequalities in Section A.7. In order to get metrics, we should identify measurable functions on X that are equal to each other almost everywhere with respect to μ, as before. We shall not be too concerned about this for the moment, and instead we can use some analogous notions.

Observe that

min(|f − g|, 1)/2 ≤ \frac{|f − g|}{1 + |f − g|} ≤ min(|f − g|, 1) (4.9.8)

on X, as in Subsection A.7.3. This implies that

d(f, g)/2 ≤ d̃(f, g) ≤ d(f, g). (4.9.9)

If f and g are integrable on X with respect to μ, then

d(f, g) ≤ ∥f − g∥_{1}. (4.9.10)

4.9.2 Connection with convergence in measure

Let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X. Of course, {f_{j}}_{j=1}^{∞} converges to f in measure on X if and only if {|f_{j} − f|}_{j=1}^{∞} converges to 0 in measure on X. It is easy to see that this happens if and only if (4.9.11) {min(|f_{j} − f|, 1)}_{j=1}^{∞} converges to 0 in measure on X. This is equivalent to the condition that

(4.9.12) \{\frac{|f_{j} − f|}{1 + |f_{j} − f|}\}_{j=1}^{∞} converge to 0 in measure on X,

because of (4.9.8).

We also have that d(f_{j}, f) → 0 as j → ∞ (4.9.13) if and only if d̃(f_{j}, f) → 0 as j → ∞, (4.9.14) because of (4.9.9). Remember that (4.9.13) implies (4.9.11), and (4.9.14) implies (4.9.12), as in Subsection 4.7.1.

Conversely, (4.9.11) implies (4.9.13), and (4.9.12) implies (4.9.14), by the version of the bounded convergence theorem mentioned in the previous section.

4.10. ABSOLUTE CONVERGENCE WITH RESPECT TO D, D̃ 119

4.9.3 Cauchy conditions associated to d and d̃

Let us say that {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d(·, ·) if

d(f_{j}, f_{l}) → 0 as j, l → ∞. (4.9.15)

Similarly, {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d̃(·, ·) if

d̃(f_{j}, f_{l}) → 0 as j, l → ∞. (4.9.16)

These conditions are equivalent to each other, because of (4.9.9), as usual.

It is easy to see that (4.9.15) implies {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure. Equivalently, this means that (4.9.16) implies that {f_{j}}_{j−1}^{∞} is a Cauchy sequence in measure. This can be obtained from Chebyshev’s inequality, as in Subsection 4.7.1.

Conversely, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure, then (4.9.15) holds, so that (4.9.16) holds. Indeed, for each ϵ > 0,

∫_{X} min(|f_{j} − f_{l}|, 1) dμ = ∫_{{x∈X:|f_{j}(x)−f_{l}(x)|≤ϵ}} min(|f_{j} − f_{l}|, 1) dμ

+ ∫_{{x∈X:|f_{j}(x)−f_{l}(x)|>ϵ}} min(|f_{j} − f_{l}|, 1) dμ (4.9.17)

≤ ϵ μ(X) + μ({x ∈ X : |f_{j}(x) − f_{l}(x)| > ϵ})

for each j and l. The second term on the right is as small as we want when j and l are sufficiently large, by hypothesis.

4.10 Absolute convergence with respect to d, d̃

Let (X, A, μ) be a measure space with μ(X) < +∞ again. Also let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X. Let us say that the corresponding infinite series ∑_{j=1}^{∞} a_{j} converges absolutely with respect to d(·, ·) if

∑_{j=1}^{∞} d(a_{j}, 0) = ∑_{j=1}^{∞} ∫_{X} min(|a_{j}|, 1) dμ < +∞. (4.10.1)

This is equivalent to the absolute convergence of ∑_{j=1}^{∞} a_{j} with respect to d̃(·, ·), in the sense that

∑_{j=1}^{∞} d̃(a_{j}, 0) < +∞, (4.10.2)

because of (4.9.9).

4.10.1 Getting pointwise convergence almost everywhere

Using (4.10.1), we get that

∑_{j=1}^{∞} min(|a_{j}(x)|, 1) < +∞ (4.10.3)

for almost every x ∈ X with respect to μ, as in Subsection 3.13.3. In particular, this implies that

lim_{j→∞} min(|a_{j}(x)|, 1) = 0 (4.10.4)

for almost every x ∈ X with respect to μ.

It follows that for almost every x ∈ X with respect to μ,

min(|a_{j}(x)|, 1) < 1 (4.10.5)

when j is sufficiently large, so that

min(|a_{j}(x)|, 1) = |a_{j}(x)|. (4.10.6)

Thus

∑_{j=1}^{∞} |a_{j}(x)| < +∞ (4.10.7)

for almost every x ∈ X with respect to μ.

4.10.2 Convergence of the sum with respect to d

Of course, this means that ∑_{j=1}^{∞} a_{j}(x) converges for almost every x ∈ X with respect to μ. If n is a positive integer and (4.10.7) holds, then

max (|∑_{j=n}^{∞} a_{j}(x)|, 1) ≤ max ( ∑_{j=n}^{∞} |a_{j}(x)|, 1) = lim_{N→∞} max ( ∑_{j=n}^{N} |a_{j}(x)|, 1)

(4.10.8)  ≤ lim_{N→∞} ∑_{j=n}^{N} max(|a_{j}(x)|, 1) = ∑_{j=n}^{∞} max(|a_{j}(x)|, 1),

using a remark at the beginning of Section A.7 in the third step.

This implies that

(4.10.9) d( ∑_{j=1}^{∞} a_{j}(x), ∑_{j=1}^{n} a_{j}(x)) = ∫_{X} max (| ∑_{j=n+1}^{∞} a_{j}(x)|, 1) dμ

≤ ∫_{X} ∑_{j=n+1}^{∞} max(|a_{j}(x)|, 1) dμ

= ∑_{j=n+1}^{∞} ∫_{X} max(|a_{j}(x)|, 1) dμ,

where the third step is as in Subsection 3.11.1. The right side tends to 0 as n → ∞, because of (4.10.1). Although we already know that the partial sums ∑_{j=1}^{n} a_{j}(x) converges to ∑_{j=1}^{∞} a_{j}(x) almost everywhere on X with respect to μ, and thus in measure, it is nice to have this more direct argument in this case as well.

4.10.3 Convergence of Cauchy sequences with respect to d

Suppose now that {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to d(·, ·). This implies that there is a subsequence {f_{j_{r}}}_{r=1}^{∞} of {f_{j}}_{j=1}^{∞} such that

∑_{r=1}^{∞} d(f_{j_{r+1}}, f_{j_{r}}) < +∞. (4.10.10)

Put

f = f_{j_{1}} + ∑_{r=1}^{∞}(f_{j_{r+1}} − f_{j_{r}}), (4.10.11)

where the sum on the right is as in the previous two subsections.

Observe that

f_{j_{1}} + ∑_{r=1}^{n}(f_{j_{r+1}} − f_{j_{r}}) = f_{j_{n+1}} (4.10.12)

converges to f pointwise almost everywhere on X and with respect to d(·, ·) as n → ∞, as before. One can use this and the Cauchy condition for {f_{j}}_{j=1}^{∞} with respect to d(·, ·) to get that {f_{j}}_{j=1}^{∞} converges to f with respect to d(·, ·), as in (4.9.13).

This is another way to look at Theorem 2.30 on p61 of [118] when μ(X) is finite. This is also related to Exercise 10.1 on p86 of [20] when μ(X) < +∞, and to part (d) of Exercise 12.47 on p182 of [159]. In particular, this shows that if μ(X) < +∞ and {f_{j}}_{j=1}^{∞} converges in measure, then there is a subsequence of {f_{j}}_{j=1}^{∞} that convergees to the same limit pointwise almost everywhere on X with respect to μ.

4.11 Pushing measures forward

Let (X, A_{X}, μ) be a measure space, let (Y, A_{Y}) be a measurable space, and let ϕ be a measurable mapping from X into Y. If E ∈ A_{Y}, then ϕ^{−1}(E) ∈ A_{X}, and we put ν(E) = ν_{ϕ}(E) = μ(ϕ^{−1}(E)). (4.11.1)

One can check that ν is a measure on Y with respect to A_{Y} under these conditions. This corresponds to a remark on p314 of [118], and to part (a) Exercise 11.38 on p161 of [159]. More precisely, in [159], one takes A_{Y} to be the collection of E ⊆ Y such that ϕ^{−1}(E) ∈ A_{X}, and part of the exercise is to check that this is a σ-algebra of subsets of Y, as mentioned near the beginning of Section 3.1.

The measure ν is called the image measure associated to μ and ϕ in [118], and the term measures on image sets is used in [159]. One may also say that ν is obtained by pushing μ forward using ϕ, and put

ϕ_{∗}(μ) = ν. (4.11.2)

Suppose for the moment that Y = X and A_{Y} = A_{X}. If ν = μ, then ϕ is said to be measure-preserving, as on p292 of [335]. This means that

μ(ϕ^{−1}(E)) = μ(E) (4.11.3)

for all E ∈ A_{X} = A_{Y}.

4.11.1 Probability distributions and random variables

Of course, ν(Y) = μ(ϕ^{−1}(Y)) = μ(X). (4.11.4)

If μ is a probability measure on X, then ν is a probability measure on Y. In this case, if Y = R and A_{Y} is the σ-algebra of Borel sets in R, then ν may be called the probability distribution on R associated to μ and ϕ, as in Definition 12.27 on p392 of [14], and on p268 of [20], and p315 of [118]. Note that ϕ may be called a random variable on X under these conditions, as in Definition 12.13 on p385 of [14], on p286 of [20], and on p314 of [118].

4.11.2 Compositions and isomorphisms

Let (Z, A_{Z}) be another measurable space, and let ψ be a measurable mapping from Y into Z. This implies that ψ ∘ ϕ is a measurable mapping from X into Z, as in Subsection 3.1.3, and one can check that

(ψ ∘ ϕ)_{∗}(μ) = ψ_{∗}(ϕ_{∗}(μ)). (4.11.5)

Suppose for the moment that ϕ is a one-to-one mapping from X onto Y. If ϕ and ϕ^{−1} are both measurable, then ϕ may be called an isomorphism from X onto Y with respect to A_{X} and A_{Y}, respectively. Using the measure ν, ϕ may also be considered as an isomorphism between the measure spaces (X, A_{X}, μ) and (Y, A_{Y} , ν).

In particular, an isomorphism from (X, A, μ) onto itself may be called an automorphism. It is easy to see that the collection of automorphisms of a measure space is a group with respect to composition of mappings.

4.11.3 Compositions and integrals

Let ϕ be any measurable mapping from X into Y again. If f is a nonnegative extended real-valued measurable function on Y, then f ∘ ϕ is measurable on X, and ∫_{X}(f ∘ ϕ) dμ = ∫_{Y} f dν. (4.11.6)

This follows from the definition of ν when f is the indicator function of a measurable set in Y. If f is a simple function on Y, then one can reduce to the previous case. Otherwise, one can approximate f by a monotonically increasing sequence of measurable simple functions, as in Subsection 3.7.1, and use the monotone convergence theorem.

If f is a measurable real or complex-valued function on Y, then f ∘ ϕ is measurable on X, and we can use the remarks in the preceding paragraph to get that

∫_{X} |f ∘ ϕ| dμ = ∫_{Y} |f| dν. (4.11.7)

This means that f is integrable on Y with respect to ν if and only if f ∘ ϕ is integrable on X with respect to μ. In this case, one can get (4.11.6) by considering the positive and negative parts of f when f is real-valued, and the real and imaginary parts of f when f is complex-valued. This corresponds to Proposition 10.1 on p314 of [118], to Proposition 21.3 on p270 of [20] when Y = R, and to a remark on p303 of [20] when Y = R^{n} for some positive integer n.

4.11.4 Pushing outer measures forward

Let X and Y be sets, and let μ_{X} be an outer measure on X, as in Subsection 1.10.4. Also let ϕ be a mapping from X into Y. If E ⊆ Y, then put

μ_{Y} (E) = μ_{X}(ϕ^{−1}(E)). (4.11.8)

One can check that this defines an outer measure on Y, as mentioned on p54 of [113], in part (b) of Exercise 11.38 on p161 of [159], and on p16 of [247].

This outer measure may be denoted ϕ_{♯}(μ_{X}), as in [113, 247]. One can check that E ⊆ Y is measurable with respect to μ_{Y} when ϕ^{−1}(E) is measurable with respect to μ_{X}, as in [113, 247]. Remember that measurability with respect to an outer measure is defined in Subsection 1.11.1.

Suppose now that X and Y are metric or topological spaces, and that ϕ is measurable with respect to the σ-algebras of Borel sets in X and Y, respectively. If μ_{X} is a Borel outer measure on X, as in Subsection 2.2.1, then it follows that μ_{Y} is a Borel outer measure on Y. This is mentioned on p16 of [247].

4.12 Some measures from measurable sets

Let (X, A, μ) be a measure space, let X_{0} be a measurable subset of X, and put

A_{X_{0}} = {E_{0} ⊆ X_{0} : E_{0} ∈ A_{X}}. (4.12.1)

It is easy to see that this is a σ-algebra of subsets of X_{0}, as in Subsection 1.6.4, and as mentioned in Definition 11.22 on p155 of [159]. The restriction μ_{0} of μ to A_{X_{0}} is a measure on X_{0} with respect to A_{X_{0}}, as in Exercise 11.37 on p161 of [159].

Note that

μ_{1}(E) = μ(E ∩ X_{0}) (4.12.2)

defines a measure on X with respect to A_{X}, as in Exercise 3.6 on p21 of [20], and Exercise 10 on p27 of [118].

Let ϕ_{0} be the obvious inclusion mapping from X_{0} into X, so that ϕ_{0}(x_{0}) = x_{0} for every x_{0} ∈ X_{0}, considered as an element of X as well. If E ⊆ X, then

ϕ_{0}^{−1}(E) = E ∩ X_{0}. (4.12.3)

Clearly ϕ_{0} is measurable as a mapping from X_{0} into X, because X_{0} ∈ A_{X}, by hypothesis. We also have that μ_{1} is the same as the measure on X obtained by pushing μ_{0} forward using ϕ_{0}.

4.13 Distribution functions

Let (X, A, μ) be a measure space with μ(X) < +∞, and let ϕ be a measurable real-valued function on X. Remember that this means that ϕ is measurable with respect to the σ-algebra of Borel sets in R, as in Section 3.1. Thus (4.11.1) defines a Borel measure on the real line with

ν(R) = μ(X), (4.13.1)

as in (4.11.4).

Put

F (t) = F_{ϕ}(t) = ν((−∞, t]) = μ(ϕ^{−1}((−∞, t])) (4.13.2)

for every t ∈ R. This is called the distribution function of ϕ with respect to μ on X, as in Definition 12.27 on p392 of [14], on p268 of [20], and on p315 of [118]. Observe that

F = α_{ν,−∞,R}, (4.13.3)

in the notation of Subsection 2.13.2. Thus F is monotonically increasing on R and right continuous at every point in R, with

F (−∞) = 0 (4.13.4)

and

F (+∞) = ν(R) = μ(X), (4.13.5)

using the notation in Subsection 2.4.2 on the left sides of these two equations. This corresponds to the “only if” part of 12.29 on p393 of [14], Proposition 21.1 on p268 of [20], and some remarks on p33 of [118].

4.13.1 The Lebesgue–Stieltjes measure associated to F

Let λ_{F} be the Lebesgue–Stieltjes measure associated to F, as in Subsection 2.5.1, considered as a Borel measure on the real line. Note that

λ_{F} ((−∞, t]) = F (t+) − F (−∞) = F (t) (4.13.6)

for every t ∈ R. In fact, ν = λ_{F} (4.13.7)

under these conditions, as in Section 2.13. This corresponds to a remark on p268 of [20] that is related to a remark on p31 of [20]. Similarly, this corresponds to a remark on p315 of [118] that is related to some remarks on p35 of [118].

More precisely, if a, b ∈ R and a < b, then

ν((−∞, a]) + ν(a, b]) = ν((−∞, b]), (4.13.8)

so that ν((a, b]) = ν((−∞, b]) − ν((−∞, a]) = F (b) − F (a). (4.13.9)

Of course, we also have that

λ_{F} ((a, b]) = F (b) − F (a). (4.13.10)

One can use this to get that

ν((a, b)) = λ_{F} ((a, b)) = F (b−) − F (a). (4.13.11)

Similarly,

(4.13.12)  ν((−∞, b)) = λ_{F} ((−∞, b)) = F (b−),
(4.13.13)  ν((a, +∞)) = λ_{F} ((a, +∞)) = F (+∞) − F (a),
(4.13.14)  ν(R) = λ_{F} (R) = F (+∞).

Thus ν and λ_{F} are the same on open intervals in the real line. If U is a nonempty open subset of R, then U can be expressed as the union of finitely or countably many pairwise-disjoint open intervals, and we get that

ν(U) = λ_{F} (U). (4.13.15)

It follows that ν and λ_{F} are the same on Borel sets in R, as before.

4.13.2 Density functions

Suppose that the restriction of F to any closed interval [a, b] is absolutely continuous, as in Section 4.4. Remember that F is differentiable almost everywhere on R with respect to Lebesgue measure, as in Section 4.3, and that F′(t) ≥ 0 when F is differentiable at t ∈ R, because F is monotonically increasing on R. The absolute continuity of F on [a, b] implies that

∫_{a}^{b} F′(t) dt = F (b) − F (a), (4.13.16) as before. In this case, F′ is called the density of F, as on p270 of [20]. This corresponds to the definition of a density function on p394 of [14], as in Subsection 4.4.2 and the next paragraph.

It follows that F′ is integrable on R under these conditions, because F is bounded on R. We also get that

∫_{−∞}^{b} F′(t) dt = F (b). (4.13.17)

Similarly,

∫_{−∞}^{+∞} F′(t) dt = F (+∞) = ν(R) = μ(X). (4.13.18)

If E is a Borel set in R, then

λ_{F} (E) = ∫_{E} F′(t) dt, (4.13.19)

as in Section 4.5, where the integral on the right is taken with respect to Lebesgue measure. More precisely, the definition of F (x) in Section 4.5 corresponds to F (x) − F (0) here. This does not affect the derivative of F, or the corresponding Lebesgue–Stieltjes measure λ_{F}. Thus we get that

ν(E) = λ_{F} (E) = ∫_{E} F′(t) dt, (4.13.20)

using (4.13.7) in the first step.

4.13.3 A slightly different type of distribution function

Let (X, A, μ) be any measure space, where μ(X) is not necessarily finite. If ϕ is a measurable real or complex-valued function on X and t is a positive real number, then consider μ({x ∈ X : |ϕ(x)| > t}), (4.13.21)

as on p197 of [118]. This is a monotonically decreasing nonnegative extended real-valued function of t > 0. In practice, one may be concerned with functions ϕ on X with some sort of integrability properties that imply that (4.13.21) is finite.

4.14 Approximate inverses of monotone functions

Let α be a monotonically increasing real-valued function on the real line, and remember that α(−∞) and α(+∞) may be defined as extended real numbers in Subsection 2.4.2. If u ∈ (α(−∞), α(+∞)), then put

ψ_{1}(u) = sup{t_{1} ∈ R : α(t_{1}) < u}. (4.14.1)

This is related to arguments in the proofs of 12.29 on p393 of [14] and Proposition 21.2 on p269 of [20]. Note that there are t_{1} ∈ R such that α(t_{1}) < u under these conditions, and that the set of these t_{1}’s has an upper bound in R.

It is easy to see that ψ_{1} is monotonically increasing on (α(−∞), α(+∞)), by construction. If α is continuous and strictly increasing on R, then one can check that ψ_{1} is the inverse of α. This corresponds to a remark on p393 of [14], concerning Exercise 13 on p399 of [14], and a remark on p269 of [20]. Otherwise, ψ_{1} has some approximately similar properties.

4.14.1 Another approximate inverse ψ_{2}

Similarly, put ψ_{2}(u) = inf{t_{2} ∈ R : α(t_{2}) > u} (4.14.2)

for each u ∈ (α(−∞), α(+∞)). As before, there are t_{2} ∈ R such that α(t_{2}) > u, and the set of these t_{2}’s has a lower bound in R.

More precisely, if α(t_{1}) < u < α(t_{2}), (4.14.3)

then it is easy to see that t_{1} < t_{2}, (4.14.4)

because α is monotonically increasing on R. This implies that

ψ_{1}(u) ≤ ψ_{2}(u). (4.14.5)

We also have that ψ_{2} is monotonically increasing on (α(−∞), α(+∞)), and that ψ_{2} is the inverse of α when α is continuous and strictly increasing on R, as before.

4.14.2 Some properties of ψ_{1}, ψ_{2}

If r_{1} ∈ R and r_{1} < ψ_{1}(u), (4.14.6)

then α(r_{1}) < u, (4.14.7)

because α is monotonically increasing on R. Similarly, if r_{2} ∈ R and

ψ_{2}(u) < r_{2}, (4.14.8)

then α(r_{2}) > u. (4.14.9)

It follows that α(ψ_{1}(u)−) ≤ u ≤ α(ψ_{2}(u)+), (4.14.10)

using the notation for one-sided limits of α in Subsection 2.4.1.

If w_{1} ∈ R and ψ_{1}(u) < w_{1}, (4.14.11)

128 CHAPTER 4. MORE ON FUNCTIONS AND MEASURES then u ≤ α(w_{1}), (4.14.12) by the definition of ψ_{1}(u). Similarly, if w_{2} ∈ R and w_{2} < ψ_{2}(u), (4.14.13) then α(w_{2}) ≤ u, (4.14.14) by the definition of ψ_{2}(u). This implies that α(ψ_{2}(u)−) ≤ u ≤ α(ψ_{1}(u)+). (4.14.15)

If α is continuous at ψ_{1}(u), then α(ψ_{1}(u)) = u, (4.14.16) as in Subsection 2.4.2. Similarly, if α is continuous at ψ_{2}(u), then α(ψ_{2}(u)) = u. (4.14.17)

4.15 Some more properties of ψ_{1}

Let us continue with the same notation and hypotheses as in the previous section. Suppose that t ∈ R and α(t) ≠ α(−∞), α(+∞), so that α(t) ∈ (α(−∞), α(+∞)). (4.15.1) Observe that ψ_{1}(α(t)) = sup{t_{1} ∈ R : α(t_{1}) < α(t)} ≤ t, (4.15.2) because α is monotonically increasing on R. If v_{1} ∈ (α(−∞), α(t)], then ψ_{1}(v_{1}) ≤ ψ_{1}(α(t)) ≤ t, (4.15.3) because ψ_{1} is monotonically increasing. This means that (α(−∞), α(t)] ⊆ ψ_{1}^{−1}((−∞, t]). (4.15.4)

4.15.1 When α is right continuous on R

If α is right continuous at every point in R, then u ≤ α(ψ_{1}(u)) (4.15.5) for every u ∈ (α(−∞), α(+∞)), because of the second inequality in (4.14.15). If ψ_{1}(u) ≤ t, then it follows that u ≤ α(t), (4.15.6)

4.15. SOME MORE PROPERTIES OF ψ_{1} 129

because ψ_{1} is monotonically increasing. Thus

ψ_{1}^{−1}((−∞, t]) ⊆ (α(−∞), α(t)]. (4.15.7)

Combining this with (4.15.4), we obtain that

ψ_{1}^{−1}((−∞, t]) = (α(−∞), α(t)]. (4.15.8)

Note that this also works when α(t) = α(−∞), because of (4.15.5).

Clearly ψ_{1}(u) ≤ sup{t_{1} ∈ R : α(t_{1}) < α(+∞)} (4.15.9)

for every u ∈ (α(−∞), α(+∞)). Of course, the right side may be +∞, so that this is trivial. If t ∈ R and α(t) = +∞, then we get that

ψ_{1}(u) ≤ t (4.15.10)

for every u ∈ (α(−∞), α(+∞)), because α is monotonically increasing. This means that ψ_{1}^{−1}((−∞, t]) = (α(−∞), α(+∞)) (4.15.11)

in this case.

Note that ψ_{1} is measurable on (α(−∞), α(+∞)), because it is monotonically increasing, as in Subsection 3.6.2. This could also be obtained from (4.15.8) and (4.15.11).

4.15.2 When α is right continuous and α(−∞) = 0

Suppose now that α(−∞) = 0, in addition to α being right continuous at every point in R, as in the previous subsection. Observe that

(4.15.12) the Lebesgue measure of ψ_{1}^{−1}((−∞, t]) is equal to α(t)

for every t ∈ R, because of (4.15.8) and (4.15.11).

Let us consider (0, α(+∞)) as a measure space, using the restriction of Lebesgue measure to the Borel sets in (0, α(+∞)), as in Subsection 4.12. If α(+∞) < +∞, then (4.15.12) is the same as saying that α(t) is the distribution function of ψ_{1} as a measurable function on (0, α(+∞)). This corresponds to the “if” part of 12.29 on p393 of [14], and Proposition 21.2 on p269 of [20], with α(+∞) = 1.

Alternatively, if λ_{α} is the Lebesgue–Stieltjes measure on the real line corresponding to α, then α is the same as the distribution function of the identity function on the real line, considered as a measurable real-valued function on the real line as a measure space with respect to λ_{α}. One can also use ψ_{1} as another way to get λ_{α}. This is related to the remarks in Subsection 2.5.2 when α is continuous and strictly increasing.

4.15.3 More on ψ_{1} and ψ_{2}

Observe that (4.14.1) is basically the same as the definition of y_{0} in Subsection 2.5.3, with different notation.

Remember that for every u ∈ R, α^{−1}({u}) is an interval in the real line, or the empty set, as in Subsection 2.5.3. There are only finitely or countably many u ∈ R such that α^{−1}({u}) has at least two elements, as before.

Suppose for the moment that u ∈ (α(−∞), α(+∞)) and α^{−1}({u}) ≠ ∅. It is easy to see that

ψ_{1}(u) ≤ inf α^{−1}({u}) ≤ sup α^{−1}({u}) ≤ ψ_{2}(u). (4.15.13)

Let t_{0} be a real number, and suppose for the moment that α(t_{0}−) < α(t_{0}+). If α(t_{0}−) < u < α(t_{0}+), (4.15.14)

then one can check that ψ_{1}(u) = ψ_{2}(u) = t_{0}. (4.15.15)

Note that α(t_{0}−) ≤ u ≤ α(t_{0}+) (4.15.16)

when ψ_{1}(u) = t_{0} and when ψ_{2}(u) = t_{0}, as in (4.14.10) and (4.14.15).

The L^{p} norm, product measures, and more

5.1 The L^{p} norm

Let (X, A, μ) be a measure space, let p be a positive real number, and let f be a real or complex-valued measurable function on X. We say that f is pth power integrable on X with respect to μ if

∫_{X} |f|^{p} dμ < +∞. (5.1.1)

In this case, we put

∥f∥_{p} = ∥f∥_{L^{p}} = ∥f∥_{L^{p}(X)} = ∥f∥_{L^{p}(X,μ)} = ( ∫_{X} |f|^{p} dμ)^{1/p}, (5.1.2)

and we may interpret this as being equal to +∞ when f is not pth-power integrable on X. Of course, this is the same as in Section 4.6 when p = 1.

If t is a real or complex number, as appropriate, then

∥t f∥_{p} = |t| ∥f∥_{p}. (5.1.3)

5.1.1 Sums of pth-power integrable functions

Let g be another real or complex-valued measurable function on X that is pthpower integrable, and let us show that f + g is pth-power integrable on X as well. Observe that

(5.1.4) |f + g|^{p} ≤ (|f| + |g|)^{p} ≤ (2 max(|f|, |g|))^{p}
= 2^{p} max(|f|^{p}, |g|^{p}) ≤ 2^{p} (|f|^{p} + |g|^{p})

132 CHAPTER 5. THE L^{P} NORM, PRODUCT MEASURES, AND MORE

on X. This implies that

∫_{X} |f + g|^{p} dμ ≤ 2^{p} ∫_{X} |f|^{p} dμ + 2^{p} ∫_{X} |g|^{p} dμ. (5.1.5)

Thus the space of all real or complex-valued measurable functions on X that are pth-power integrable is a linear subspace of the space of all real or complexvalued measurable functions on X, as appropriate.

5.1.2 Minkowski’s inequality

If p ≥ 1, then Minkowski’s inequality states that

∥f + g∥_{p} ≤ ∥f∥_{p} + ∥g∥_{p}. (5.1.6)

This corresponds to 7.14 on p199 of [14], Proposition 15.3 on p143 of [20], 6.5 on p183 of [118], Theorem 13.7 on p191 of [159], and Theorem 3.9 on p68 of [298]. This also corresponds to Theorem 11.36 on p326 of [299] and part (iv) of Proposition 1.1 on p157 of [335] when p = 2.

5.1.3 A related inequality when p < 1

If 0 < p ≤ 1, then

∥f + g∥_{p}^{p} = ∫_{X} |f + g|^{p} dμ ≤ ∫_{X}(|f| + |g|)^{p} dμ
(5.1.7)  ≤ ∫_{X}(|f|^{p} + |g|^{p}) dμ = ∥f∥_{p}^{p} + ∥g∥_{p}^{p},

where the third step is as in Subsection A.10.1. This corresponds to Exercise 6 on p208 of [14] and (3) in 1.47 on p35 of [297], and it is mentioned in part (a) of Exercise 13.25 on p199 of [159]. A version of the inequality used in the third step in (5.1.7) is mentioned in part (a) of Exercise 17 on p76 of [298].

5.1.4 The corresponding metrics and norms

Note that for any p > 0, ∥f∥_{p} = 0 (5.1.8)

if and only if f = 0 almost everywhere on X with respect to μ. If p ≥ 1, then (5.1.2) determines a norm on the space of pth-power integrable real or complexvalued measurable functions on X if we identify two functions that are equal almost everywhere with respect to μ. In particular, this means that

∥f − g∥_{p} (5.1.9)

defines a metric on the space of pth-power integrable functions on X when we identify functions that are equal almost everywhere.

5.2. CONVERGENCE IN L^{P} 133

Similarly, if 0 < p ≤ 1, then

∥f − g∥_{p}^{p} (5.1.10)

defines a metric on the space of pth-power integrable real or complex-valued measurable functions on X if we identify two functions that are equal almost everywhere again. This corresponds to part of Exercise 16 on p187 of [118], part of part (b) of Exercise 13.25 on p200 of [159], part of Problem N on p55 of [189], part of 1.47 on p35 of [297], and to part of part (a) of Exercise 24 on p78 of [298]. As before, we shall not be too concerned about this right now, and use similar arguments in practice.

A pth-power integrable function on X is also said to be square integrable when p = 2. This case is discussed further in Section 5.5.

5.2 Convergence in L^{p}

Let (X, A, μ) be a measure space, and let p be a positive real number again. Let us say that a sequence {f_{j}}_{j=1}^{∞} of pth-power integrable real or complex-valued measurable functions on X converges in L^{p} to a pth-power integrable real or complex-valued measurable function f on X, as appropriate, if

∥f_{j} − f∥_{p} → 0 as j → ∞, (5.2.1)

as in Definition 10.1 on p83 of [20]. Equivalently, this means that

∥f_{j} − f∥_{p}^{p} = ∫_{X} |f_{j} − f|^{p} dμ → 0 as j → ∞. (5.2.2)

As when p = 1, one can check that if {f_{j}}_{j=1}^{∞} also converges in L^{p} to another pth-power integrable real or complex-valued measurable function f̃ on X, as appropriate, then f = f̃ almost everywhere on X with respect to μ.

Let us say that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p} if

∥f_{j} − f_{l}∥_{p} → 0 as j, l → ∞. (5.2.3)

This is equivalent to saying that

∥f_{j} − f_{l}∥_{p}^{p} = ∫_{X} |f_{j} − f_{l}|^{p} dμ → 0 as j, l → ∞. (5.2.4)

If {f_{j}}_{j=1}^{∞} converges in L^{p}, then one can verify that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p}.

5.2.1 Another version of Chebyshev’s inequality

If f is a pth-power integrable function on X and a is a positive real number, then another version of Chebyshev’s inequality states that

μ({x ∈ X : |f(x)| ≥ a}) ≤ a^{−p} ∫_{X} |f|^{p} dμ, (5.2.5) as in Lemma 10.4 on p85 of [20] and 6.17 on p193 of [118]. This can be shown in the same way as when p = 1, in Subsection 4.7.1, and one can also reduce to the p = 1 case.

If {f_{j}}_{j=1}^{∞} is a sequence of pth-power integrable functions on X that converges in L^{p} to a pth-power integrable function f on X, then

(5.2.6) {f_{j}}_{j=1}^{∞} converges to f in measure on X,

as in Proposition 10.5 on p85 of [20], and part (b) of Exercise 18 on p77 of [298]. Similarly, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p}, then

(5.2.7) {f_{j}}_{j=1}^{∞} is a Cauchy sequence in measure,

as in Subsection 4.7.4.

5.2.2 Absolutely convergent series in L^{p}

Let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that are pth-power integrable with respect to μ. Let us say that the corresponding infinite series ∑_{j=1}^{∞} a_{j} converges absolutely in L^{p} if p ≥ 1 and

∑_{j=1}^{∞} ∥a_{j}∥_{p} < +∞. (5.2.8)

If 0 < p ≤ 1, then let us say that ∑_{j=1}^{∞} a_{j} is p-summable in L^{p} when

∑_{j=1}^{∞} ∥a_{j}∥_{p}^{p} < +∞. (5.2.9)

Observe that

∥∑_{j=1}^{n} |a_{j}|∥_{p} ≤ ∑_{j=1}^{n} ∥a_{j}∥_{p} ≤ ∑_{j=1}^{∞} ∥a_{j}∥_{p} (5.2.10)

for every positive integer n when p ≥ 1, and that

∥∑_{j=1}^{n} |a_{j}|∥_{p}^{p} ≤ ∑_{j=1}^{n} ∥a_{j}∥_{p}^{p} ≤ ∑_{j=1}^{∞} ∥a_{j}∥_{p}^{p} (5.2.11)

when p ≤ 1. The monotone convergence theorem implies that

∥∑_{j=1}^{∞} |a_{j}|∥_{p} ≤ ∑_{j=1}^{∞} ∥a_{j}∥_{p} (5.2.12)

when p ≥ 1, and

∥∑_{j=1}^{∞} |a_{j}|∥_{p}^{p} ≤ ∑_{j=1}^{∞} ∥a_{j}∥_{p}^{p} (5.2.13)

when p ≤ 1.

If (5.2.8) holds when p ≥ 1, or (5.2.9) holds when p ≤ 1, then it follows that

∑_{j=1}^{∞} |a_{j}| (5.2.14)

is pth-power integrable on X with respect to μ. In particular, this means that (5.2.14) is finite almost everywhere with respect to μ on X.

5.2.3 Convergence of the partial sums in L^{p}

Thus ∑_{j=1}^{∞} a_{j}(x) converges absolutely for almost every x ∈ X with respect to μ. Let us interpret the sum as being equal to 0 at points where it does not converge, to get a function on X that is pth-power integrable with respect to μ. Note that

∥∑_{j=1}^{∞} a_{j}∥_{p} ≤ ∥∑_{j=1}^{∞} |a_{j}|∥_{p}. (5.2.15)

One can use the dominated convergence theorem to get that the sequence of partial sums ∑_{j=1}^{n} a_{j} converges to ∑_{j=1}^{∞} a_{j} in L^{p}.

Alternatively,

∥∑_{j=1}^{∞} a_{j} − ∑_{j=1}^{n} a_{j}∥_{p} = ∥ ∑_{j=n+1}^{∞} a_{j}∥_{p} ≤ ∑_{j=n+1}^{∞} ∥a_{j}∥_{p} (5.2.16)

when p ≥ 1, and

∥∑_{j=1}^{∞} a_{j} − ∑_{j=1}^{n} a_{j}∥_{p}^{p} = ∥ ∑_{j=n+1}^{∞} a_{j}∥_{p}^{p} ≤ ∑_{j=n+1}^{∞} ∥a_{j}∥_{p}^{p} (5.2.17)

when p ≤ 1, as before. The right sides of these inequalities tend to 0 as n → ∞ when (5.2.8) or (5.2.9) holds, as appropriate.

5.2.4 Convergence in L^{p} of Cauchy sequences in L^{p}

If {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p}, then it is well known that

(5.2.18) {f_{j}}_{j=1}^{∞} converges in L^{p},

as in 7.20 on p204 of [14], Theorem 15.4 on p145 of [20], Theorem 6.6 on p183 of [118], Theorem 13.11 on p192 of [159], and Theorem 3.11 on p69 of [298]. This corresponds to Theorem 11.42 on p329, Theorem 1.2 on p159 of [335], and a remark on p276 of [335] when p = 2. The case where p < 1 is mentioned in Exercise 9 on p208 of [14], Exercise 16 on p187 of [118], part (b) of Exercise 13.25 on p200 of [159], part of Problem M on p67 of [189], and in 1.47 on p35 of [297].

One can start with a subsequence {f_{j_{r}}}_{r=1}^{∞} of {f_{j}}_{j=1}^{∞} such that

∑_{r=1}^{∞} ∥f_{j_{r+1}} − f_{j_{r}}∥_{p} < +∞ (5.2.19)

when p ≥ 1, and

∑_{r=1}^{∞} ∥f_{j_{r+1}} − f_{j_{r}}∥_{p}^{p} < +∞ (5.2.20)

when p ≤ 1. Using the remarks in the previous two subsections,

f = f_{j_{1}} + ∑_{r=1}^{∞}(f_{j_{r+1}} − f_{j_{r}}) (5.2.21)

defines a pth-power integrable function on X, and {f_{j_{n}}}_{n=1}^{∞} converges to f pointwise almost everywhere on X and in L^{p}. It follows that {f_{j}}_{j=1}^{∞} converges to f in L^{p}, because {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p}, by a standard argument.

If {f_{j}}_{j=1}^{∞} converges in L^{p} to a pth-power integrable function f̃ on X, then {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{p}, so that we can use the previous argument. If f is as in the preceding paragraph, then f̃ = f almost everywhere on X with respect to μ, and thus {f_{j_{r}}}_{r=1}^{∞} converges to f̃ pointwise almost everywhere on X with respect to μ, as in 7.23 on p205 of [14], Exercise 10 on p208 of [14], and Theorem 3.12 on p70 of [298].

5.3 Hölder’s inequality

Let (X, A, μ) be a measure space, and let p, q be real numbers with p, q > 1. Suppose that p and q are conjugate exponents, in the sense that

1/p + 1/q = 1. (5.3.1)

Let f and g be real or complex-valued measurable functions on X such that f is pth-power integrable and g is qth-power integrable on X with respect to μ. Under these conditions, Hölder’s inequality states that

(5.3.2) f g is integrable on X with respect to μ,

with ∥f g∥_{1} ≤ ∥f∥_{p} ∥g∥_{q}. (5.3.3)

This corresponds to 7.9 on p196 of [14], Proposition 15.1 on p142 of [20], 6.2 on p182 of [118], Theorem 13.4 on p190 of [159], and Theorem 3.8 on p67 of [298].

If p = q = 2, then (5.3.1) holds, and (5.3.3) is known as the Cauchy–Schwarz inequality. This case is also mentioned in Theorem 11.35 on p326 of [299], and in part (ii) of Proposition 1.1 on p157 of [335].

5.3.1 A bit of the proof

If a and b are nonnegative real numbers, then it is well known that

a b ≤ a^{p}/p + b^{q}/q. (5.3.4)

This is simpler when p = q = 2, using the fact that (a−b)^{2} ≥ 0. Otherwise, this is often obtained using calculus, or the convexity of the exponential function on the real line.

Using (5.3.4), we get that

∫_{X} |f| |g| dμ ≤ p^{−1} ∫_{X} |f|^{p} dμ + q^{−1} ∫_{X} |g|^{q} dμ. (5.3.5)

One can use this to get (5.3.3) when ∥f∥_{p} = ∥g∥_{q} = 1, and it is not too difficult to reduce to that case.

Hölder’s inequality is often used to obtain Minkowski’s inequality when p is strictly larger than 1. This is simpler when p = 2, and there are analogous arguments for inner products more broadly in that case.

5.3.2 An extension of Hölder’s inequality

Let p_{1}, q_{1}, and r_{1} be positive real numbers such that

1/p_{1} + 1/q_{1} = 1/r_{1}. (5.3.6)

Also let f_{1} and g_{1} be real or complex-valued measurable functions on X such that f_{1} is p_{1}th-power integrable and g_{1} is q_{1}th-power integrable on X with respect to μ. One can use Hölder’s inequality to get that f_{1} g_{1} is r_{1}th-power integrable on X, with ∥f_{1} g_{1}∥_{r_{1}} ≤ ∥f_{1}∥_{p_{1}} ∥g_{1}∥_{q_{1}}. (5.3.7)

This corresponds to Exercise 7 on p200 of [14] and Exercise 31 on p196 of [118] with n = 2, and it is related to Exercise 4 on p73 of [298].

5.3.3 Using Hölder’s inequality when μ(X) is finite

If p_{1} and r_{1} are positive real numbers with p_{1} < r_{1}, then (5.3.6) holds for some q_{1} > 0. Let f_{1} be a real or complex-valued measurable function on X that is p_{1}th-power integrable again. If μ(X) is finite, then we can take g_{1} ≡ 1 on X to get that f_{1} is r_{1}th-power integrable on X with respect to μ, with

∥f_{1}∥_{r_{1}} ≤ ∥f_{1}∥_{p_{1}} μ(X)^{1/q_{1}}, (5.3.8)

as in (5.3.7). This corresponds to Proposition 6.12 on p186 of [118], Theorem 13.17 on p196 of [159], and parts of Exercise 5 on p73 of [298]. This also corresponds to part of Exercise 10 on p333 of [299] when p_{1} = 2 and r_{1} = 1.

5.3.4 Another extension of Hölder’s inequality

Let p_{1}, …, p_{n} be finitely many positive real numbers, and let r_{0} be the positive real number such that

∑_{j=1}^{n} 1/p_{j} = 1/r_{0}. (5.3.9)

If f_{1}, …, f_{n} are real or complex-valued measurable functions on X such that f_{j} is p_{j}th-power integrable with respect to μ for j = 1, …, n, then

(5.3.10) ∏_{j=1}^{n} f_{j} is r_{0}th-power integrable on X with respect to μ,

with

∥∏_{j=1}^{n} f_{j}∥_{r_{0}} ≤ ∏_{j=1}^{n} ∥f_{j}∥_{p_{j}}. (5.3.11)

This is the same as before when n = 2, and otherwise one can repeat the process. This corresponds to Exercise 8 on p200 of [14], Exercise 23.1 on p342 of [20], Exercise 31 on p196 of [118], and Exercise 13.26 on p200 of [159].

5.4 Essentially bounded functions

Let (X, A, μ) be a measure space again. A real or complex-valued measurable function f on X is said to be essentially bounded with respect to μ if there is a nonnegative real number b such that

|f(x)| ≤ b (5.4.1)

for almost every x ∈ X with respect to μ. In this case,

∥f∥_{∞} = ∥f∥_{L^{∞}} = ∥f∥_{L^{∞}(X)} = ∥f∥_{L^{∞}(X,μ)} (5.4.2)

is defined to be the infimum of the set of b ≥ 0 with this property.

One can check that this infimum is always attained, using the fact that the union of a sequence of measurable sets with measure 0 with respect to μ also has measure 0 with respect to μ. This means that (5.4.1) holds with b = ∥f∥_{∞} almost everywhere on X with respect to μ.

It is sometimes convenient to interpret ∥f∥_{∞} as being +∞ when f is not essentially bounded on X, as usual. Observe that

(5.4.3) ∥f∥_{∞} is the same as the usual supremum norm when μ is counting measure on X.

5.4.1 More on the L^{∞} norm

If f is essentially bounded on X and t is a real or complex number, as appropriate, then it is easy to see that t f is essentially bounded on X too, with

∥t f∥_{∞} = |t| ∥f∥_{∞}. (5.4.4)

If g is another real or complex-valued measurable function on X that is essentially bounded with respect to μ, then one can check that f + g is essentially bounded on X with respect to μ as well, with

∥f + g∥_{∞} ≤ ∥f∥_{∞} + ∥g∥_{∞}. (5.4.5)

Thus the space of all real or complex-valued measurable functions on X that are essentially bounded with respect to μ is a linear subspace of the space of all real or complex-valued measurable functions on X, as appropriate.

Of course, ∥f∥_{∞} = 0 if and only if f = 0 almost everywhere on X with respect to μ. As usual, ∥·∥_{∞} determines a norm on the space of real or complexvalued measurable functions on X that are essentially bounded with respect to μ if we identify two functions that are equal almost everywhere on X with respect to μ.

5.4.2 Convergence in L^{∞}

Let us say that a sequence {f_{j}}_{j=1}^{∞} of essentially bounded real or complex-valued measurable functions on X converges in L^{∞} to an essentially bounded real or complex-valued function f on X, as appropriate, if

∥f_{j} − f∥_{∞} → 0 as j → ∞. (5.4.6)

One can check that this is equivalent to uniform onvergence on the complement of a measurable set in X with measure 0 with respect to μ. In particular, this implies convergence in measure.

Let us say that {f_{j}}_{j=1}^{∞} is a Cauchy sequence in L^{∞} if

∥f_{j} − f_{l}∥_{∞} → 0 as j, l → ∞. (5.4.7)

One can check that this is equivalent to having a Cauchy sequence with respect to the supremum norm on the complement of a measurable set in X with measure 0 with respect to μ. Thus convergence in L^{∞} follows from the analogous statement for the supremum norm.

5.4.3 More on essentially bounded functions

Let p_{1} be a positive real number, and suppose that f_{1} and g_{1} are real or complexvalued measurable functions on X with f_{1} p_{1}th-power integrable and g_{1} essentially bounded with respect to μ. Under these conditions, one can check that f_{1} g_{1} is p_{1}th-power integrable on X with respect to μ, with

∥f_{1} g_{1}∥_{p_{1}} ≤ ∥f_{1}∥_{p_{1}} ∥g_{1}∥_{∞}. (5.4.8)

This is the analogue of (5.3.7) with q_{1} = ∞ and r_{1} = p_{1}.

If f is also essentially bounded on X with respect to μ, then it is easy to see that f g is essentially bounded on X with respect to μ, with

∥f_{1} g_{1}∥_{∞} ≤ ∥f_{1}∥_{∞} ∥g_{1}∥_{∞}. (5.4.9)

Of course, this is the analogue of (5.4.8) with p_{1} = ∞.

Let r_{1} be a positive real number, and suppose that f_{1} is essentially bounded on X again. If μ(X) < +∞, then f_{1} is r_{1}th-power integrable on X with respect to μ, with ∥f_{1}∥_{r_{1}} ≤ ∥f_{1}∥_{∞} μ(X)^{1/r_{1}}. (5.4.10)

This is the analogue of (5.3.8) with p_{1} = ∞ and q_{1} = r_{1}.

5.5 Integral inner products

Let (X, A, μ) be a measure space, and let f and g be real or complex-valued measurable functions on X that are square-integrable with respect to μ. This implies that |f| |g| is integrable on X, as in Section 5.3. Put

⟨f, g⟩ = ∫_{X} f g dμ (5.5.1)

when f and g are real-valued, and

⟨f, g⟩ = ∫_{X} f \overline{g} dμ (5.5.2)

when they are complex-valued. Note that

⟨f, g⟩ = ⟨g, f⟩ (5.5.3)

when f and g are real-valued, and that

⟨f, g⟩ = \overline{⟨g, f⟩} (5.5.4)

when they are complex-valued. In both cases, we have that

⟨f, f⟩ = ∥f∥_{2}^{2}. (5.5.5)

These are almost like inner products on the spaces of real and complexvalued square-integrable functions on X with respect to μ, because (5.5.5) is equal to 0 if and only if f = 0 almost everywhere on X with respect to μ. As usual, one can get actual inner products by identifying two functions that are equal almost everywhere on X with respect to μ. This is related to an example on p213 of [14], Example 19.5 on p199 of [20], the example on p172 of [118], Example 13.15 on p195 of [159], part of Problem M on p67 of [189], Example 4.5 (b) on p81 of [298], and some remarks on p157, 276 of [335].

5.5.1 Orthogonal functions

Let f and g be real or complex-valued measurable functions on X that are square-integrable with respect to μ again. If

⟨f, g⟩ = 0, (5.5.6)

then f and g are said to be orthogonal with respect to (5.5.1) or (5.5.2), as appropriate. In this case, we get that

∥f + g∥_{2}^{2} = ⟨f + g, f + g⟩ = ⟨f, f⟩ + ⟨g, g⟩ = ∥f∥_{2}^{2} + ∥g∥_{2}^{2}. (5.5.7)

Similarly, let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that are square-integrable with respect to μ and pairwiseorthogonal, so that ⟨f_{j}, f_{l}⟩ = 0 (5.5.8)

when j ≠ l. This implies that

∥∑_{j=l}^{n} f_{j}∥_{2}^{2} = ∑_{j=l}^{n} ∥f_{j}∥_{2}^{2} (5.5.9)

for all integers n ≥ l ≥ 1, as before. If

∑_{j=1}^{∞} ∥f_{j}∥_{2}^{2} < +∞, (5.5.10)

then the sequence of partial sums ∑_{j=1}^{n} f_{j} is a Cauchy sequence in L^{2}, as in Section 5.2. This implies that the sequence of partial sums converges in L^{2}, as in Subsection 5.2.4. Note that (5.5.10) is less restrictive than absolute convergence in L^{2}, as in Subsection 5.2.2.

5.6 Products of outer measures

Let X and Y be sets, and let μ_{X} and μ_{Y} be outer measures on X and Y, respectively, as in Subsection 1.10.4. If E ⊆ X × Y, then let μ_{X×Y} (E) be the infimum of the sums

∑_{j=1}^{∞} μ_{X}(A_{j}) μ_{Y} (B_{j}), (5.6.1)

where the A_{j}’s are measurable subsets of X with respect to μ_{X}, the B_{j}’s are measurable subsets of Y with respect to μ_{Y}, and

E ⊆ ⋃_{j=1}^{∞}(A_{j} × B_{j}). (5.6.2)

Remember that measurability of a set with respect to an outer measure was defined in Subsection 1.11.1.

This definition of μ_{X×Y} (E) is mentioned on p22 of [110], beginning on p114 of [113], and on p14 of [247]. Note that the usual convention that the product of 0 and +∞ is equal to 0 is used in (5.6.1).

One can check that

(5.6.3) μ_{X×Y} defines an outer measure on X × Y.

This may be considered as a class of examples of the construction in Subsection 1.11.3. Let us call μ_{X×Y} the product outer measure on X × Y associated to μ_{X} and μ_{Y}.

If A ⊆ X and B ⊆ Y are measurable with respect to μ_{X} and μ_{Y}, respectively, then μ_{X×Y} (A × B) ≤ μ_{X}(A) μ_{Y} (B), (5.6.4)

by construction. In fact, μ_{X×Y} is the largest outer measure on X × Y with this property, as mentioned on p115 of [113].

It is well known that A × B is measurable with respect to μ_{X×Y} under these conditions, with μ_{X×Y} (A × B) = μ_{Y} (A) μ_{Y} (B), (5.6.5)

as in part (ii) of Theorem 1 on p22 of [110], and part (2) of 2.6.2 on p115 of [113]. One can also show that μ_{X×Y} is regular as an outer measure on X × Y, in the sense discussed in Section 2.7, as in part (i) of Theorem 1 on p22 of [110], and part (1) of 2.6.2 on p115 of [113].

5.6.1 Countable measurability and σ-finiteness

Let Z be a set, and let μ_{Z} be an outer measure on Z. A subset E of Z is said to be countably measurable with respect to μ_{Z} if E can be expressed as the union of a sequence of measurable sets with respect to μ_{Z} that also have finite measure. This is the first part of 2.3.4 on p76 of [113]. One may also say that E is σ-finite with respect to μ in this case, as on p4 of [110]. This is a bit different from the way that this term was used in Subsections 2.1.2 and 2.7.2.

Some additional properties of the product outer measure in Theorem 1 on p22 of [110] and 2.6.2 on p115 of [113] involve subsets of X×Y that are countably measurable with respect to μ_{X×Y}. We shall consider slightly different constructions of product measures here, in the next sections, with similar properties.

5.7 Elementary families of sets

Let Z be a set, and let E be a collection of subsets of Z. Let us say that E is an elementary family if it has the following three properties, as on p23 of [118]. First, the empty set is an element of E. Second, if E_{1}, E_{2} ∈ E, then

E_{1} ∩ E_{2} ∈ E. (5.7.1)

Third, if E ∈ E, then Z \ E may be expressed as the union of finitely many pairwise-disjoint elements of E.

Suppose that E is an elementary family of subsets of Z, and let A be the collection of subsets of Z that can be expressed as the union of finitely many pairwise-disjoint elements of E. Under these conditions, Proposition 1.7 on p23 of [118] says that (5.7.2) A is an algebra of subsets of Z.

In the proof, one starts by showing that if A, B ∈ E, then A \ B ∈ A. This implies that A ∪ B = (A \ B) ∪ B ∈ A. (5.7.3)

One can use induction to show that the union of finitely many elements of E is an element of A, as in [118].

It follows that A is the same as the collection of subsets of Z that can be expressed as the union of finitely many elements of E, and in particular that the union of finitely many elements of A is an element of A. One can also check that the complement of an element of A is in A as well, as in [118].

5.7.1 Products of two measurable spaces

Let (X, A_{X}) and (Y, A_{Y}) be measurable spaces. A subset of X × Y of the form A × B, with A ∈ A_{X} and B ∈ A_{Y}, is called a measurable rectangle.

Let A_{X×Y} be the σ-algebra of subsets of X ×Y generated by the collection of measurable rectangles, as in Section 2.2. This is sometimes denoted A_{X} × A_{Y}, as on p90 of [20], in 21.2 on p379 of [159], and p145 of [298], and it is denoted A_{X} ⊗A_{Y} in Definition 5.2 on p117 of [14], and on p22 of [118]. This is the same as the σ-algebra defined on p277 of [335], using different notation.

5.7.2 The obvious mappings from X × Y into X, Y

Let p_{X} and p_{Y} be the obvious mappings from X ×Y into X and Y, respectively, with p_{X}((x, y)) = x and p_{Y} ((x, y)) = y (5.7.4)

for every (x, y) ∈ X × Y. Of course, p_{X} and p_{Y} map X × Y onto X and Y, respectively, when X, Y ≠ ∅.

Note that p_{X}^{−1}(A) = A × Y and p_{Y}^{−1}(B) = X × B (5.7.5)

for every A ⊆ X and B ⊆ Y. More precisely, A_{X×Y} is defined in [118] as the σ-algebra of subsets of X × Y generated by measurable rectangles of the forms in (5.7.5) with A ∈ A_{X} and B ∈ A_{Y}. It is easy to see that this is the same as the σ-algebra generated by all measurable rectangles.

Clearly p_{X} and p_{Y} are measurable with respect to A_{X×Y}. In fact, A_{X×Y} is the smallest σ-algebra of subsets of X ×Y with this property, as in the preceding paragraph.

5.7.3 More on measurable rectangles

The collection of measurable rectangles obviously satisfies the first condition in the definition of an elementary family. If A_{1} × B_{1} and A_{2} × B_{2} are measurable rectangles, then

(A_{1} × B_{1}) ∩ (A_{2} × B_{2}) = (A_{1} ∩ A_{2}) × (B_{1} ∩ B_{2}) (5.7.6)

is a measurable rectangle. In order to get the third condition, observe that

(X × Y) \ (A × B) = ((X \ A) × B) ∪ ((X \ A) × (Y \ B)) ∪ (A × (Y \ B))
(5.7.7)  = ((X \ A) × Y) ∪ (A × (Y \ B))
= ((X \ A) × B) ∪ (X × (Y \ B)).

Any of the three expressions on the right sides would work here. This corresponds to a remark on p64 of [118], although similar arguments are used in other treatments.

This implies that the collection A_{X×Y,0} of subsets of X × Y that can be expressed as the union of finitely many pairwise-disjoint measurable rectangles is an algebra of subsets of X × Y, as in [118]. This also corresponds to 5.13 on p121 of [14], a remark on p90 of [20], Theorem 21.3 on p379 of [159], and a remark on p277 of [335]. This collection is called the class of elementary sets on p145 of [298].

5.7.4 Monotone classes and measurable rectangles

Note that A_{X×Y} is the same as the σ-algebra of subsets of X × Y generated by A_{X×Y,0}. Because A_{X×Y,0} is an algebra of subsets of X × Y, it follows that A_{X×Y} is the same as the monotone class of subsets of X × Y generated by A_{X×Y,0}, as in Subsection 2.6.1. This corresponds to Corollary 21.7 on p381 of [159], and Theorem 7.3 on p146 of [298].

5.8 More on A_{X×Y}

Let (X, A_{X}) and (Y, A_{Y}) be measurable spaces again, and let E_{X} and E_{Y} be collections of subsets of X and Y that generate A_{X} and A_{Y} as σ-algebras, respectively, as in Section 2.2. Consider the σ-algebra Ã_{X×Y} of subsets of X × Y generated by measurable rectangles of the forms in (5.7.5), with A ∈ E_{X} and B ∈ E_{Y}. Proposition 1.4 on p23 of [118] implies that

Ã_{X×Y} = A_{X×Y} . (5.8.1)

Of course, Ã_{X×Y} ⊆ A_{X×Y} (5.8.2)

automatically. Note that

{A ⊆ X : p_{X}^{−1}(A) = A × Y ∈ Ã_{X×Y} } (5.8.3)

5.8. MORE ON A_{X×Y} 145

and {B ⊆ Y : p_{Y}^{−1}(B) = X × B ∈ Ã_{X×Y} } (5.8.4)

are σ-algebras of subsets of X and Y, respectively, as mentioned near the beginning of Section 3.1. These σ-algebras contain E_{X} and E_{Y}, respectively, by construction, and thus contain A_{X} and A_{Y}, respectively, by hypothesis. This implies that A_{X×Y} ⊆ Ã_{X×Y}.

5.8.1 When X and Y are metric or topological spaces

If X and Y are metric spaces, then X × Y may be considered as a metric space, using a suitable product metric, as in Section A.4. Similarly, if X and Y are topological spaces, then X × Y may be considered as a topological space, using the corresponding product topology. Suppose for the moment that A_{X} and A_{Y} are the corresponding σ-algebras of Borel sets in X and Y, respectively, so that we can take E_{X} and E_{Y} to be the collections of open sets in X and Y, respectively. In this case, (5.8.1) implies that

(5.8.5) A_{X×Y} is contained in the σ-algebra of Borel sets in X × Y,

because products of open sets in X and Y are open sets in X × Y. This corresponds to part of Proposition 1.5 on p23 of [118].

Suppose that there are bases for the topologies of X and Y with only finitely or countably many elements, as in Section 2.3. This implies that there is a base for the product topology on X × Y with only finitely or countably many elements, and whose elements are products of open sets in X and Y, as in Subsection A.14.1. It follows in particular that every open set in X × Y can be expressed as the union of finitely or countably many products of open sets in X and Y. This means that

(5.8.6) open sets in X × Y are elements of A_{X×Y} ,

so that A_{X×Y} contains all of the Borel sets in X × Y. Thus

(5.8.7) A_{X×Y} is the collection of all Borel sets in X × Y

under these conditions, as in Proposition 1.5 on p23 of [118], Exercise 21.19 on p391 of [159], and Exercise 11 on p161 of [298].

In particular, if X = Y = R with the standard Euclidean metric, then one can take X × Y = R^{2} to be equipped with the standard Euclidean metric.

5.8.2 Measurability of mappings into X × Y

Let (X, A_{X}) and (Y, A_{Y}) be any measurable spaces again, and let (W, A_{W}) be another measurable space. It is well known that a mapping ϕ from W into X × Y is measurable with respect to A_{X×Y} on X × Y if and only if

(5.8.8) the compositions of ϕ with p_{X} and p_{Y} are measurable as mappings from W into X and Y, respectively, as in Proposition 2.4 on p44 of [118]. Of course, measurability of ϕ implies that p_{X} ∘ ϕ and p_{Y} ∘ ϕ are measurable, as in Subsection 3.1.3.

Conversely, if p_{X} ∘ ϕ and p_{Y} ∘ ϕ are measurable, then

(p_{X} ∘ ϕ)^{−1}(A) = ϕ^{−1}(p_{X}^{−1}(A)), (p_{Y} ∘ ϕ)^{−1}(B) = ϕ^{−1}(p_{Y}^{−1}(B)) ∈ A_{W} (5.8.9)

for all A ∈ A_{X} and B ∈ A_{Y}. Note that

{E ⊆ X × Y : ϕ^{−1}(E) ∈ A_{W} } (5.8.10)

is a σ-algebra of subsets of X × Y, as mentioned near the beginning of Section 3.1. This σ-algebra contains p_{X}^{−1}(A) and p_{Y}^{−1}(B) for all A ∈ A_{X} and B ∈ A_{Y}, as before. It follows that this σ-algebra contains A_{X×Y}, as in Subsection 5.7.2. This means that ϕ is measurable with respect to A_{X×Y} on X × Y.

5.8.3 Sections of subsets of X × Y

If E ⊆ X × Y, x ∈ X, and y ∈ Y, then put

E_{x} = {y′ ∈ Y : (x, y′) ∈ E} and E^{y} = {x′ ∈ X : (x′, y) ∈ E}. (5.8.11)

It is well known that E_{x} ∈ A_{Y} and E^{y} ∈ A_{X} (5.8.12)

for every x ∈ X and y ∈ Y when E ∈ A_{X×Y}, as in 5.6 on p118 of [14], part (1) of Lemma 11.1 on p90 of [20], part a of Proposition 2.34 on p65 of [118], Theorem 21.4 on p380 of [159], and Theorem 7.2 on p146 of [298].

Although this can be obtained fairly directly, one can also use the remarks in the preceding subsection. Let ϕ_{x} and ψ_{y} be the mappings from X and Y into X × Y, respectively, defined by

ϕ_{x}(y′) = (x, y′) and ψ_{y}(x′) = (x′, y) (5.8.13)

for all x′ ∈ X and y′ ∈ Y. It is easy to see that ϕ_{x} and ψ_{y} are measurable with respect to A_{X×Y}, because their compositions with p_{X} and p_{Y} are measurable. Clearly E_{x} = ϕ_{x}^{−1}(E) and E^{y} = ψ_{y}^{−1}(E). (5.8.14)

This implies (5.8.12) when E ∈ A_{X×Y}.

5.8.4 Sections of functions on X × Y

Let (Z, A_{Z}) be another measurable space, and let f be a function on X × Y with values in Z. If x ∈ X and y ∈ Y, then let f_{x} and f^{y} be the functions on X and Y, respectively, with values in Z defined by

f_{x}(y′) = f((x, y)) and f^{y}(x′) = f((x′, y)) (5.8.15)

for all x′ ∈ X and y′ ∈ Y. It is well known that

(5.8.16) f_{x} and f^{y} are measurable for every x ∈ X and y ∈ Y when f is measurable with respect to A_{X×Y} on X ×Y, as in 5.9 on p119 of [14], part (2) of Lemma 11.1 on p90 of [20], part b of Proposition 2.34 on p65 of [118], Theorem 21.5 on p380 of [159], and Theorem 7.5 on p147 of [298].

As before, this can be obtained fairly directly, or by observing that

f_{x} = f ∘ ϕ_{x} and f^{y} = f ∘ ψ_{y}. (5.8.17)

Thus (5.8.16) follows from the measurability of ϕ_{x} and ψ_{y} when f is measurable, as in Subsection 3.1.3.

5.9 Product measures

Let (X, A_{X}, μ) and (Y, A_{Y} , ν) be σ-finite measure spaces, and let A_{X×Y} be as in Subsection 5.7.1. If E ∈ A_{X×Y}, then it is well known that

(5.9.1) ν(E_{x}) and μ(E^{y}) are measurable

as nonnegative extended real-valued functions of x ∈ X and y ∈ Y. This corresponds to 5.20 on p124 of [14], part (1) of Proposition 11.2 on p91 of [20], parts (i) and (ii) of Theorem 21.8 on p381 of [159], and part of Theorem 7.6 on p148 of [298].

It is also well known that

∫_{X} ν(E_{x}) dμ(x) = ∫_{Y} μ(E^{y}) dν(y), (5.9.2)

as in part (2) of Proposition 11.2 on p91 of [20], part (iii) of Theorem 21.8 on p381 of [159], and another part of Theorem 7.6 on p148 of [298].

One can define the product measure μ × ν on X × Y with respect to A_{X×Y} by

(μ × ν)(E) = ∫_{X} ν(E_{x}) dμ(x) = ∫_{Y} μ(E^{y}) dν(y), (5.9.3)

as on p93 of [20], Definition 21.9 on p384 of [159], and Definition 7.7 on p149 of [298]. Definition 5.25 on p126 of [14] only uses the first equality in (5.9.3), and the second equality may be obtained from Tonelli’s theorem 5.28 on p129 of [14].

The fact that μ × ν is a measure corresponds to 5.27 on p127 of [14], some remarks on p93 of [20], part of Theorem 21.10 on p384 of [159], and a remark on p149 of [298]. We also have that μ × ν is σ-finite on X × Y, as in Exercise 9 on p128 of [14], another part of Theorem 21.10 on p384 of [159], and another remark on p149 of [298].

If A ∈ A_{X} and B ∈ A_{Y}, then

(μ × ν)(A × B) = μ(A) ν(B), (5.9.4)

where the product of 0 and +∞ is considered to be 0, as before. This is Example 5.26 on p127 of [14], and it mentioned in the proofs of (5.9.2) in [20, 159, 298].

5.9.1 Another approach to μ × ν

Remember that the collection A_{X×Y,0} of unions of finitely many pairwisedisjoint measurable rectangles is an algebra of subsets of X ×Y, as in Subsection 5.7.3. If E ∈ A_{X×Y,0} is expressed as the union of finitely many pairwise-disjoint measurable rectangles, then one would like to define (μ × ν)(E) to be the corresponding sum of the measures of the measurable rectangles, using (5.9.4).

To do this, one should check that this sum does not depend on the way that E is expressed as the union of finitely many pairwise-disjoint measurable rectangles, so that (μ × ν)(E) is well-defined in this way. One would also like to verify that μ × ν is countably additive on A_{X×Y,0}. This is the approach used on p64 of [118], and on p277 of [335].

If E ∈ A_{X×Y,0}, then it is easy to see directly that (5.9.1) and (5.9.2) hold. This is simpler for measurable rectangles, and one can use that to get the same result for any element of A_{X×Y,0}. More precisely, one gets measurable simple functions in (5.9.1) when E ∈ A_{X×Y,0}. In this case, (5.9.3) is the same as the preliminary definition of (μ × ν)(E) mentioned in the previous two paragraphs, and it does not depend on the way that E is expressed as a union of finitely many pairwise-disjoint measurable rectangles. One can also use this to get that this defines a countably additive measure on A_{X×Y,0}, as in [118, 335].

This permits one to extend μ × ν to a countably additive measure on A_{X×Y} as in Subsection 1.11.4. This is the way that μ × ν is defined on A_{X×Y} in [118, 335]. The equivalence of this with (5.9.3) is shown in Theorem 2.36 on p66 of [118], and Proposition 3.2 on p278 of [335].

5.9.2 Integration on X × Y

Let f be a nonnegative extended real-valued function on X × Y that is measurable with respect to A_{X×Y}. Tonelli’s theorem states that

x ↦ ∫_{Y} f(x, y) dν(y) (5.9.5)

is measurable on X with respect to A_{X},

y ↦ ∫_{X} f(x, y) dμ(x) (5.9.6)

is measurable on Y with respect to A_{Y}, and

(5.9.7)  ∫_{X×Y} f d(μ × ν) = ∫_{X} ( ∫_{Y} f(x, y) dν(y)) dμ(x)
= ∫_{Y} ( ∫_{X} f(x, y) dμ(x)) dν(y).

5.10. PRODUCTS OF N MEASURABLE SPACES 149

This is 5.28 on p129 of [14], part of Theorem 11.3 beginning on p93 of [20], part a of 2.37 on p67 of [118], Theorem 21.12 beginning on p384 of [159], and part (a) of Theorem 7.8 on p150 of [298]. Theorem 3.2 beginning on p80 of [335] is a version of this using Lebesgue measure on Euclidean spaces.

Now let f be a real or complex-valued function on X × Y that is measurable with respect to A_{X×Y} and integrable with respect to μ × ν. Fubini’s theorem states that ∫_{Y} |f(x, y)| dν(y) < +∞ (5.9.8)

for almost every x ∈ X with respect to μ,

∫_{X} |f(x, y)| dμ(x) < +∞ (5.9.9)

for almost every y ∈ Y with respect to ν, that (5.9.5) and (5.9.6) are measurable and in fact integrable with respect to μ and ν, respectively, and that (5.9.7) holds. More precisely, the integrals in (5.9.5) and (5.9.6) are defined when (5.9.8) and (5.9.9) hold, respectively, and otherwise they can be taken to be 0. This corresponds to 5.32 on p132 of [14], another part of Theorem 11.3 beginning on p93 of [20], part b of 2.37 on p67 of [118], 21.13 on p386 of [159], part (c) of Theorem 7.8 on p150 of [298], and Theorem 3.3 on p279 of [335]. Theorem 3.1 on p76 of [335] is a version of this using Lebesgue measure on Euclidean spaces.

5.9.3 Uniqueness of the product measure

Let us mention that μ × ν is the unique measure on X × Y with respect to A_{X×Y} that satisfies (5.9.4). It is easy to see that any such measure is σ-finite, because μ and ν are σ-finite, by hypothesis. It is also clear that any such measure is uniquely determined on A_{X×Y,0} by (5.9.4). Uniqueness on A_{X×Y} may be obtained as in Subsection 2.6.4 and as in Section 2.9. This corresponds to Exercise 10 on p128 of [14], Exercise 11.17 on p100 of [20], a remark on p64 of [118], 21.11 on p384 of [159], and Exercise 7 on p160 of [298].

5.10 Products of n measurable spaces

Let n be a positive integer, and let (X_{j}, A_{X_{j}}) be a measurable space for j = 1, …, n. Put X = ∏_{j=1}^{n} X_{j}, and let p_{l} be the obvious mapping from X into X_{l} for each l = 1, …, n, so that p_{l}(x) = x_{l} (5.10.1)

for each l. More precisely, p_{l} maps X onto X_{l} for each l as long as X_{j} ≠ ∅ for each j. If A_{l} ⊆ X_{l}, then

p_{l}^{−1}(A_{l}) = ∏_{j=1}^{n} A_{j,l}, (5.10.2)

where A_{j,l} = X_{j} when j ≠ l, and A_{l,l} = A_{l}. If A_{l} ⊆ X_{l} for each l, then

∏_{l=1}^{n} A_{l} = ⋂_{l=1}^{n} p_{l}^{−1}(A_{l}), (5.10.3)

Let A_{X} be the σ-algebra of subsets of X generated by sets of the form (5.10.3), with A_{l} ∈ A_{X_{l}} for each l. This corresponds to part of 22.2 beginning on p429 of [159] for finite products, and of course is the same as in Subsection 5.7.1 when n = 2. Equivalently, we can take A_{X} to be the σ-algebra of subsets of X generated by sets of the form p_{l}^{−1}(A_{l}), with A_{l} ∈ A_{X_{l}} and 1 ≤ l ≤ n, as on p22 of [118], and as in Subsection 5.7.2 when n = 2. This σ-algebra may be denoted ⨂_{l=1}^{n} A_{X_{l}} or A_{1} ⊗ ⋯ ⊗ A_{n}, as in [118].

Thus p_{l} is measurable with respect to A_{X} on X for each l, and A_{X} is the smallest σ-algebra on X with this property, as in Subsection 5.7.2 when n = 2.

5.10.1 Collections of subsets of X_{j} that generate A_{X_{j}}

Let E_{X_{j}} be a collection of subsets of X_{j} that generates A_{X_{j}} as a σ-algebra for each j, and let Ã_{X} be the σ-algebra of subsets of X generated by sets of the form p_{l}^{−1}(A_{l}), with A_{l} ∈ E_{X_{l}} and 1 ≤ l ≤ n. Under these conditions, Proposition 1.4 on p23 of [118] implies that Ã_{X} = A_{X}. (5.10.4)

This was discussed at the beginning of Section 5.8 when n = 2.

As before, {A_{l} ⊆ X_{l} : p_{l}^{−1}(A_{l}) ∈ Ã_{X}} (5.10.5)

is a σ-algebra of subsets of X_{l} that contains E_{X_{l}} and thus A_{X_{l}} for each l. This implies that A_{X} ⊆ Ã_{X}, and the opposite inclusion follows directly from the definitions.

5.10.2 When the X_{j}’s are metric or topological spaces

If the X_{j}’s are metric spaces, then X may be considered as a metric space with respect to a suitable product metric, as in Section A.4. Similarly, if the X_{j}’s are topological spaces, then X may be considered as a topological space, using the corresponding product topology. Suppose for the moment that A_{X_{j}} is the σ-algebra of Borel sets in X_{j} for each j, so that we can take E_{X_{j}} to be the collection of open sets in X_{j}. Under these conditions, (5.10.4) implies that

(5.10.6) A_{X} is contained in the σ-algebra of Borel sets in X,

because p_{l}^{−1}(A_{l}) is an open set in X when A_{l} is an open set in X_{l}. This corresponds to the first part of Proposition 1.5 on p23 of [118], as in Subsection 5.8.1 when n = 2.

If there is a base for the topology of X_{j} with only finitely or countably many elements for each j, then there is a base for the product topology on X with only finitely or countably many elements, and whose elements are products of open sets in the X_{j}’s, as in Subsection A.14.1. This implies in particular that every open set in X may be expressed as the union of finitely or countably many products of open sets in the X_{j}’s. It follows that

(5.10.7) open sets in X are elements of A_{X}, so that A_{X} contains all of the Borel sets in X. This means that

(5.10.8) A_{X} is the collection of all Borel sets in X

in this case, as in Proposition 1.5 on p23 of [118], and as in Subsection 5.8.1 when n = 2.

If X_{j} = R with the standard Euclidean metric for each j = 1, …, n, then one can take X = R^{n} to be equipped with the standard Euclidean metric.

5.10.3 Measurability of mappings into X

Let (X_{j}, A_{X_{j}}) be any measurable space for j = 1, …, n again, and let (W, A_{W}) be another measurable space. A mapping ϕ from W into X is measurable with respect to A_{X} on X if and only if

(5.10.9) p_{l} ∘ ϕ is measurable as a mapping from W into X_{l} for each l,

as in Proposition 2.4 on p44 of [118]. This was discussed in Subsection 5.8.2 when n = 2, and the argument is analogous when n ≥ 3. As before, if p_{l} ∘ ϕ is measurable, then

(p_{l} ∘ ϕ)^{−1}(A_{l}) = ϕ^{−1}(p_{l}^{−1}(A_{l})) ∈ A_{W} (5.10.10)

for every A_{l} ∈ A_{X_{l}}. We also have that

{E ⊆ X : ϕ^{−1}(E) ∈ A_{W} } (5.10.11)

is a σ-algebra of subsets of X. One can use these two facts to get that ϕ is measurable when p_{l} ∘ ϕ is measurable for each l. The converse follows from the measurability of p_{l} for each l.

5.10.4 Products of n measurable sets

If A_{l} and B_{l} are subsets of X_{l} for each l = 1, …, n, then

( ∏_{l=1}^{n} A_{l}) ∩ ( ∏_{l=1}^{n} B_{l}) = ∏_{l=1}^{n}(A_{l} ∩ B_{l}). (5.10.12)

Equivalently,

( ⋂_{l=1}^{n} p_{l}^{−1}(A_{l})) ∩ ( ⋂_{l=1}^{n} p_{l}^{−1}(B_{l})) = ⋂_{l=1}^{n}(p_{l}^{−1}(A_{l}) ∩ p_{l}^{−1}(B_{l}))

(5.10.13)  = ⋂_{l=1}^{n} p_{l}^{−1}(A_{l} ∩ B_{l}).

Let R_{X} be the collection of subsets of X of the form (5.10.3) with A_{l} ∈ A_{X_{l}} for each l. We would like to check that

(5.10.14) R_{X} is an elementary family, as in Section 5.7. The first condition in the definition of an elementary family is clear, and the second condition follows from the remarks in the preceding paragraph. To get the third condition, observe that

(5.10.15) X \ ( ∏_{l=1}^{n} A_{l}) = X \ ( ⋂_{l=1}^{n} p_{l}^{−1}(A_{l})) = ⋃_{l=1}^{n}(X \ p_{l}^{−1}(A_{l}))

= ⋃_{l=1}^{n} p_{l}^{−1}(X_{l} \ A_{l})

may be expressed as a union of finitely many pairwise-disjoint sets of the form

∏_{l=1}^{n} C_{l}, (5.10.16)

where C_{l} = A_{l} or X_{l} \ A_{l} for each l, and C_{l} = X_{l} \ A_{l} for at least one l. One can use fewer sets by taking C_{l} = A_{l} when l < r, C_{r} = X_{r} \ A_{r}, and C_{l} = X_{l} when l > r, for r = 1, …, n.

Let A_{X,0} be the collection of subsets of X that can be expressed as the union of finitely many pairwise-disjoint elements of R_{X}. This is an algebra of subsets of X, by the remarks at the beginning of Section 5.7. It follows that A_{X} is the same as the monotone class of subsets of X generated by A_{X,0}, as in Subsection 2.6.1. This is the same as in Subsection 5.7.4 when n = 2.

5.11 Some related Cartesian products

Let us continue with the same notation and hypotheses as in the previous section. Let n_{1} and n_{2} be positive integers such that n_{1} + n_{2} = n, and consider the Cartesian products ∏_{j=1}^{n_{1}} X_{j}, ∏_{l=n_{1}+1}^{n} X_{l}, and their product

( ∏_{j=1}^{n_{1}} X_{j}) × ( ∏_{l=n_{1}+1}^{n} X_{l}). (5.11.1)

There is an obvious bijection between X and (5.11.1), that sends x ∈ X to

((x_{1}, …, x_{n_{1}}), (x_{n_{1}+1}, …, x_{n})). (5.11.2)

It is often convenient to identify X with (5.11.1) in this way, so that subsets of (5.11.1) correspond to subsets of X. In particular, if A_{j} ⊆ X_{j} for each j = 1, …, n, then

( ∏_{j=1}^{n_{1}} A_{j}) × ( ∏_{l=n_{1}+1}^{n} A_{l}) (5.11.3)

is a subset of (5.11.1) that corresponds to ∏_{j=1}^{n} A_{j} as a subset of X.

Similarly, x ↦ (x_{1}, …, x_{n_{1}}) (5.11.4)

5.12. PRODUCTS OF N MEASURES 153

defines a mapping from X into ∏_{j=1}^{n_{1}} X_{j}, and

x ↦ (x_{n_{1}+1}, …, x_{n}) (5.11.5)

defines a mapping from X into ∏_{l=n_{1}+1}^{n} X_{l}. These mappings correspond to the usual mappings from (5.11.1) into ∏_{j=1}^{n_{1}} X_{j} and ∏_{l=n_{1}+1}^{n} X_{l}, as in Subsection 5.7.2, using the identification between X and (5.11.1) mentioned in the preceding paragraph.

One can define σ-algebras of measurable subsets of ∏_{j=1}^{n_{1}} X_{j}, ∏_{l=n_{1}+1}^{n} X_{l}, and thus (5.11.1) in the same way as before. The σ-algebra of measurable subsets of (5.11.1) corresponds exactly to A_{X}, using the identification mentioned earlier. This corresponds to a remark on p65 of [118], and the case where n_{1} = 2, n_{2} = 1 is the first part of Exercise 45 on p68 of [118]. Equivalently, this means that the bijection from X onto (5.11.1) defined by (5.11.2) is measurable, and its inverse mapping is measurable.

The measurability of this mapping is equivalent to the measurability of (5.11.4) and (5.11.5), as in Subsection 5.8.2. The measurability of each of these two mappings reduces to the measurability of the p_{l}’s on X, as before.

Similarly, to get the measurability of the inverse mapping from (5.11.1) onto X, it suffices to check the measurability of the mappings from (5.11.1) into X_{r} obtained by composing the mapping onto X with p_{r} for each r = 1, …, n. This mapping is the same as the composition of one of the usual mappings from (5.11.1) into ∏_{j=1}^{n_{1}} X_{j} or ∏_{l=n_{1}+1}^{n} X_{l} with the analogous mapping from ∏_{j=1}^{n_{1}} X_{j} or ∏_{l=n_{1}+1}^{n} X_{l}, as appropriate, into X_{r}, depending on whether r ≤ n_{1} or r ≥ n_{1} + 1. These mappings are measurable, because they are compositions of measurable mappings.

5.12 Products of n measures

Let n be a positive integer again, and let (X_{j}, A_{X_{j}}, μ_{j}) be a measure space for j = 1, …, n. Put X = ∏_{j=1}^{n} X_{j}, and let us continue with the other notation from the previous two sections, for the X_{j}’s as measurable spaces. We would like to define a product measure

μ = μ_{1} × ⋯ × μ_{n} (5.12.1)

on X with respect to A_{X}, with

μ( ∏_{j=1}^{n} A_{j}) = ∏_{j=1}^{n} μ_{j}(A_{j}) (5.12.2)

when A_{j} ∈ A_{X_{j}} for each j.

If E ∈ A_{X,0} is expressed as the union of finitely many pairwise-disjoint measurable Cartesian products as in (5.10.3), then one would like to define μ(E) to be the corresponding sum of the measures of these Cartesian products, using (5.12.2). As in the n = 2 case, one should check that μ is well-defined on A_{X,0} in this way, and countably additive on A_{X,0}.

5.12.1 Some iterated integrals

More precisely, one can verify that μ(E) is equal to the iterated integral

∫_{X_{1}} ( ∫_{X_{2}} ⋯ ( ∫_{X_{n}} 1_{E}(x_{1}, x_{2}, …, x_{n}) dμ_{n}(x_{n})) ⋯ dμ_{2}(x_{2})) dμ_{1}(x_{1}). (5.12.3)

If E is an element of R_{X}, then it is easy to see that at each step in the iterated integral, one gets a multiple of the indicator function of the Cartesian product of measurable sets in the variables that have not been integrated yet.

If E is any element of A_{X,0}, then 1_{E} is the sum of finitely many indicator functions of elements of R_{X}, as in Subsection 1.5.3. In this case, it follows that at each step in the iterated integral, one is integrating a measurable simple function in one variable, and getting a measurable simple function of the variables to be integrated. Of course, the iterated integral of 1_{E} is equal to the sum of the iterated integrals of the indicators functions of the corresponding elements of R_{X} under these conditions.

5.12.2 Extending μ to A_{X}

Once μ is defined as a countably additive measure on X with respect to A_{X,0}, it can be extended to a countably additive measure on X with respect to A_{X} as in Subsection 1.11.4. This is the approach used on p65 of [118], and in Exercise 14 on p315 of [335].

Let us suppose from now on in this section that

(5.12.4) X_{j} is σ-finite with respect to μ_{j} for each j.

This implies that (5.12.5) X is σ-finite with respect to μ,

because of (5.12.2).

5.12.3 Uniqueness of μ on A_{X}

One can use this to get that μ is uniquely determined as a measure on X with respect to A_{X} by (5.12.2), as in Subsection 2.6.4 and Section 2.9, and as in the n = 2 case in Subsection 5.9.3. More precisely, it is easy to see that μ is uniquely determined on A_{X,0} by (5.12.2), and the previous arguments imply that μ is uniquely determined on A_{X} by its restriction to A_{X,0}. as before.

If n_{1} and n_{2} are positive integers with n_{1} + n_{2} = n, then one can consider analogous product measures on ∏_{j=1}^{n_{1}} X_{j}, ∏_{l=n_{1}+1}^{n} X_{l}, and (5.11.1). The uniqueness result mentioned in the preceding paragraph implies that the product measure on (5.11.1) corresponds exactly to μ on X. This corresponds to a remark on p65 of [118], and to the second part of Exercise 45 on p68 of [118] when n_{1} = 2 and n_{2} = 1.

5.12.4 Another approach to the product measure on X

Alternatively, one can obtain the product measure μ on X by reducing to the n = 2 case, using induction, as in Lemma 22.5 on p431 of [159]. The simplest way to do this is to define the appropriate product measure on ∏_{j=1}^{r} X_{j} for r = 1, …, n, one step at a time, as in [159]. As before, one can use uniqueness to get that the resulting product measure on X is the same as the product measure obtained analogously on (5.11.1). This is related to Lemma 22.6 on p432 of [159]. Note that products of n measures are mentioned on p95 of [20] too.

5.13 Sequences of measurable spaces

Let (X_{1}, A_{X_{1}}), (X_{2}, A_{X_{2}}), (X_{3}, A_{X_{3}}), … be an infinite sequence of measurable spaces, and put X = ∏_{j=1}^{∞} X_{j}. This is the space of infinite sequences x = {x_{j}}_{j=1}^{∞} with x_{j} ∈ X_{j} for each j, as in Section A.8. If l is a positive integer, then let p_{l} be the mapping from X into X_{l} defined by p_{l}(x) = x_{l}. If A_{l} ⊆ X_{l}, then

p_{l}^{−1}(A_{l}) = ∏_{j=1}^{∞} A_{j,l}, (5.13.1)

where A_{j,l} = X_{j} when j ≠ l, and A_{l,l} = A_{l}. Note that

∏_{l=1}^{∞} A_{l} = ⋂_{l=1}^{∞} p_{l}^{−1}(A_{l}) (5.13.2)

when A_{l} ⊆ X_{l} for each l.

5.13.1 Cylinder sets

Let us call a subset of X of the form (5.13.2) a cylinder set if A_{l} ∈ A_{X_{l}} for each l, and (5.13.3) A_{l} = X_{l} for all but finitely many l,

as in Exercise 15 on p316 of [335]. This is the same as a set of the form

⋂_{l=1}^{n} p_{l}^{−1}(A_{l}) (5.13.4)

for some positive integer n, with A_{l} ∈ A_{X_{l}} for l = 1, …, n.

Let A_{X} be the σ-algebra of subsets of X generated by these cylinder sets. This corresponds to part of 22.2 beginning on p429 of [159], although it is formulated a bit differently here. Equivalently, we can take A_{X} to be the σalgebra of subsets of X generated by sets of the form p_{l}^{−1}(A_{l}), with A_{l} ∈ A_{X_{l}}, as on p22 of [118]. This σ-algebra may be denoted ⨂_{l=1}^{∞} A_{X_{l}}, as in [118]. Note that p_{l} is measurable with respect to A_{X} on X for each l, and that A_{X} is the smallest σ-algebra of subsets of X with this property.

If A_{l} ∈ A_{X_{l}} for each l ≥ 1, then (5.13.2) is an element of A_{X}, because it is the intersection of a sequence of cylinder sets. This is related to Proposition 1.3 on p23 of [118].

5.13.2 Collections of subsets of the X_{j}’s generating A_{X_{j}}

Let E_{X_{j}} be a collection of subsets of X_{j} that generates A_{X_{j}} as a σ-algebra for each j, and let Ã_{X} be the σ-algebra of subsets of X generated by sets of the form p_{l}^{−1}(A_{l}), with A_{l} ∈ E_{X_{l}} and l ≥ 1. Of course, Ã_{X} is contained in A_{X}, by construction, and Proposition 1.4 on p23 of [118] implies that they are equal. This is because the collection of A_{l} ⊆ X_{l} such that p_{l}^{−1}(A_{l}) ∈ Ã_{X} is a σ-algebra of subsets of X_{l} that contains E_{X_{l}} and thus A_{X_{l}} for each l.

5.13.3 Borel sets in X

If X_{j} is a metric space for each j, then X may be considered as a metric space with respect to a suitable product metric, as in Sections A.8 and A.9. More precisely, we may suppose that the condition mentioned at the beginning of Section A.9 holds, by adjusting the metrics as in Subsection A.7.1, if necessary. This implies that the corresponding topology on X is the same as the product topology, as in Subsection A.9.1. Similarly, if X_{j} is a topological space for each j, then X may be considered as a topological space with respect to the corresponding product topology.

Suppose for the moment in this subsection that A_{X_{l}} is the σ-algebra of Borel sets in X_{l} for each l. This means that we can take E_{X_{l}} to be the collection of open sets in X_{l} for each l in the remarks preceding this subsection. Note that p_{l}^{−1}(A_{l}) is an open set in X when A_{l} is an open set in X_{l} for any l. One can use this to get that

(5.13.5) A_{X} is contained in the σ-algebra of Borel sets in X,

for essentially the same reasons as in Subsection 5.10.2.

If there is a base for the topology of X_{j} with only finitely or countably many elements for each j, then there is a base for the product topology on X with only finitely or countably many elements, and whose elements are products of open sets in the X_{j}’s, as in Subsections A.14.3 and A.14.4. This implies in particular that every open set in X may be expressed as the union of finitely or countably many sets of the form (5.13.2), where A_{l} is an open set in X_{l} for each l, and (5.13.3) holds. This means that

(5.13.6) every open set in X is an element of A_{X}.

It follows that A_{X} contains all of the Borel sets in X under these conditions, so that (5.13.7) A_{X} is the collection of all Borel sets in X.

5.14. A HELPFUL FAMILY OF RELATED CARTESIAN PRODUCTS 157

5.13.4 Measurable mappings into X

Let (X_{j}, A_{j}) be any measurable space for each j again, and let (W, A_{W}) be another measurable space. One can check that a mapping ϕ from W into X is measurable with respect to A_{X} on X if and only if

(5.13.8) p_{l} ∘ ϕ is measurable as a mapping from W into X_{l} for each l,

as in Proposition 2.4 on p44 of [118]. The argument is analogous to the one for mappings into finite products in Subsection 5.10.3.

5.14 A helpful family of related Cartesian products

Let us continue with the same notation and hypotheses as in the previous section. Let n be a positive integer, and consider the Cartesian products ∏_{j=1}^{n} X_{j}, ∏_{l=n+1}^{∞} X_{l}, and their product

( ∏_{j=1}^{n} X_{j}) × ( ∏_{l=n+1}^{∞} X_{l}). (5.14.1)

More precisely, ∏_{l=n+1}^{∞} X_{l} may be considered as the set of infinite sequences {x_{l}}_{l=n+1}^{∞} with x_{l} ∈ X_{l} for each l ≥ n + 1.

Observe that x ↦ ((x_{1}, …, x_{n}), {x_{l}}_{l=n+1}^{∞}) (5.14.2)

defines a one-to-one mapping from X onto (5.14.1). We shall sometimes want to identify X with (5.14.1) in this way, so that subsets of (5.14.1) correspond to subsets of X. If A_{j} ⊆ X_{j} for each j, then

( ∏_{j=1}^{n} A_{j}) × ( ∏_{l=n+1}^{∞} A_{l}) (5.14.3)

is a subset of (5.14.1) that corresponds to ∏_{j=1}^{∞} A_{j}.

We also have mappings

x ↦ (x_{1}, …, x_{n}) (5.14.4)

from X into ∏_{j=1}^{n} X_{j}, and x ↦ {x_{l}}_{l=n+1}^{∞} (5.14.5)

from X into ∏_{l=n+1}^{∞} X_{l}. These mappings correspond to the usual ones from (5.14.1) into ∏_{j=1}^{n} X_{j} and ∏_{l=n+1}^{∞} X_{l}, as in Subsection 5.7.2, using (5.14.2).

5.14.1 σ-Algebras of measurable sets in these products

One can define a σ-algebra of measurable subsets of ∏_{j=1}^{n} X_{j} as in Section 5.10.4, and a σ-algebra of measurable subsets of ∏_{l=n+1}^{∞} X_{l} as in the previous section. This leads to a σ-algebra of measurable sets in (5.14.1), as in Subsection 5.7.1. This corresponds exactly to A_{X}, using (5.14.2). This corresponds to part of Lemma 22.4 on p430 of [159]. This is the same as saying that (5.14.2) and its inverse are both measurable.

The measurability of (5.14.2) is equivalent to the measurability of (5.14.4) and (5.14.5), as in Subsection 5.8.2. The measurability of these two mappings reduces to the measurability of the p_{l}’s, as in Subsection 5.10.3 and Subsection 5.13.4.

In order to get the measurability of the inverse of (5.14.2), it suffices to verify the measurability of the composition of the inverse mapping with p_{r} for each r, as in Subsection 5.13.4 again. This is the same as the composition of one of the usual mappings from (5.14.1) into ∏_{j=1}^{n} X_{j} or ∏_{l=n+1}^{∞} X_{l} with the analogous mapping from ∏_{j=1}^{n} X_{j} or ∏_{l=n+1}^{∞} X_{l}, as appropriate, into X_{r}, depending on whether r ≤ n or r ≥ n + 1. The measurability of these mappings follows from the fact that compositions of measurable mappings are measurable, as in Subsection 3.1.3.

5.14.2 Some remarks about products of measurable sets

If A_{l} and B_{l} are subsets of X_{l} for each l, then

( ∏_{l=1}^{∞} A_{l}) ∩ ( ∏_{l=1}^{∞} B_{l}) = ∏_{l=1}^{∞}(A_{l} ∩ B_{l}). (5.14.6)

This is the same as saying that

( ⋂_{l=1}^{∞} p_{l}^{−1}(A_{l})) ∩ ( ⋂_{l=1}^{∞} p_{l}^{−1}(B_{l})) = ⋂_{l=1}^{∞}(p_{l}^{−1}(A_{l}) ∩ p_{l}^{−1}(B_{l}))

(5.14.7)  = ⋂_{l=1}^{∞} p_{l}^{−1}(A_{l} ∩ B_{l}).

Let n be a positive integer again, and let

C_{X,n} (5.14.8)

be the collection of cylinder sets ∏_{l=1}^{∞} A_{l} with A_{l} ∈ A_{X_{l}} for each l and

A_{l} = X_{l} for all l ≥ n + 1. (5.14.9)

Note that C_{X,n} ⊆ C_{X,n+1} (5.14.10) for each n, and that

C_{X} = ⋃_{n=1}^{∞} C_{X,n} (5.14.11)

is the collection of all cylinder sets in X.

One can check that

(5.14.12) C_{X,n} is an elementary family of subsets of X

for each n, as in Section 5.7, and thus that C_{X} is an elementary family of sets as well. More precisely, the fact that C_{X,n} satisfies the third condition in the definition of an elementary family of sets is essentially the same as in Subsection 5.10.4. One can also use (5.14.4) to reduce to the analogous statement for ∏_{j=1}^{n} X_{j}.

Let AC_{X,n} (5.14.13)

be the collection of subsets of X that can be expressed as the union of finitely many pairwise-disjoint elements of C_{X,n}. This is an algebra of subsets of X, as in Section 5.7. This is the same as the collection of subsets of X that can be expressed as the inverse image with respect to (5.14.4) of an analogous type of subset of ∏_{j=1}^{n} X_{j}, as in Subsection 5.10.4.

Note that AC_{X,n} ⊆ AC_{X,n+1} (5.14.14)

for each n, because of (5.14.10). It is easy to see that

AC_{X} = ⋃_{n=1}^{∞} AC_{X,n} (5.14.15)

is an algebra of subsets of X, consisting of the subsets of X that can be expressed as the union of finitely many pairwise-disjoint elements of C_{X}. We also have that

(5.14.16) A_{X} is the monotone class of subsets of X generated by AC_{X},

as in Subsection 2.6.1, because A_{X} is the same as the σ-algebra of subsets of X generated by AC_{X}. This is related to 22.2 beginning on p429 of [159].

5.15 Sequences of probability spaces

Let {(X_{j}, A_{X_{j}}, μ_{j})}_{j=1}^{∞} be an infinite sequence of probability spaces, and put X = ∏_{j=1}^{∞} X_{j}. Let us also continue with the other notation from the previous two sections, for the X_{j}’s as measurable spaces. We would like to define a product probability measure μ on X with respect to A_{X}, with

μ( ∏_{j=1}^{∞} A_{j}) = ∏_{j=1}^{∞} μ_{j}(A_{j}) (5.15.1)

when A_{j} ∈ A_{X_{j}} for each j, and A_{j} = X_{j} for all but finitely many j. Note that

(5.15.2) μ_{j}(A_{j}) = 1 for all but finitely many j

in this case, so that the product on the right side of (5.15.1) reduces to a finite product.

Let us put

X(n) = ∏_{j=1}^{n} X_{j} (5.15.3)

for each positive integer n, and let R_{X(n)} be the collection of subsets of X(n) of the form ∏_{j=1}^{n} A_{j}, with A_{j} ∈ A_{X_{j}} for j = 1, …, n, as in Subsection 5.10.4. Also let A_{X(n)} be the σ-algebra of subsets of X(n) generated by R_{X(n)}, and let A_{X(n),0} be the collection of subsets of X(n) that can be expressed as the union of finitely many pairwise-disjoint elements of R_{X(n)}, as before. Remember that A_{X(n),0} is an algebra of subsets of X(n), and that A_{X(n)} is generated by A_{X(n),0} as a σ-algebra, and thus as a monotone class of subsets of X(n).

5.15.1 Using the standard mapping from X onto X(n)

Let ϕ_{n} be the mapping from X onto X(n) defined by (5.14.4). Of course, X_{j} ≠ ∅ for each j, because μ_{j}(X_{j}) = 1, by hypothesis. If A_{j} ∈ A_{X_{j}} for j = 1, …, n, then

ϕ_{n}^{−1}( ∏_{j=1}^{n} A_{j}) = ∏_{j=1}^{∞} A_{j}, (5.15.4)

where A_{j} = X_{j} for j ≥ n + 1. In this case, (5.15.1) is the same as saying that

μ(ϕ_{n}^{−1}( ∏_{j=1}^{n} A_{j})) = ∏_{j=1}^{n} μ_{j}(A_{j}) = (μ_{1} × ⋯ × μ_{n})( ∏_{j=1}^{n} A_{j}), (5.15.5)

where μ_{1} × ⋯ × μ_{n} is the product measure defined on X(n) with respect to A_{X(n)} as in Section 5.12.

Note that E(n) ↦ ϕ_{n}^{−1}(E(n)) (5.15.6)

defines a one-to-one mapping from R_{X(n)} onto C_{X,n}. This also defines a oneto-one mapping from A_{X(n),0} onto AC_{X,n}. Put

μ(ϕ_{n}^{−1}(E(n))) = (μ_{1} × ⋯ × μ_{n})(E(n)) (5.15.7)

for each E(n) ∈ A_{X(n),0}. This defines μ as a countably additive measure on AC_{X,n}. This is uniquely determined as a finitely additive measure on AC_{X,n} by its restriction to C_{X,n}, as in (5.15.5).

5.15.2 Defining μ on AC_{X}, and on A_{X}

Remember that AC_{X,n} is contained in AC_{X,n+1}, as in (5.14.14). Of course, one can define μ on AC_{X,n+1} in the same way as before. One can check that this is compatible with the definition of μ on AC_{X,n}, because μ_{n+1}(X_{n+1}) = 1, by hypothesis.

Thus we can use this to define μ on AC_{X}, because of (5.14.15). It is easy to see that μ is finitely additive on AC_{X}, and that μ is uniquely determined as a finitely-additive measure on AC_{X} by its restriction to C_{X}, as in (5.15.5). This corresponds to part of Theorem 22.7 on p432 of [159].

It is well known that μ is countably additive on AC_{X}, so that μ can be extended to a countably additive probability measure on X with respect to A_{X}, as in Subsection 1.11.4. This corresponds to Theorem 22.8 on p433 of [159], and another argument may be found in [302]. This is also mentioned in Exercise 15 on p316 of [335]. This extension is unique, as in Subsection 2.6.2 and Section 2.9. This means that μ is uniquely determined by (5.15.1).

5.15.3 Some more properties of these product measures

One can check that (5.15.7) holds for every n and E(n) ∈ A_{X(n)}. This is the same as saying that (ϕ_{n})_{∗}(μ) = μ_{1} × ⋯ × μ_{n}, (5.15.8)

in the notation of Section 4.11. This uses the fact that the right side is uniquely determined by its restriction to R_{X(n)}, as in Subsection 5.12.3.

If f_{n} is nonnegative extended real-valued measurable function on X(n), then f_{n} ∘ ϕ_{n} is a nonnegative extended real-valued measurable function on X, and

∫_{X}(f_{n} ∘ ϕ_{n}) dμ = ∫_{X(n)} f_{n} d(μ_{1} × ⋯ × μ_{n}), (5.15.9)

as in Subsection 4.11.3. If f_{n} is a measurable real or complex-valued function on X(n), then f_{n} ∘ ϕ_{n} is measurable on X, and the previous statement implies that

∫_{X} |f_{n} ∘ ϕ_{n}| dμ = ∫_{X(n)} |f_{n}| d(μ_{1} × ⋯ × μ_{n}). (5.15.10)

In particular, f_{n} is integrable on X(n) if and only of f_{n} ∘ ϕ_{n} is integrable on X. In this case, (5.15.9) holds, as before.

5.15.4 Some other approaches to product measures like these

Suppose that X_{j} is a compact metric space, or a compact Hausdorff topological space, for each j. This implies that X = ∏_{j=1}^{∞} X_{j} is compact with respect to the product topology, by Tychonoff’s theorem. If the X_{j}’s are metric spaces, then we may suppose that the condition mentioned at the beginning of Section A.9 holds, by adjusting the metrics as in Section A.7, if necessary, and use a suitable product metric on X as in Section A.8.

It is well known that finite regular Borel measures on compact Hausdorff topological spaces correspond to nonnegative linear functionals on the associated spaces of continuous real-valued functions. One can use this to get a product probability measure on X from regular Borel probability measures on the X_{j}’s. More precisely, one can define a nonnegative linear functional on the space of continuous real-valued functions on X by integrating over the X_{j}’s and taking a suitable limit. This corresponds to part of Theorem 7.28 on p230 of [118], and to the discussion beginning in 13.15 on p157 of [158].

If one starts with locally compact Hausdorff topological spaces that are not compact, then one can get compact Hausdorff spaces using one-point compactifications. It is easy to extend a Borel measure on such a space to the one-point compactification, where the measure of the set consisting only of the point at infinity is equal to 0. The one-point compactification of the real line is homeomorphic to a circle, for instance. This can be used to reduce to the case of products of compact spaces, as mentioned on p328 of [118].

Let us take X_{j} = R and A_{X_{j}} to be the σ-algebra of Borel sets in R for each j. In this case, the existence of a suitable product probability measure is mentioned on p309 of [20], as a consequence of a theorem of Kolmogorov. This theorem deals more broadly with probability measures on R^{n} for each n, with a suitable compatibility condition for the measures on R^{n} and R^{n+1}. This condition holds automatically when these probability measures are obtained from the products of the first n terms of a sequence of probability measures on R. Theorem 10.18 on p328 of [118] is another result of this type.

Independence in probability theory

6.1 Some terminology in probability theory

Let (X, A, μ) be a probability space, i.e., a measure space with μ(X) = 1. An element of A may be called an event. The term almost surely may be used instead of almost everywhere with respect to μ. A measurable real-valued function f on X is called a random variable, as mentioned in Subsection 4.11.1. If f is integrable on X with respect to μ, then

E(f) = ∫_{X} f dμ (6.1.1)

may be called the expectation, expected value, or mean of f. We may also use this notation when f is a nonnegative extended real-valued measurable function on X that is not necessarily integrable, so that E(f) may be +∞. This is discussed beginning on p381 of [14], beginning on p267 of [20], and on p314 of [118].

6.1.1 Independent pairs of events

Two events A, B in X are said to be independent if

μ(A ∩ B) = μ(A) μ(B), (6.1.2)

as in Definition 12.7 on p383 of [14], and on p272 of [20].

Suppose for the moment that (X_{1}, A_{1}, μ_{1}) and (X_{2}, A_{2}, μ_{2}) are probability spaces, and that (X, A, μ) corresponds to their product, as in Section 5.9. If A_{1} ∈ A_{1} and B_{2} ∈ A_{2}, then one can check that

A = A_{1} × X_{2} and B = X_{1} × B_{2} (6.1.3)

are independent events in X. This corresponds to Example 12.9 on p383 of [14].

6.1.2 Independence of n events

Let n be a positive integer, and let A_{1}, …, A_{n} be n events in X. These events are said to be independent if

μ(A_{j_{1}} ∩ A_{j_{2}} ∩ ⋯ ∩ A_{j_{r}}) = μ(A_{j_{1}}) μ(A_{j_{2}}) ⋯ μ(A_{j_{r}}) (6.1.4)

whenever 1 ≤ j_{1} < j_{2} < ⋯ < j_{r} ≤ n, (6.1.5)

as on p272 of [20]. Note that this implies that every subcollection of these events is independent too.

Suppose for the moment that (X_{j}, A_{j}, μ_{j}) is a probability space for j = 1, …, n, and that (X, A, μ) corresponds to their product, as in Section 5.12. Let p_{l} be the obvious mapping from X onto X_{l} for each l = 1, …, n, as in Section 5.10. If B_{l} ∈ A_{l} for l = 1, …, n, then one can check that

p_{l}^{−1}(B_{l}), 1 ≤ l ≤ n, (6.1.6)

are independent events in X. More precisely, if 1 ≤ l_{1} < l_{2} < ⋯ < l_{r} ≤ n, then

p_{l_{1}}^{−1}(B_{l_{1}}) ∩ p_{l_{2}}^{−1}(B_{l_{2}}) ∩ ⋯ ∩ p_{l_{r}}^{−1}(B_{l_{r}}) = ∏_{j=1}^{n} C_{j}, (6.1.7)

where C_{l_{k}} = B_{l_{k}} when k = 1, …, r, and C_{j} = X_{j} when j ≠ l_{1}, …, l_{r}. This implies that

μ(p_{l_{1}}^{−1}(B_{l_{1}}) ∩ p_{l_{2}}^{−1}(B_{l_{2}}) ∩ ⋯ ∩ p_{l_{r}}^{−1}(B_{l_{r}})) (6.1.8)

= μ( ∏_{j=1}^{n} C_{j}) = ∏_{j=1}^{n} μ_{j}(C_{j})
= μ_{l_{1}}(B_{l_{1}}) μ_{l_{2}}(B_{l_{2}}) ⋯ μ_{l_{r}}(B_{l_{r}})
= μ(p_{l_{1}}^{−1}(B_{l_{1}})) μ(p_{l_{2}}^{−1}(B_{l_{2}})) ⋯ μ(p_{l_{r}}^{−1}(B_{l_{r}})).

6.1.3 Independent sequences of events

An infinite sequence A_{1}, A_{2}, A_{3}, … of events in X is said to be independent if any finite subcollection of these events is independent, as on p273 of [20]. The independence of any family of events can be defined in the same way, as in Definition 12.7 on p383 of [14]. An equivalent definition may be found on p315 of [118]. In the case of sequences, this is the same as saying that (6.1.4) holds when 1 ≤ j_{1} < j_{2} < ⋯ < j_{r}. (6.1.9)

Suppose for the moment again that {(X_{j}, A_{j}, μ_{j})}_{j=1}^{∞} is an infinite sequence of probability spaces, and that (X, A, μ) corresponds to their product, as in Section 5.15. Let p_{l} be the usual mapping from X onto X_{l} for each positive integer l, as in Section 5.13. If B_{l} ∈ A_{l} for each l, then one can verify that

p_{l}^{−1}(B_{l}), l ≥ 1, (6.1.10) are independent events in X. This is basically the same as in the previous subsection.

6.1.4 Independence and complements

If A_{1}, …, A_{n} are independent events in X, then it is well known that

(6.1.11) A_{1}, …, A_{n−1}, X \ A_{n} are independent in X.

This corresponds to part of Exercise 3 on p398 of [14] and Proposition 21.7 on p273 of [20] when n = 2, and to part a of Exercise 3 on p319 of [118] for any n. To see this, we should check that

μ(A_{j_{1}} ∩ ⋯ ∩ A_{j_{r}} ∩ (X \ A_{n})) = μ(A_{j_{1}}) ⋯ μ(A_{j_{r}}) μ(X \ A_{n}) (6.1.12)

when 1 ≤ j_{1} < ⋯ < j_{r} < n. Note that

μ(A_{j_{1}} ∩ ⋯ ∩ A_{j_{r}} ∩ X) = μ(A_{j_{1}}) ⋯ μ(A_{j_{r}}) μ(X), (6.1.13)

as in (6.1.4), and that

μ(A_{j_{1}} ∩ ⋯ ∩ A_{j_{r}} ∩ A_{n}) = μ(A_{j_{1}}) ⋯ μ(A_{j_{r}}) μ(A_{n}), (6.1.14)

using independence again. One can get (6.1.12) by subtracting both sides of (6.1.14) from the corresponding side of (6.1.13).

The same argument can be used to replace any one of A_{1}, …, A_{n} with its complement in X, to get independent events in X. One can repeat the argument, to replace as many of A_{1}, …, A_{n} by their complements in X as one would like, and get independent events. This implies the analogous statement for infinite sequences of independent events in X, or arbitrary families of independent events in X, as in part b of Exercise 3 on p319 of [118]. In particular, one can replace all of the events by their complements in X, as in Exercise 4 on p398 of [14].

6.2 Infinite products of real numbers

Let {t_{j}}_{j=1}^{∞} be an infinite sequence of real numbers. Let us say that the infinite product

∏_{j=1}^{∞} t_{j} (6.2.1)

converges if the corresponding sequence of partial products

∏_{j=1}^{n} t_{j} (6.2.2)

converges in R. In this case, the value of (6.2.1) is defined to be the limit of the sequence of partial products.

Of course, if t_{l} = 0 for some l, then (6.2.2) is equal to 0 when n ≥ l, so that (6.2.1) converges in this sense and is equal to 0. Sometimes (6.2.1) is not considered to be convergent if t_{l} ≠ 0 for each l and the sequence of partial products converges to 0.

Suppose for the moment that (6.2.1) converges and is not 0, so that t_{l} ≠ 0 for each l in particular. Observe that

t_{n} = ( ∏_{j=1}^{n} t_{j})/( ∏_{j=1}^{n−1} t_{j}) → 1 as n → ∞ (6.2.3)

under these conditions.

Clearly

|∏_{j=1}^{n} t_{j}| = ∏_{j=1}^{n} |t_{j}| (6.2.4)

for each n. If (6.2.1) converges, then it follows that ∏_{j=1}^{∞} |t_{j}| converges, with

∏_{j=1}^{∞} |t_{j}| = |∏_{j=1}^{∞} t_{j}|. (6.2.5)

Suppose for the moment that 0 ≤ t_{j} ≤ 1 for each j, so that the sequence of partial products (6.2.2) is a monotonically decreasing sequence of elements of [0, 1]. This implies that (6.2.1) converges, and is an element of [0, 1].

6.2.1 Intersections of independent events

Let (X, A, μ) be a probability space, and let A_{1}, A_{2}, A_{3}, … be an infinite sequence of independent events in X. Observe that

∏_{j=1}^{n} μ(A_{j}) = μ( ⋂_{j=1}^{n} A_{j}) → μ( ⋂_{j=1}^{∞} A_{j}) as n → ∞, (6.2.6)

where the second step is as in Subsection 1.9.2. This means that

∏_{j=1}^{∞} μ(A_{j}) = μ( ⋂_{j=1}^{∞} A_{j}). (6.2.7)

Let {(X_{j}, A_{j}, μ_{j})}_{j=1}^{∞} be an infinite sequence of probability spaces, and suppose that (X, A, μ) corresponds to their product, as in Section 5.15. Also let p_{l} be the usual mapping from X onto X_{l} for each l, as in Section 5.13, and let B_{l} be an element of A_{l} for each l. Remember that

∏_{l=1}^{∞} B_{l} = ⋂_{l=1}^{∞} p_{l}^{−1}(B_{l}) ∈ A, (6.2.8)

as in Subsection 5.13.1, and that

μ(p_{l}^{−1}(B_{l})) = μ_{l}(B_{l}) (6.2.9)

for each l, by construction. Thus

μ( ∏_{l=1}^{∞} B_{l}) = ∏_{l=1}^{∞} μ_{l}(B_{l}), (6.2.10)

as in (6.2.7), with A_{l} = p_{l}^{−1}(B_{l}) for each l. This can be seen a bit more directly using the fact that

∏_{l=1}^{∞} B_{l} = ⋂_{n=1}^{∞} ( ∏_{l=1}^{∞} C_{l,n}), (6.2.11)

with (6.2.12) C_{l,n} = B_{l} when l ≤ n, = X_{l} when l > n.

6.2.2 Products of positive real numbers

Let {t_{j}}_{j=1}^{∞} be an infinite sequence of positive real numbers, so that (6.2.2) is a positive real number for each n. Of course,

log ( ∏_{j=1}^{n} t_{j}) = ∑_{j=1}^{n} log t_{j} (6.2.13)

for each n. It is easy to see that (6.2.1) converges and is not 0 if and only if

∑_{j=1}^{∞} log t_{j} (6.2.14)

converges as an infinite series of real numbers, with the logarithm of (6.2.1) equal to (6.2.14) in this case. If (6.2.14) converges absolutely, then one may say that (6.2.1) converges absolutely as an infinite product.

It is well known and not difficult to show that (6.2.14) converges absolutely if and only if

∑_{j=1}^{∞} |t_{j} − 1| < +∞. (6.2.15)

Of course, (6.2.15) implies that t_{j} → 1 as j → ∞. Similarly, the convergence of (6.2.14) implies that log t_{j} → 0 and j → ∞, so that t_{j} → 1.

6.2.3 More on products with 0 ≤ t_{j} ≤ 1 for each j

Suppose that 0 ≤ t_{j} ≤ 1 for each j again, so that

t_{j} = 1 − x_{j} (6.2.16) for each j, with 0 ≤ x_{j} ≤ 1. If x_{j} < 1 for each j and

∑_{j=1}^{∞} x_{j} < +∞, (6.2.17)

then t_{j} > 0 for each j and (6.2.15) holds, so that (6.2.1) is positive, as in the previous subsection. This corresponds to the second part of Lemma 22.25 on p445 of [159].

The first part of Lemma 22.25 on p445 of [159] states that

∏_{j=1}^{∞}(1 − x_{j}) = 0 (6.2.18)

when

∑_{j=1}^{∞} x_{j} = +∞. (6.2.19)

Of course, (6.2.18) holds automatically when x_{j} = 1 for some j, and so we may as well suppose that x_{j} < 1 for each j. This implies that

1 + x_{j} ≤ \frac{1}{1 − x_{j}} (6.2.20)

for each j, because (1 + x_{j}) (1 − x_{j}) = 1 − x_{j}^{2} ≤ 1. Observe that

1 + ∑_{j=1}^{n} x_{j} ≤ ∏_{j=1}^{n}(1 + x_{j}) (6.2.21)

for each n, so that

1 + ∑_{j=1}^{n} x_{j} ≤ ( ∏_{j=1}^{n}(1 − x_{j}))^{−1}. (6.2.22)

If ∑_{j=1}^{n} x_{j} → +∞ as n → ∞, then it follows that ∏_{j=1}^{n}(1 − x_{j}) → 0. Alternatively, one can check that

1 − x ≤ exp(−x) (6.2.23)

for all nonnegative real numbers x using calculus, as mentioned on p385 of [14], p274 of [20], and p321 of [118]. This implies that

∏_{j=1}^{n}(1 − x_{j}) ≤ exp ( − ∑_{j=1}^{n} x_{j}) (6.2.24)

for each n. This can be used in the same way as before.

6.3 The Borel–Cantelli lemma

Let (X, A, μ) be a measure space, and let A_{1}, A_{2}, A_{3}, … be an infinite sequence of measurable subsets of X. Remember that lim sup_{j→∞} A_{j} is the set of x ∈ X such that x ∈ A_{j} for infinitely many j, as in Subsection 1.4.2. It is easy to see that this is a measurable subset of X, using the characterization in Subsection 1.4.3. If

∑_{j=1}^{∞} μ(A_{j}) < +∞, (6.3.1)

then part of the Borel–Cantelli lemma states that

μ( lim sup_{j→∞} A_{j}) = 0. (6.3.2)

This corresponds to 12.6 on p383 of [14], part (1) of Lemma 21.9 on p274 of [20], part a of 10.10 on p321 of [118], part of 22.26 on p445 of [159], Theorem 1.41 on p32 of [298], and Exercise 16 on p42 of [335]. More precisely, the result is initially stated in [159] for certain types of sequences of measurable sets in the product of a sequence of probability spaces, as in the previous section, but the proof of this part does not use that, as mentioned in [159]. Similarly, this is stated in [335] for Lebesgue measure on Euclidean spaces, but the argument works for arbitrary measure spaces.

6.3.1 Two proofs

There are two basic types of arguments that are commonly used to show this. In one argument, we observe that

∫_{X} ( ∑_{j=1}^{∞} 1_{A_{j}}) dμ = ∑_{j=1}^{∞} ∫_{X} 1_{A_{j}} dμ = ∑_{j=1}^{∞} μ(A_{j}) < +∞, (6.3.3)

where the first step is as in Subsection 3.11.1. This implies that

∑_{j=1}^{∞} 1_{A_{j}}(x) < +∞ (6.3.4)

for almost every x ∈ X with respect to μ, as in Subsection 3.10.2. This is equivalent to (6.3.2). This argument is used in [14, 298], and it is also mentioned on p63 of [335].

In the second argument, we use the fact that

μ( ⋃_{j=l}^{∞} A_{j}) ≤ ∑_{j=l}^{∞} μ(A_{j}), (6.3.5)

as in Subsection 1.10.2. We also have that

(6.3.6) μ( ⋃_{j=l}^{∞} A_{j}) → μ( lim sup_{j→∞} A_{j}) as l → ∞,

because of the characterization of lim sup_{j→∞} A_{j} in Subsection 1.4.3, and the remarks in Subsection 1.9.2. Of course,

∑_{j=l}^{∞} μ(A_{j}) → 0 as l → ∞, (6.3.7)

because of (6.3.1). Thus (6.3.2) holds. This is the argument used in [20, 118, 159, 335], and it corresponds to Exercise 11 on p33 of [298] as well.

6.3.2 When the A_{j}’s are independent

Suppose now that (X, A, μ) is a probability space, and that A_{1}, A_{2}, A_{3}, … are independent events in X. If

∑_{j=1}^{∞} μ(A_{j}) = +∞, (6.3.8)

then

μ( lim sup_{j→∞} A_{j}) = 1, (6.3.9)

as in 12.10 on p385 of [14], part (2) of Lemma 21.9 on p274 of [20], and part b of 10.10 on p321 of [118]. This also corresponds to 22.26 on p445 of [159] when the A_{j}’s are the type of independent events in the product of a sequence of probabilty spaces mentioned in Subsection 6.1.3, although the proof is essentially the same as for arbitrary sequences of independent events.

Of course, (6.3.9) is the same as saying that

μ(X \ ( lim sup_{j→∞} A_{j})) = 0. (6.3.10)

It is easy to see that

X \ ( lim sup_{j→∞} A_{j}) = ⋃_{l=1}^{∞} ( ⋂_{j=l}^{∞}(X \ A_{j})), (6.3.11)

using the characterization of lim sup_{j→∞} A_{j} mentioned in Subsection 1.4.3. Thus it suffices to show that

μ( ⋂_{j=l}^{∞}(X \ A_{j})) = 0 (6.3.12)

for each l.

Remember that the events X \ A_{j} are independent, as in Subsection 6.1.4. This implies that

μ( ⋂_{j=l}^{∞}(X \ A_{j})) = ∏_{j=l}^{∞} μ(X \ A_{j}) = ∏_{j=l}^{∞}(1 − μ(A_{j})) (6.3.13)

for each l, where the first step is as in Subsection 6.2.1.

Note that

∑_{j=l}^{∞} μ(A_{j}) = +∞ (6.3.14)

for each l, because of (6.3.8). It follows that the right side of (6.3.13) is equal to 0, as in Subsection 6.2.3.

6.4 Independent random variables

Let (X, A, μ) be a probability space, and let f and g be random variables on X, i.e., measurable real-valued functions on X. If

(6.4.1) f^{−1}(U) and g^{−1}(V) are independent events in X

for all Borel sets U, V ⊆ R, then f and g are said to be independent as random variables on X, as in Definition 12.14 on p386 of [14]. This is described in an equivalent way on p273 of [20].

Similarly, let n be a positive integer, and let f_{1}, …, f_{n} be n random variables on X. If (6.4.2) f_{1}^{−1}(U_{1}), …, f_{n}^{−1}(U_{n}) are independent events in X

for all Borel sets U_{1}, …, U_{n} ⊆ R, then f_{1}, …, f_{n} are said to be independent as random variables on X. Of course, (6.4.2) implies that

μ(f_{1}^{−1}(U_{1}) ∩ … ∩ f_{n}^{−1}(U_{n})) = μ(f_{1}^{−1}(U_{1})) ⋯ μ(f_{n}^{−1}(U_{n})), (6.4.3)

as in Subsection 6.1.2. In fact, it suffices to check that this holds for all Borel sets U_{1}, …, U_{n} ⊆ R, because this implies the analogous condition for all subcollections of these events, by taking U_{l} = R for some l’s, so that f_{l}^{−1}(U_{l}) = X. In particular, if f_{1}, …, f_{n} are independent random variables on X, then any subcollection of f_{1}, …, f_{n} is independent as well.

An infinite sequence {f_{j}}_{j=1}^{∞} of random variables on X is said to be independent if

(6.4.4) {f_{j}^{−1}(U_{j})}_{j=1}^{∞} is a sequence of independent events in X

for every sequence {U_{j}}_{j=1}^{∞} of Borel sets in R. One can check that this is the same as saying that any finite subcollection of the f_{j}’s is independent. One can define independence of arbitrary families of random variables on X in the same way, as in Definition 12.14 on p386 of [14], on p273 of [20], and on p315 of [118].

6.4.1 Indicator functions of independent events

One can check that events A_{1}, …, A_{n} in X are independent if and only if

(6.4.5) their indicator functions 1_{A_{1}}, …, 1_{A_{n}} are independent as random variables on X.

The “if” part follows directly from the definition of independence of random variables, and the “only if” part uses the remarks in Subsection 6.1.4. There are analogous statements for infinite sequences of events and arbitrary families of events. This corresponds to the first example on p386 of [14] for pairs of events, and to part of Exercise 3 on p319 of [118] for arbitrary families of events.

6.4.2 Some independent random variables

Let (X_{1}, A_{1}, μ_{1}), …, (X_{n}, A_{n}, μ_{n}) be n probability spaces, and suppose for the moment that (X, A, μ) corresponds to their product, as in Section 5.12. Also let p_{l} be the obvious mapping from X onto X_{l} for each l = 1, …, n, as in Section 5.10, and let f_{l} be a measurable real-valued function on X_{l} for each l = 1, …, n. Thus f_{l} ∘ p_{l} is a measurable real-valued function on X for each l = 1, …, n, and one can check that

(6.4.6) f_{1} ∘ p_{1}, …, f_{n} ∘ p_{n} are independent on X.

Indeed, if U_{1}, …, U_{n} are Borel sets in R, then f_{l}^{−1}(U_{l}) ∈ A_{l} for l = 1, …, n, and (f_{l} ∘ p_{l})^{−1}(U_{l}) = p_{l}^{−1}(f_{l}^{−1}(U_{l})), 1 ≤ l ≤ n, (6.4.7)

are independent events in X, as in Subsection 6.1.2. This corresponds to the third example on p386 of [14] when n = 2, it is related to part (2) of Exercise 21.39 on p320 of [20], and it also corresponds to some remarks on p318 of [118].

6.4.3 Some more independent random variables

Similarly, let {(X_{j}, A_{j}, μ_{j})}_{j=1}^{∞} be an infinite sequence of probability spaces, and suppose for the moment again that (X, A, μ) corresponds to their product, as in Section 5.15. Let p_{l} be the usual mapping from X onto X_{l} for each l, as in Section 5.13, and let f_{l} be a measurable real-valued function on X_{l} for each l. One can check that

(6.4.8) {f_{l} ∘ p_{l}}_{l=1}^{∞} is an independent sequence of random variables on X,

using the analgous remark for independent sequences of events in X in Subsection 6.1.3, as in the preceding subsection. This is related to a remark on p309 of [20], and to Corollary 10.19 on p329 of [118]. Sequences of independent random variables like these are also considered in Section 22 of [159], without using this terminology.

6.4.4 Compositions with other functions

Suppose that f_{1}, …, f_{n} are independent random variables on X, and that α_{1}, …, α_{n} are real-valued Borel measurable functions on the real line. Thus α_{1} ∘ f_{1}, …, α_{n} ∘ f_{n} are also measurable on X, as in Subsection 3.1.3. It is easy to see that

(6.4.9) α_{1} ∘ f_{1}, …, α_{n} ∘ f_{n} are independent random variables on X, as in 12.16 on p387 of [14] and Remark 21.8 on p273 of [20] when n = 2, and part of Proposition 10.2 on p316 of [118]. There are analogous statements for sequences or arbitrary families of independent random variables, as usual.

6.5 Rademacher functions and related topics

Let k be a positive integer, and let r_{k} be the real-valued function defined on [0, 1) by

(6.5.1) r_{k}(x) = 1 when (l − 1) 2^{−k} ≤ x < l 2^{−k} and l is odd
= −1 when (l − 1) 2^{−k} ≤ x < l 2^{−k} and l is even,

as on p238 of [14]. More precisely, 1 ≤ l ≤ 2^{k} in both cases. These are the Rademacher functions on the unit interval. These functions are also defined in 5.2 on p104 of [331] and on p6 of Volume I of [388], with slightly different conventions. We may take r_{k}(1) = −1 for each k, so that r_{k} is defined on [0, 1].

Of course, [0, 1) and [0, 1] may be considered as probability spaces, using Lebesgue measure on the corresponding σ-algebra of Borel sets. Note that the Rademacher functions are Borel measurable. In fact, it is well known that the Rademacher functions are independent as random variables with respect to Lebesgue measure, as in Exercise 16 on p399 of [14]. This is related to the remarks in Subsection A.12.2.

More precisely, let B be the set of all binary sequences, as in Section A.12, and let ϕ be the usual mapping from B onto [0, 1], as in Subsection A.12.2. Consider the function y ↦ (−1)^{y_{k}} (6.5.2)

on B. Let A ⊆ B be as in Section A.13. One can check that

r_{k}(ϕ(y)) = (−1)^{y_{k}} (6.5.3)

when y ∈ B \ A.

In fact, A is countably infinite, ϕ(A) = E is countably infinite, and the restriction of ϕ to B \ A is a one-to-one mapping onto [0, 1] \ E, as in Section A.13. Let us use the metric d_{B,θ} on B as in Section A.12, with 0 < θ < 1, and one might as well take θ = 1/2. One can show that ϕ is continuous with respect to this metric, as in Subsection A.12.2. One can also show that ϕ^{−1} is continuous on [0, 1] \ E, as in Subsection A.13.2.

This implies that Borel measures on B \ A correspond exactly to Borel measures on [0, 1] \ E using ϕ and its inverse, as in Section 4.11. One can use this to get that Borel measures on B that are equal to 0 on sets with one element correspond exactly to Borel measures on [0, 1] that are equal to 0 on sets with one element. In particular, Lebesgue measure on [0, 1] corresponds to a Borel probability measure on B with this property in this way.

As in Section A.12, B may be considered as the Cartesian product ∏_{j=1}^{∞} B_{j}, with B_{j} = {0, 1} for each j. Let μ_{j} be the probability measure on the collection of all subsets of B_{j} with μ_{j}({0}) = μ_{j}({1}) = 1/2 for each j. This leads to a probability measure μ on B, as in Section 5.15. More precisely, μ is defined on the corresponding σ-algebra of Borel sets in B, as in Subsection 5.13.3. It is well known and not too difficult to see that μ corresponds to Lebesgue measure on [0, 1], as in the preceding paragraph.

The functions (6.5.2) are independent random variables on B with respect to any product probability measure on B, as in the previous section. This implies that the Rademacher functions are independent random variables on [0, 1] with respect to Lebesgue measure.

Some related results for expansions with base b ≥ 2 are discussed in Exercise 23 on p329 of [118]. The case of ordinary decimal expanions, b = 10, is also mentioned on p396 of [14].

6.6 More on pushing measures forward

Let (X, A_{X}, μ) be a probability space, let (Y, A_{Y}) be a measurable space, let ϕ be a measurable mapping from X into Y, and let ν = ϕ_{∗}(μ) be the measure on Y with respect to A_{Y} obtained by pushing μ forward using ϕ, as in Section 4.11. Note that ν is a probability measure on Y, and let B_{1}, …, B_{n} be measurable subsets of Y. One can check that

(6.6.1) B_{1}, …, B_{n} are independent in Y with respect to ν

if and only if

(6.6.2) ϕ^{−1}(B_{1}), …, ϕ^{−1}(B_{n}) are independent in X with respect to μ.

There are analogous statements for infinite sequences and arbitrary families of events in Y.

Similarly,

(6.6.3) random variables β_{1}, …, β_{n} on Y are independent with respect to ν

if and only if

(6.6.4) β_{1} ∘ ϕ, …, β_{n} ∘ ϕ are independent as random variables on X with respect to μ.

This follows from the previous statement, with

B_{l} = β_{l}^{−1}(U_{l}), (6.6.5)

1 ≤ l ≤ n, where U_{1}, …, U_{n} are Borel sets in R. There are analogous statements for infinite sequences and arbitrary families of random variables on Y, as before.

6.6.1 Measurable mappings into R^{n}

Let f_{1}, …, f_{n} be measurable real-valued functions on X, and let F be the corresponding mapping from X into R^{n}, so that the lth component of F is f_{l} for l = 1, …, n. Note that F is measurable with respect to the σ-algebra of Borel sets in R^{n}, as in Subsections 5.10.2 and 5.10.3. A measurable mapping from X into R^{n} may be called a random vector, as on p302 of [20].

Let ν_{F} be the Borel probability measure on R^{n} obtained by pushing μ forward using F, and let ν_{f_{l}} be the Borel probability measure on R obtained by pushing μ forward using f_{l} for each l. The measure ν_{F} is called the joint distribution of f_{1}, …, f_{n} with respect to μ, as on p303 of [20], and p315 of [118].

We would like to check that

(6.6.6) f_{1}, …, f_{n} are independent random variables on X

if and only if ν_{F} = ν_{f_{1}} × ⋯ × ν_{f_{n}} (6.6.7)

as Borel probability measures on R^{n}, where the right side is as in Section 5.12. This corresponds to some remarks on p316 of [118], and it is related to part (2) of Exercise 21.39 on p320 of [20].

Note that (6.6.7) holds if and only if

ν_{F} ( ∏_{l=1}^{n} A_{l}) = ∏_{l=1}^{n} ν_{f_{l}}(A_{l}) (6.6.8)

for all Borel sets A_{1}, …, A_{n} ⊆ R, as in Subsection 5.12.3. We also have that

F^{−1}( ∏_{l=1}^{n} A_{l}) = ⋂_{l=1}^{n} f_{l}^{−1}(A_{l}), (6.6.9)

by the definition of F. Thus

ν_{F} ( ∏_{l=1}^{n} A_{l}) = μ(F^{−1}( ∏_{l=1}^{n} A_{l})) = μ( ⋂_{l=1}^{n} f_{l}^{−1}(A_{l})). (6.6.10)

This means that (6.6.8) is equivalent to (6.4.3), by the definition ν_{f_{l}}, 1 ≤ l ≤ n.

6.6.2 Sequences of random variables

Let {f_{j}}_{j=1}^{∞} be an infinite sequence of measurable real-valued functions on X, and let us take Y_{j} = R for each j ≥ 1, considered as a measurable space with respect to the σ-algebra of Borel sets. Let us also take Y = ∏_{j=1}^{∞} Y_{j}, and remember that the corresponding product σ-algebra A_{Y} of measurable subsets of Y is the same as the σ-algebra of Borel sets in Y with respect to the product topology, or a suitable product metric, as in Subsection 5.13.3. The mapping F from X into Y whose components are the f_{j}’s is measurable, as in Subsection 5.13.4.

Let ν_{F} be the probability measure on Y obtained by pushing μ forward using F, and let ν_{f_{l}} be the probability measure on Y_{l} = R obtained by pushing μ forward using f_{l} for each l, as before. One can check that f_{1}, f_{2}, f_{3}, … are independent random variables on X with respect to μ if and only if ν_{F} is the same as the product measure on Y associated to the ν_{f_{l}}’s, as in Section 5.15.

More precisely, these two measures are the same if and only if they are the same on cylinder sets in Y, as before. One can verify that this happens if and only if the sequence of f_{l}’s is independent on X, using an argument like the one in the previous subsection.

6.7 Pushing some product measures forward

Let n be a positive integer, and let (X_{j}, A_{X_{j}}, μ_{j}) be a probability space for j = 1, …, n. This leads to a σ-algebra A_{X} on X = ∏_{j=1}^{n} X_{j} as in Section 5.10, and to a product probability measure μ on X with respect to A_{X} as in Section 5.12. If p_{l} is the obvious mapping from X onto X_{l} for each l, as in Section 5.10, then μ_{l}(A_{l}) = μ(p_{l}^{−1}(A_{l})) (6.7.1)

for every A_{l} ∈ A_{X_{l}}, by construction. More precisely, the right side is equal to μ_{l}(A_{l}) times the product of μ_{j}(X_{j}) with j ≠ l, and μ_{j}(X_{j}) = 1 for each j, by hypothesis. This means that

μ_{l} = (p_{l})_{∗}(μ), (6.7.2)

as in Section 4.11.

If γ_{l} is a nonnegative extended real-valued measurable function on X_{l}, then it follows that ∫_{X}(γ_{l} ∘ p_{l}) dμ = ∫_{X_{l}} γ_{l} dμ_{l}, (6.7.3)

as in Subsection 4.11.3. If γ_{l} is a measurable real or complex-valued function on X_{l}, then

∫_{X} |γ_{l} ∘ p_{l}| dμ = ∫_{X_{l}} |γ_{l}| dμ_{l}. (6.7.4)

Thus γ_{l} is integrable on X_{l} with respect to μ_{l} if and only if γ_{l} ∘p_{l} is integrable on X with respect to μ, and in this case, (6.7.3) holds, as before. There are analogous statements for products of sequences of probability spaces, as in Section 5.15.

6.7.1 Some mappings between product spaces

Let (Y_{j}, A_{Y_{j}}) be a measurable space for j = 1, …, n, and let A_{Y} be the corresponding σ-algebra of subsets of Y = ∏_{j=1}^{n} Y_{j}, as in Section 5.10. If ϕ_{j} is a mapping from X_{j} into Y_{j} for each j, then we get a mapping ϕ from X into Y by taking ϕ(x) = (ϕ_{1}(x_{1}), …, ϕ_{n}(x_{n})) (6.7.5) for each x ∈ X. Equivalently, ϕ_{l} ∘ p_{l} maps X into Y_{l} for l = 1, …, n, and this is the same as the composition of ϕ with the obvious mapping from Y into Y_{l}.

If ϕ_{l} is measurable for each l, then ϕ_{l} ∘ p_{l} is measurable as a mapping from X into Y_{l}, because it is a composition of measurable mappings. This implies that (6.7.6) ϕ is measurable as a mapping from X into Y,

as in Subsection 5.10.3.

The measure (ϕ_{j})_{∗}(μ_{j}) on Y_{j} with respect to A_{Y_{j}} obtained by pushing μ_{j} forward using ϕ_{j} as in Section 4.11 is a probability measure, as before. Similarly, the measure ϕ_{∗}(μ) on Y with respect to A_{Y} obtained by pushing μ forward using ϕ is a probability measure. One can check that

(6.7.7) ϕ_{∗}(μ) is the same as the product measure on Y associated to (ϕ_{j})_{∗}(μ_{j}), 1 ≤ j ≤ n.

More precisely, if B_{j} ∈ A_{Y_{j}} for j = 1, …, n, then the measure of ∏_{j=1}^{n} B_{j} with respect to the product measure is equal to the product of the measures of the B_{j}’s. The measure of B_{j} is equal to the measure of ϕ_{j}^{−1}(B_{j}) in X_{j} for each j, by construction. The product of these measures is the same as the measure of ∏_{j=1}^{n} ϕ_{j}^{−1}(B_{j}) with respect to the product measure on X. This subset of X is the same as

ϕ^{−1}( ∏_{j=1}^{n} B_{j}), (6.7.8)

by definition of ϕ, and the measure of this set with respect to μ is the same as the measure of ∏_{j=1}^{n} B_{j} with respect to ϕ_{∗}(μ).

There are analogous statements for sequences of probability spaces and measurable mappings between them, as usual.

6.8 Some identifications with finite products

Let N be a positive integer, and let L(n) be a positive integer for each n = 1, …, N. Also let Z_{n,l} be a nonempty set for each n = 1, …, N and l = 1, …, L(n). Put

W_{n} = ∏_{l=1}^{L(n)} Z_{n,l} (6.8.1)

for each n = 1, …, N, and

W = ∏_{n=1}^{N} W_{n}. (6.8.2)

An element of W may be expressed as w = (w_{1}, …, w_{N}), where

w_{n} = (w_{n,1}, …, w_{n,L(n)}) (6.8.3)

is an element of W_{n} for each n. Of course, this means that w_{n,l} ∈ Z_{n,l} for l = 1, …, L(n).

6.8.1 Another Cartesian product

We can also consider the Cartesian product of the Z_{n,l}’s more directly, as follows. Put

L = ∑_{n=1}^{N} L(n), (6.8.4)

and observe that each r = 1, …, L may be expressed in a unique way as

r = L(n − 1) + l, (6.8.5)

where 1 ≤ n ≤ N, 1 ≤ l ≤ L(n), and L(0) is interpreted as being equal to 0. In this case, we put Z(r) = Z_{n,l}, (6.8.6) and consider the Cartesian product

Z = ∏_{r=1}^{L} Z(r) (6.8.7)

of these sets. Let us express an element of Z as z = (z(1), …, z(L)). There is an obvious one-to-one correspondence of Z with W, where z ∈ Z corresponds to w ∈ W with w_{n,l} = z(r) (6.8.8) when r is as in (6.8.5).

6.8.2 Associated product σ-algebras

Let A_{Z_{n,l}} be a σ-algebra of subsets of Z_{n,l} for each n = 1, …, N and l = 1, …, L(n). This may also be expressed as A_{Z(r)} when r is as in (6.8.5). Using these σ-algebras, we get product σ-algebras A_{W_{n}} on W_{n} for each n = 1, …, N, A_{W} on W, and A_{Z} on Z, as in Section 5.10. In fact, A_{W} corresponds to A_{Z}, using the identification between W and Z mentioned in the preceding paragraph. This is related to the associativity properties of the product σalgebras mentioned on p65 of [118].

This corresponds to the remarks in Section 5.11 when N = 2, and one can use analogous arguments here. The measurability of this mapping from Z onto W is equivalent to the measurability of the corresponding mappings from Z onto W_{n} for each n = 1, …, N, as in Subsection 5.10.3. The measurability of these mappings reduces to the measurability of the corresponding mappings from Z onto Z_{n,l} for each n = 1, …, N and l = 1, …, L(n). This is the same as the measurability of the obvious coordinate projection from Z onto Z(r) for r = 1, …, L.

Similarly, the measurability of the inverse mapping from W onto Z is equivalent to the measurability of the corresponding mappings from W onto Z(r) for each r = 1, …, L. If r is as in (6.8.5), then this is the same as the mapping from W onto Z_{n,l} obtained by composing the obvious projection from W onto W_{n} with the coordinate projection from W_{n} onto Z_{n,l}. These mappings are measurable, by construction, so that their composition is measurable.

6.8.3 Associated product measures

Let μ_{Z_{n,l}} be a σ-finite measure on Z_{n,l} with respect to A_{Z_{n,l}} for each n = 1, …, N and l = 1, …, L(n), and let us also express this measure as μ_{Z(r)} when r is as in (6.8.5). This leads to product measures μ_{W_{n}} on W_{n} with respect to A_{W_{n}} for each n = 1, …, N, μ_{W} on W with respect to A_{W}, and μ_{Z} on Z with respect to A_{Z}, as in Section 5.12.

One can check that μ_{W} corresponds to μ_{Z} with respect to the identification between W and Z mentioned earlier, because of the characterization of μ_{Z} in terms of products of measurable sets in Subsection 5.12.3, and the analogous properties of μ_{W_{n}} and μ_{W}. This corresponds to a remark in Subsection 5.12.3 when N = 2, and it is related to the associativity properties of product measures mentioned on p65 of [118].

6.9 Some combinations of random variables

Let N be a positive integer, and let L(n) be a positive integer for each n = 1, …, N, as in the previous section. Also let (X, A_{X}, μ_{X}) be a probability space, and let f_{r} be a measurable real-valued function on X for r = 1, …, L, where L is as in (6.8.4). This may be expressed as f_{n,l} when r is as in (6.8.5). Let F_{n} be the mapping from X into R^{L(n)} whose lth component is f_{n,l} for n = 1, …, N and l = 1, …, L(n). Remember that F_{n} is measurable with respect to the σ-algebra of Borel sets in R^{L(n)}, as in Subsection 6.6.1.

Let α_{n} be a Borel measurable real-valued function on R^{L(n)} for n = 1, …, N. This implies that β_{n} = α_{n} ∘ F_{n} (6.9.1)

is a measurable real-valued function on X for each n. Equivalently,

β_{n}(x) = α_{n}(f_{n,1}(x), …, f_{n,L(n)}(x)) (6.9.2)

for every x ∈ X. Let β be the mapping from X into R^{N} whose nth coordinate is equal to β_{n} for each n = 1, …, N. As before, β is measurable with respect to the σ-algebra of Borel sets in R^{N}.

We can identify

∏_{n=1}^{N} R^{L(n)} (6.9.3)

with R^{L}, using the remarks in the previous section, with Z_{n,l} = R for n = 1, …, N and l = 1, …, L(n). Let F be the mapping from X into (6.9.3) whose nth component is F_{n} for n = 1, …, N. This corresponds to the mapping from X into R^{L} whose rth component is f_{r} for r = 1, …, L.

Let α be the mapping from (6.9.3) into R^{N} obtained from the α_{n}’s as in Subsection 6.7.1. Thus β = α ∘ F, (6.9.4)

by construction.

If f_{1}, …, f_{L} are independent as random variables on X, then Proposition 10.2 on p316 of [118] states that

(6.9.5) β_{1}, …, β_{N} are independent

as random variables on X as well. This corresponds to the analogous statement in Subsection 6.4.4 when L(n) = 1 for each n.

6.9.1 The measure β_{∗}(μ) on R^{N}

To see this, it suffices to check that the measure β_{∗}(μ) on R^{N} obtained by pushing μ forward using β is the same as the product measure corresponding to the measures (β_{n})_{∗}(μ) on R obtained by pushing μ forward using β_{n} for n = 1, …, N, as in Subsection 6.6.1. Note that β_{∗}(μ) is the same as the measure α_{∗}(F_{∗}(μ)) obtained by first pushing μ forward to a Borel measure on (6.9.3) using F, and then pushing that measure forward to a Borel measure on R^{N} using α, as in Subsection 4.11.2.

We can identify F_{∗}(μ) with a Borel measure on R^{L}, by identifying R^{L} with (6.9.3), as before. This is the same as the Borel measure on R^{L} obtained by pushing μ forward using the mapping from X into R^{L} whose rth component is f_{r} for r = 1, …, L. The hypothesis that f_{1}, …, f_{L} be independent as random variables on X implies that this Borel measure on R^{L} is the same as the product measure associated to the Borel measures (f_{r})_{∗}(μ) on R obtained by pushing μ forward using f_{r} for r = 1, …, L, as in Subsection 6.6.1.

Similarly, for each n = 1, …, N, the Borel measure (F_{n})_{∗}(μ) on R^{L(n)} obtained by pushing μ forward using F_{n} is the same as the product measure associated to the Borel measures (f_{n,l})_{∗}(μ) on R obtained by pusing μ forward using f_{n,l} for l = 1, …, L(n). Using (F_{n})_{∗}(μ) for n = 1, …, N, we get a product measure on (6.9.3). This corresponds to the product measure on R^{L} associated to (f_{r})_{∗}(μ) for r = 1, …, L, as in Subsection 6.8.3.

This means that F_{∗}(μ) is the same as the product measure on (6.9.3) associated to (F_{n})_{∗}(μ) on R^{L(n)} for n = 1, …, N. The measure α_{∗}(F_{∗}(μ)) on R^{N} obtained by pushing F_{∗}(μ) forward using α is the same as the product measure associated to the measures (α_{n})_{∗}((F_{n})_{∗}(μ)) on R obtained by pushing (F_{n})_{∗}(μ) forward using α_{n} for n = 1, …, N, as in Subsection 6.7.1. These measures on R are the same as (β_{n})_{∗}(μ), as in Subsection 4.11.2. This shows that β_{∗}(μ) is the same as the product measure on R^{N} associated to (β_{n})_{∗}(μ) on R for n = 1, …, N, so that (6.9.5) holds.

6.10 The multiplication theorem

Let (X, A, μ) be a probability space, and suppose for the moment that (X, A, μ) is the product of n probability spaces (X_{l}, A_{X_{l}}, μ_{l}), 1 ≤ l ≤ n, as in Section 5.12. Also let γ_{l} be a nonnegative extended real-valued measurable function on X_{l} for each l, and let p_{l} be the usual coordinate mapping from X onto X_{l} for each l, as in Sections 5.10. Thus γ_{l} ∘ p_{l} is measurable on X for each l, and we have that

∫_{X} ( ∏_{l=1}^{n}(γ_{l} ∘ p_{l})) dμ = ∏_{l=1}^{n} ( ∫_{X_{l}} γ_{l} dμ_{l}), (6.10.1)

by a suitable version of Tonelli’s theorem for products of n measure spaces.

More precisely, if n = 2, then one can use Tonelli’s theorem, as in Subsection 5.9.2. Otherwise, one can reduce to the n = 2 case, by considering (X, A, μ) as the product of two probability spaces, as in Subsection 5.12.4. These other two probability spaces may be products of fewer probability spaces, and one can use Tonelli’s theorem repeatedly, as needed.

Note that (6.10.1) is the same as saying that

∫_{X} ( ∏_{l=1}^{n}(γ_{l} ∘ p_{l})) dμ = ∏_{l=1}^{n} ( ∫_{X}(γ_{l} ∘ p_{l})dμ), (6.10.2)

because of (6.7.3).

If γ_{l} is a measurable real-valued function on X_{l} for each l, then

∫_{X} ( ∏_{l=1}^{n} |γ_{l} ∘ p_{l}|) dμ = ∏_{l=1}^{n} ( ∫_{X_{l}} |γ_{l}| dμ_{l}) = ∏_{l=1}^{n} ( ∫_{X} |γ_{l} ∘ p_{l}| dμ), (6.10.3)

as before. Remember that γ_{l} is integrable on X_{l} with respect to μ_{l} if and only if γ_{l} ∘ p_{l} is integrable on X with respect to μ for each l, as in Section 6.7. If this holds for all l = 1, …, n, then it follows from (6.10.3) that

(6.10.4)  ∏_{l=1}^{n}(γ_{l} ∘ p_{l}) is integrable on X with respect to μ.

In this case, (6.10.1) and thus (6.10.2) hold, using Fubini’s theorem in an argument like the earlier one, or by reducing to the previous statement for nonnegative functions.

6.10.1 Reducing to the case of product measures

Now let (X, A, μ) be any probability space, let f_{1}, …, f_{n} be measurable realvalued functions on X, and let F be the corresponding mapping from X into R^{n}, as in Subsection 6.6.1. Also let ν_{F} be the Borel probability measure on R^{n} obtained by pushing μ forward using F, as before, and let a be a nonnegative extended real-valued Borel measurable function on R^{n}. Remember that a ∘ F is measurable on X, as in Subsection 3.1.3, and that

∫_{X}(a ∘ F) dμ = ∫_{R^{n}} a dν_{F} , (6.10.5)

as in Subsection 4.11.3. Similarly, if a is a real or complex-valued Borel measurable function on R^{n}, then a ∘ F is measurable on X, and a is integrable on R^{n} with respect to ν_{F} if and only if a ∘ F is integrable on X with respect to μ, by the analogue of the previous statement for |a|. In this case, (6.10.5) holds again, as before.

6.10.2 Using nonnegative functions on the real line

Suppose that

(6.10.6) f_{1}, …, f_{n} are independent as random variables on X,

so that (6.6.7) holds, where ν_{f_{l}} is the Borel probability measure on R obtained by pushing μ forward using f_{l} for each l, as before. Let a_{1}, …, a_{n} be nonnegative extended real-valued Borel measurable functions on the real line, and note that a_{l}(y_{l}) is a Borel measurable function of y ∈ R^{n} for each l, because it is the composition of the lth coordinate projection on R^{n} with a_{l}. This implies that

a(y) = a_{1}(y_{1}) ⋯ a_{n}(y_{n}) (6.10.7)

is Borel measurable as a function of y ∈ R^{n}, and

(a ∘ F)(x) = a_{1}(f_{1}(x)) ⋯ a_{n}(f_{n}(x)) (6.10.8)

for each x ∈ X. Thus

∫_{X}(a ∘ F) dμ = ∫_{X} ( ∏_{l=1}^{n}(a_{l} ∘ f_{l})) dμ (6.10.9)

under these conditions. This means that

∫_{X} ( ∏_{l=1}^{n}(a_{l} ∘ f_{l})) dμ = ∫_{R^{n}} a dν_{F} , (6.10.10)

because of (6.10.5).

We also have that

∫_{R^{n}} a dν_{F} = ∏_{l=1}^{n} ( ∫_{R} a_{l}(y_{l}) dν_{f_{l}}(y_{l})), (6.10.11)

because of (6.6.7), as in (6.10.1). This means that

∫_{R^{n}} a dν_{F} = ∏_{l=1}^{n} ( ∫_{X}(a_{l} ∘ f_{l}) dμ), (6.10.12)

as in Section 4.11.3, by definition of ν_{f_{l}}. Combining this with (6.10.10), we obtain that

∫_{X} ( ∏_{l=1}^{n}(a_{l} ∘ f_{l})) dμ = ∏_{l=1}^{n} ( ∫_{X}(a_{l} ∘ f_{l}) dμ). (6.10.13)

6.10.3 Using other functions on the real line

If a_{1}, …, a_{n} are Borel measurable real-valued functions on the real line, then we can use the previous argument with |a_{l}| in place of a_{l} to get that

∫_{X} ( ∏_{l=1}^{n} |a_{l} ∘ f_{l}|) dμ = ∏_{l=1}^{n} ( ∫_{X} |a_{l} ∘ f_{l}| dμ). (6.10.14)

If a_{l} ∘ f_{l} is integrable on X with respect to μ for each l, then we get that

(6.10.15)  ∏_{l=1}^{n}(a_{l} ∘ f_{l}) is integrable on X with respect to μ.

Under these conditions, one can get (6.10.13), by reducing to the case of nonnegative functions. This corresponds to 12.17 on p387 of [14], Theorem 21.10 on p274 of [20], and Proposition 10.5 on p317 of [118]. This may be called the multiplication theorem, as in [20].

More precisely, this is normally stated with a_{j} equal to the identity function on R for each j. One can reduce to that case using the remarks in Subsection 6.4.4.

6.10.4 A more direct approach

Remember that indicator functions are independent as random variables if and only if the corresponding events are independent, as in Subsection 6.4.1. It is easy to see that the multiplication theorem holds in this case.

One can use this to verify the multiplication theorem for independent random variables that are also simple functions. In this case, the product of these functions may be expressed as a linear combination of products of indicator functions of independent events.

One can approximate measurable functions by measurable simple functions, as mentioned in Subsection 3.7.1. In fact, these approximations may be obtained using compositions with Borel measurable simple functions on the real line that approximate the identity function. This implies that these approximations preserve independence, as in Subsection 6.4.4.

One can use this to reduce the multiplication theorem to the case of independent random variables that are simple functions. This is the type of argument used in [14, 20], and the argument in the previous subsections is essentially the same as in [118].

6.11 Some remarks about finite moments

Let (X, A, μ) be a probability space, let f be a measurable real-valued function on X, and let ν_{f} = f_{∗}(μ) be the Borel probability measure on the real line obtained by pushing μ forward using f, as in Section 4.11. If p is a positive real number, then

E(|f|^{p}) = ∫_{X} |f|^{p} dμ = ∫_{R} |t|^{p} dν_{f}(t), (6.11.1)

where the second step is as in Subsection 4.11.3. One may say that f has finite pth moment with respect to μ on X when this is finite, as mentioned on p314 of [118]. Similarly, if f is integrable on X with respect to μ, then

E(f) = ∫_{X} f dμ = ∫_{R} t dν_{f}(t), (6.11.2) as before.

If 0 < p_{1} < p_{2} < ∞, and f has finite p_{2}-th moment, then f has finite p_{1}-th moment, with E(|f|^{p_{1}})^{1/p_{1}} ≤ E(|f|^{p_{2}})^{1/p_{2}}, (6.11.3)

as in Subsection 5.3.3.

6.11.1 The variance of f

If f is integrable on X with respect to μ, then the variance of f may be defined by Var(f) = E(f − E(f))^{2}, (6.11.4)

as on p388 of [14], p271 of [20], and p314 of [118]. This is finite exactly when f is square-integrable with respect to μ on X. In this case, the standard deviation of f is defined to be Var(f)^{1/2}, (6.11.5)

using the nonnegative square root, as in [14, 20, 118]. This may be denoted σ(f), as in [14, 118], although the same notation is used on p268 of [20] for the σ-algebra of subsets of X of the form f^{−1}(V), where V is a Borel set in R.

Observe that

(6.11.6) Var(f) = E(f^{2} − 2 E(f) f + E(f)^{2})
= E(f^{2}) − 2 E(f)^{2} + E(f)^{2} = E(f^{2}) − E(f)^{2},

as in 12.20 on p389 of [14], 21.3 on p271 of [20], and on p314 of [118]. In particular, Var(f) ≤ E(f^{2}), (6.11.7)

as mentioned in [20]. We also have that

Var(f + c) = Var(f) (6.11.8)

for every c ∈ R, considered as a constant function on X, as mentioned in [20].

6.11.2 Another description of the variance

The variance is initially defined on p314 of [118] by

Var(f) = inf_{a∈R} E((f − a)^{2}), (6.11.9)

without asking that f be integrable on X. This is equal to +∞ unless f is square-integrable on X. If f is integrable on X, then one can check that this definition of the variance is equivalent to the previous one, using (6.11.7) and (6.11.8). Alternatively, if a ∈ R, then

E((f − a)^{2}) = E(f^{2}) − 2 a E(f) + a^{2}. (6.11.10)

If f is square-integrable on X, then one can verify that (6.11.10) is minimized exactly when a = E(f), as mentioned in [118].

6.11.3 More on the variance

If f is integrable on X with respect to μ, then

Var(f) = ∫_{R}(t − E(f))^{2} dν_{f}(t), (6.11.11)

by the remarks in Subsection 4.11.3. We also have that

Var(f) = ∫_{R} t^{2} dν_{f}(t) − ( ∫_{R} t dν_{f}(t))^{2}, (6.11.12)

as in (6.11.6), using (6.11.1) and (6.11.2). This shows in particular that the variance of f depends only on the probability distribution ν_{f} in R associated to f and μ.

If r is a positive real number, then another version of Chebyshev’s inequality states that μ({x ∈ X : |f(x) − E(f)| ≥ r}) ≤ r^{−2} Var(f), (6.11.13)

as in 12.21 on p389 of [14], and (21.4) on p272 of [20]. This follows from the version in Subsection 5.2.1, with p = 2.

6.12 The variance of a sum

Let (X, A, μ) be a probability space. If f, g are real-valued measurable functions on X that are square-integrable with respect to μ, then put

⟨f, g⟩ = E(f g) = ∫_{X} f g dμ, (6.12.1)

as in Section 5.5. Thus E(f^{2}) = ⟨f, f⟩ = ∥f∥_{2}^{2}, (6.12.2)

where the right side is as in Section 5.1. In particular,

Var(f) = ∥f − E(f)∥_{2}^{2} = ⟨f − E(f), f − E(f)⟩. (6.12.3)

Let f_{1}, …, f_{n} be finitely many real-valued measurable functions on X that are square-integrable with respect to μ. Suppose that

E((f_{j} − E(f_{j})) (f_{l} − E(f_{l}))) = 0 (6.12.4)

when j ≠ l. This means that the functions f_{j} − E(f_{j}) are pairwise orthogonal with respect to (6.12.1), as in Subsection 5.5.1. It follows that

(6.12.5)  Var ( ∑_{j=1}^{n} f_{j}) = ∥∑_{j=1}^{n}(f_{j} − E(f_{j}))∥_{2}^{2}

= ∑_{j=1}^{n} ∥f_{j} − E(f_{j})∥_{2}^{2} = ∑_{j=1}^{n} Var(f_{j}),

as before.

If f_{j} and f_{l} are independent as random variables, then it is easy to see that

(6.12.6) f_{j} − E(f_{j}) and f_{l} − E(f_{l}) are independent as random variables

too. In this case,

E((f_{j} − E(f_{j})) (f_{l} − E(f_{l}))) = E(f_{j} − E(f_{j})) E(f_{l} − E(f_{l})) = 0, (6.12.7)

using the multiplication theorem in the first step. In particular, this holds when j ≠ l and f_{1}, …, f_{n} are independent random variables on X. Under these conditions, (6.12.5) corresponds to 12.22 on p390 of [14], Remark 21.11 on p275 of [20], and Corollary 10.6 on p317 of [118].

6.12.1 The variance of an average

Let {f_{j}}_{j=1}^{∞} be an infinite sequence of measurable real-valued functions on X that are square-integrable with respect to μ, and suppose that (6.12.4) holds when j ≠ l. This implies that

∥n^{−1} ∑_{j=1}^{n}(f_{j} − E(f_{j}))∥_{2}^{2} = n^{−2} ∑_{j=1}^{n} Var(f_{j}) (6.12.8)

for every positive integer n, as in (6.12.5). If

lim_{n→∞} n^{−2} ∑_{j=1}^{n} Var(f_{j}) = 0, (6.12.9)

then

n^{−1} ∑_{j=1}^{n}(f_{j} − E(f_{j})) → 0 as n → ∞ in L^{2}, (6.12.10)

as in Section 5.2. In particular, this means that

(6.12.11) n^{−1} ∑_{j=1}^{n}(f_{j} − E(f_{j})) → 0 as n → ∞ in measure,

as in Subsection 5.2.1. This corresponds to the formulation of the weak law of large numbers in 10.9 on p321 of [118] when {f_{j}}_{j=1}^{∞} is a sequence of independent random variables on X. The sufficiency of (6.12.4) when j ≠ l in place of independence is mentioned in Exercise 13 on p324 of [118].

Suppose that there is a real number a such that

E(f_{j}) = a (6.12.12)

for each j, so that

n^{−1} ∑_{j=1}^{n}(f_{j} − E(f_{j})) = n^{−1} ∑_{j=1}^{n} f_{j} − a (6.12.13)

for each n ≥ 1. In this case, (6.12.10) and (6.12.11) are the same as saying that

(6.12.14) n^{−1} ∑_{j=1}^{n} f_{j} → a as n → ∞ in L^{2}, and thus in measure.

This corresponds to the version of the weak law of large numbers in part (b) of Exercise 22.32 on p452 of [159] when a = 0, and for a sequence of independent random variables as in Subsection 6.4.3.

Suppose for the moment that the variances of the f_{j}’s are bounded, so that

Var(f_{j}) ≤ C (6.12.15)

for some nonnegative real number C and each j. This implies that

n^{−2} ∑_{j=1}^{n} Var(f_{j}) ≤ C/n (6.12.16)

for each n, so that (6.12.9) holds.

6.12.2 Identically distributed random variables

A family of random variables on X is said to be identically distributed with respect to μ if

(6.12.17) they determine the same probability distributions on R with respect to μ,

as in Subsection 4.11.1. This is equivalent to saying that these random variables have the same distribution functions, as in Section 4.13. This corresponds to Definition 12.35 on p396 of [14], and some remarks on p276 of [20] and p315 of [118]. In this case, the random variables have the same integrability properties, means, and variances, for instance.

The weak law of large numbers is stated for sequences of independent identically distributed square-integrable random variables in 12.38 on p397 of [14] and Theorem 21.12 on p276 of [20].

6.13 A criterion for convergence to 0 a.e.

Let (X, A, μ) be a measure space, and let p be a positive real number. Also let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X that are pth-power integrable with respect to μ, as in Section 5.1, and suppose that

∑_{j=1}^{∞} ∥a_{j}∥_{p}^{p} = ∑_{j=1}^{∞} ∫_{X} |a_{j}|^{p} dμ < +∞. (6.13.1)

This means that

∫_{X} ( ∑_{j=1}^{∞} |a_{j}|^{p}) dμ < +∞, (6.13.2)

as in Subsection 3.11.1, so that

∑_{j=1}^{∞} |a_{j}(x)|^{p} < +∞ (6.13.3)

for almost every x ∈ X with respect to μ, as in Subsection 3.10.2. It follows in particular that

lim_{j→∞} a_{j}(x) = 0 (6.13.4)

for almost every x ∈ X with respect to μ.

6.13.1 Some L^{4} norms of sums

Let f_{1}, …, f_{n} be finitely many measurable real-valued functions on X that are 4th power integrable on X with respect to μ, as in Section 5.1 again. Observe that

∥∑_{j=1}^{n} f_{j}∥_{4}^{4} = ∫_{X} ( ∑_{j=1}^{n} f_{j})^{4} dμ = ∫_{X} ∑_{j=1}^{n} ∑_{k=1}^{n} ∑_{l=1}^{n} ∑_{m=1}^{n} f_{j} f_{k} f_{l} f_{m} dμ

(6.13.5)  = ∑_{j=1}^{n} ∑_{k=1}^{n} ∑_{l=1}^{n} ∑_{m=1}^{n} ∫_{X} f_{j} f_{k} f_{l} f_{m} dμ.

More precisely, f_{j} f_{k} f_{l} f_{m} is integrable on X with respect to μ for all j, k, l, m = 1, …, n, as in Subsection 5.3.4.

6.13.2 Using independence and mean 0

Suppose now that μ is a probability measure on X, and that f_{1}, …, f_{n} are also independent as random variables on X, and that they all have mean 0. If j, k, l, and m are all distinct, then it is easy to see that

∫_{X} f_{j} f_{k} f_{l} f_{m} dμ = 0, (6.13.6)

using the multiplication theorem. In fact, one can verify that this holds when at least one of the indices is distinct from each of the other three.

If all of the indices are the same, then the sum of the corresponding terms on the right side of (6.13.5) is equal to

∑_{j=1}^{n} ∫_{X} f_{j}^{4} dμ = ∑_{j=1}^{n} ∥f_{j}∥_{4}^{4}. (6.13.7)

Otherwise, if the four indices occur in pairs of equal indices that are distinct from the other pair, then the sum of the corresponding terms on the right side of (6.13.5) is equal to

3 ∑_{j=1}^{n} ∑_{{1≤r≤n \atop r≠j}} ∫_{X} f_{j}^{2} f_{r}^{2} dμ. (6.13.8)

If j ≠ r, then

∫_{X} f_{j}^{2} f_{r}^{2} dμ = ( ∫_{X} f_{j}^{2} dμ) ( ∫_{X} f_{r}^{2} dμ) = ∥f_{j}∥_{2}^{2} ∥f_{r}∥_{2}^{2}, (6.13.9)

using the multiplication theorem in the first step. Thus (6.13.8) is equal to

3 ∑_{j=1}^{n} ∑_{{1≤r≤n \atop r≠j}} ∥f_{j}∥_{2}^{2} ∥f_{r}∥_{2}^{2}. (6.13.10)

This is less than or equal to

3 ∑_{j=1}^{n} ∑_{r=1}^{n} ∥f_{j}∥_{2}^{2} ∥f_{r}∥_{2}^{2} = 3 ( ∑_{j=1}^{n} ∥f_{j}∥_{2}^{2})^{2}. (6.13.11)

It follows that

∥∑_{j=1}^{n} f_{j}∥_{4}^{4} ≤ ∑_{j=1}^{n} ∥f_{j}∥_{4}^{4} + 3 ( ∑_{j=1}^{n} ∥f_{j}∥_{2}^{2})^{2} (6.13.12)

under these conditions.

6.13.3 Using a bound on ∥f_{j}∥_{4}

Suppose that ∥f_{j}∥_{4} ≤ C (6.13.13)

for some nonnegative real number C and each j, so that

∥f_{j}∥_{2} ≤ C (6.13.14)

for each j, as in Subsection 5.3.3 and (6.11.3). In this case, (6.13.7) and (6.13.10) are less than or equal to

C^{4} n and 3 C^{4} n (n − 1), (6.13.15)

respectively. This means that

∥∑_{j=1}^{n} f_{j}∥_{4}^{4} ≤ C^{4} n + 3 C^{4} n (n − 1) ≤ 3 C^{4} n^{2}, (6.13.16)

using the same arguments as before. This corresponds to part (1) of Exercise 21.9 on p315 of [20].

6.13.4 Another convergence property

Now let {f_{j}}_{j=1}^{∞} be a sequence of measurable real-valued functions on X that are 4th-power integrable with respect to μ, independent as random variables on X, and have mean 0. If (6.13.13) holds for some C ≥ 0 and each j, then

∥n^{−1} ∑_{j=1}^{n} f_{j}∥_{4}^{4} ≤ 3 C^{4} n^{−2} (6.13.17)

for every positive integer n, as in (6.13.16). In particular, this means that

∑_{n=1}^{∞} ∥n^{−1} ∑_{j=1}^{n} f_{j}∥_{4}^{4} < +∞, (6.13.18)

so that

lim_{n→∞} n^{−1} ∑_{j=1}^{n} f_{j}(x) = 0 (6.13.19)

for almost every x ∈ X with respect to μ, by the remarks at the beginning of the section. This corresponds to part (2) of Exercise 21.9 on p315 of [20], and may be considered as a version of the strong law of large numbers.

6.14 An inequality of Kolmogorov

Let (X, A, μ) be a probability space, and let f_{1}, …, f_{n} be finitely many measurable real-valued functions on X that are independent as random variables on X, square-integrable with respect to μ, and have mean 0. If ϵ is a positive real number, then Kolmogorov’s inequality states that

μ({x ∈ X : max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}) ≤ ϵ^{−2} ∑_{l=1}^{n} ∥f_{l}∥_{2}^{2}. (6.14.1)

This is 10.11 on p322 of [118].

Note that

max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≤ ∑_{j=1}^{n} |f_{j}(x)| (6.14.2)

for every x ∈ X. This implies that

μ({x ∈ X : max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}) (6.14.3)

≤ μ({x ∈ X : ∑_{j=1}^{n} |f_{j}(x)| ≥ ϵ}).

It follows that

μ({x ∈ X : max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}) ≤ ϵ^{−1} ∑_{l=1}^{n} ∥f_{l}∥_{1}, (6.14.4)

because of Chebyshev’s inequality, as in Subsection 4.7.1. This does not use independence of the f_{j}’s, or that they have mean 0.

6.14.1 Some measurable sets A_{l}

Put A_{1} = {x ∈ X : |f_{1}(x)| ≥ ϵ}. (6.14.5)

If 2 ≤ l ≤ n, then put

(6.14.6) A_{l} = {x ∈ X : |∑_{j=1}^{l} f_{j}(x)| ≥ ϵ and

|∑_{j=1}^{k} f_{j}(x)| < ϵ for k = 1, …, l − 1}.

One can check that A_{1}, …, A_{n} are pairwise-disjoint measurable subsets of X such that

⋃_{l=1}^{n} A_{l} = {x ∈ X : max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}. (6.14.7)

More precisely, if x is an element of the set on the right, then x ∈ A_{l} when l is the smallest k such that

| ∑_{j=1}^{k} f_{j}(x)| ≥ ϵ. (6.14.8)

6.14.2 Some related independence properties

One can express 1_{A_{l}} as

1_{A_{l}}(x) = β_{l}(f_{1}(x), …, f_{l}(x)), (6.14.9)

where β_{l} is the indicator function of a Borel set in R^{l}. This implies that

1_{A_{l}}(x) ( ∑_{j=1}^{l} f_{j}(x)) = α_{l}(f_{1}(x), …, f_{l}(x)) (6.14.10)

where α_{l} is a Borel measurable function on R^{l}. If l < n, then we get that

(6.14.11) 1_{A_{l}} ( ∑_{j=1}^{l} f_{j}), f_{l+1}, …, f_{n} are independent

as random variables on X, as in Section 6.9.

If l < r ≤ n, then we can use the multiplication theorem to get that

∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j}) f_{r} dμ = ( ∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j}) dμ) ( ∫_{X} f_{r} dμ) = 0, (6.14.12)

because f_{r} has mean 0, by hypothesis. This means that

∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j}) ( ∑_{r=l+1}^{n} f_{r}) dμ = 0 (6.14.13)

when l < n.

6.14.3 Using Chebyshev’s inequality another way

Clearly

μ({x ∈ X : max_{1≤k≤n} |∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}) = ∑_{l=1}^{n} μ(A_{l}), (6.14.14)

because of (6.14.7) and the pairwise-disjointness of the A_{l}’s. We also have that

μ(A_{l}) ≤ ϵ^{−2} ∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j})^{2} dμ (6.14.15)

for each l, using Chebyshev’s inequality, as in Subsection 5.2.1, with p = 2. This uses the fact that | ∑_{j=1}^{l} f_{j}| ≥ ϵ on A_{l}, by construction, as well.

Of course,

( ∑_{j=1}^{n} f_{j})^{2} = ( ∑_{j=1}^{l} f_{j} + ∑_{j=l+1}^{n} f_{j})^{2}

(6.14.16)  = ( ∑_{j=1}^{l} f_{j})^{2} + 2 ( ∑_{j=1}^{l} f_{j}) ( ∑_{r=l+1}^{n} f_{r})^{2} + ( ∑_{r=l+1}^{n} f_{r})^{2}

≥ ( ∑_{j=1}^{l} f_{j})^{2} + 2 ( ∑_{j=1}^{l} f_{j}) ( ∑_{r=l+1}^{n} f_{r})

when l < n. This implies that

∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j})^{2} dμ ≤ ∫_{X} 1_{A_{l}} ( ∑_{j=1}^{n} f_{j})^{2} dμ, (6.14.17)

using (6.14.13) when l < n. Thus

(6.14.18) ∑_{l=1}^{n} ∫_{X} 1_{A_{l}} ( ∑_{j=1}^{l} f_{j})^{2} dμ ≤ ∑_{l=1}^{n} ∫_{X} 1_{A_{l}} ( ∑_{j=1}^{n} f_{j})^{2} dμ

≤ ∫_{X} ( ∑_{j=1}^{n} f_{j})^{2} dμ,

using the fact that the A_{l}’s are pairwise-disjoint in the second step. Combining (6.14.14), (6.14.15), and (6.14.18), we get that

μ({x ∈ X : max_{1≤k≤n} | ∑_{j=1}^{k} f_{j}(x)| ≥ ϵ}) ≤ ϵ^{−2} ∫_{X} ( ∑_{j=1}^{n} f_{j})^{2} dμ. (6.14.19)

Note that

∫_{X} ( ∑_{j=1}^{n} f_{j})^{2} dμ = ∥∑_{j=1}^{n} f_{j}∥_{2}^{2} = ∑_{j=1}^{n} ∥f_{j}∥_{2}^{2}, (6.14.20)

because the f_{j}’s are independent with mean 0, and are thus pairwise orthogonal, as in Section 6.12. This implies (6.14.1).

6.15 Some more convergence results

Let (X, A, μ) be a measure space, and let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X. Put

g_{n}(x) = sup_{m≥n} |f_{m}(x) − f_{n}(x)| (6.15.1)

for each positive integer n and x ∈ X, where more precisely the supremum is taken over all integers m ≥ n. If {g_{n}}_{n=1}^{∞} converges to 0 in measure on X with respect to μ, then Exercise 10.7 on p87 of [20] asks one to show that {f_{j}}_{j=1}^{∞} converges pointwise almost everywhere on X.

To see this, put

b_{n}(x) = sup_{l,r≥n} |f_{l}(x) − f_{r}(x)| (6.15.2)

for each n ≥ 1 and x ∈ X, where the supremum is taken over all integers l, r ≥ n. One can check that {b_{n}}_{n=1}^{∞} converges to 0 in measure on X with respect to μ when {g_{n}}_{n=1}^{∞} has this property.

One can also verify that {b_{n}(x)}_{n=1}^{∞} converges to 0 for almost every x ∈ X with respect to μ when {b_{n}}_{n=1}^{∞} converges to 0 in meagure on X with respect to μ. This means that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in R or C, as appropriate, for almost every x ∈ X with respect to μ.

6.15.1 Some convergent series

Suppose from now on in this section that μ(X) = 1. Let {a_{j}}_{j=1}^{∞} be a sequence of real-valued measurable functions on X that are square-integrable with respect to μ, independent as random variables on X, and have mean 0. If

∑_{j=1}^{∞} ∥a_{j}∥_{2}^{2} < +∞, (6.15.3)

then Exercise 14 on p324 of [118] asks one to show that ∑_{j=1}^{∞} a_{j}(x) converges for almost every x ∈ X with respect to μ. More precisely, it is suggested in [118] that one use Kolmogorov’s inequality to show that the sequence of partial sums is a Cauchy sequence for almost every x ∈ X with respect to μ. Note that the sequence of partial sums converges in L^{2}, as in Subsection 5.5.1, because the a_{j}’s are pairwise orthogonal in the appropriate sense, as in Section 6.12.

6.15.2 Kolmogorov’s strong law

Let {f_{j}}_{j=1}^{∞} be an infinite sequence of measurable real-valued functions on X that are square-integrable with respect to μ and independent as random variables on X. If

∑_{j=1}^{∞} Var(f_{j})/j^{2} < +∞, (6.15.4)

then 10.12 on p322 of [118] says that

(6.15.5) n^{−1} ∑_{j=1}^{n}(f_{j} − E(f_{j})) → 0 as n → ∞ almost everywhere on X

with respect to μ. This is Komogorov’s strong law of large numbers, and it is shown using Kolmogorov’s inequality in the previous section. This corresponds to Theorem 22.29 on p447 of [159] for independent random variables as in Subsection 6.4.3.

6.15.3 Some sequences and series

Exercise 11 on p324 of [118] asks one to show that (6.15.4) implies the hypothesis (6.12.9) of the weak law of large numbers.

If {α_{k}}_{k=1}^{∞} is a sequence of complex numbers such that

(6.15.6)  ∑_{k=1}^{∞} α_{k}/k converges,

then Lemma 22.28 on p447 of [159] states that

lim_{n→∞} n^{−1} ∑_{k=1}^{n} α_{k} = 0. (6.15.7)

This is used to obtain the strong law of large numbers from the convergence result in the subsection before that.

6.15.4 Khinchine’s strong law

Suppose now that {f_{j}}_{j=1}^{∞} is an infinite sequence of real-valued measurable functions on X that are integrable with respect to μ and independent as random variables on X. If the f_{j}’s are also identically distributed as random variables on X, as in Subsection 6.12.2, then Khinchine’s strong law of large numbers says that

(6.15.8) n^{−1} ∑_{j=1}^{n} f_{j} converges to the mean of the f_{l}’s as n → ∞

almost everywhere on X with respect to μ. This is 10.13 on p323 of [118] and Theorem 21.16 on p280 of [20], and it corresponds to Theorem 22.31 on p449 of [159] for independent random variables as in Subsection 6.4.3.

Weak^{∗} limits and Fourier transforms

7.1 Some spaces of continuous functions on R^{n}

Let n be a positive integer, and let

(7.1.1) C_{b}(R^{n}) = C_{b}(R^{n}, R) and C_{b}(R^{n}, C)

be the spaces of bounded real and complex-valued continuous functions on R^{n}, respectively, using the standard Euclidean metrics on R, C, and R^{n}. If f is a bounded real or complex-valued function on R^{n}, then the supremum norm of f is defined by

∥f∥_{sup} = sup_{x∈R^{n}} |f(x)|. (7.1.2)

The supremum metric is defined on the spaces of bounded real and complexvalued function on R^{n} as the supremum norm of the difference of two bounded functions on R^{n}.

It is well known and not too difficult to show that the spaces of bounded continuous real and complex-valued functions on R^{n} are complete with respect to the supremum metric.

7.1.1 Compact support in R^{n}

The support of a real or complex-valued function f on R^{n} is the closure of the set of x ∈ R^{n} such that f(x) ≠ 0. Of course, this is a closed set in R^{n}, by constuction. Thus the support of f is a compact set in R^{n} if and only if it is a bounded set, because closed and bounded sets in R^{n} are compact, and compact sets in any metric space are bounded. Equivalently, this happens if and only if f(x) = 0 for all x ∈ R^{n} such that |x| is sufficiently large. The spaces of continuous real and complex-valued functions on R^{n} with compact support 7.1. SOME SPACES OF CONTINUOUS FUNCTIONS ON R^{N} 197

may be denoted

(7.1.3) C_{com}(R^{n}) = C_{com}(R^{n}, R) and C_{com}(R^{n}, C),

respectively.

Note that continuous real and complex-valued functions on R^{n} with compact support are bounded, because continuous real and complex-valued functions on compact subsets of R^{n} are bounded. Similarly, it is easy to see that continuous real and complex-valued functions on R^{n} with compact support are uniformly continuous, because continuous real and complex-valued functions on compact subsets of R^{n} are uniformly continuous.

7.1.2 Vanishing at infinity on R^{n}

A real or complex-valued function f on R^{n} is said to vanish at infinity if

f(x) → 0 as |x| → +∞. (7.1.4)

The spaces of continuous real and complex-valued functions on R^{n} that vanish at infinity may be denoted

(7.1.5) C_{0}(R^{n}) = C_{0}(R^{n}, R) and C_{0}(R^{n}, C),

respectively. One can check that continuous real and complex-valued functions on R^{n} that vanish at infinity are bounded and uniformly continuous on R^{n}, using the analogous statements for continuous functions on compact subsets of R^{n}.

It is well known and not too difficult to show that the spaces of continuous real and complex-valued functions on R^{n} that vanish at infinity are closed sets in the corresponding spaces of bounded continuous functions on R^{n}, with respect to the supremum metric. In particular, this implies that these spaces are also complete with respect to the supremum metric.

If f is a continuous real or complex-valued function on R^{n} that vanishes at infinitly, then it is not too difficult to show that f can be approximated by continuous functions with compact support in R^{n}, uniformly on R^{n}. This means that the spaces of continuous real and complex-valued functions on R^{n} that vanish at infinity are the same as the closures of the spaces of continuous functions with compact support in R^{n} in the corresponding spaces of bounded continuous functions on R^{n}, with respect to the supremum metric.

7.1.3 More on functions with compact support in R^{n}

Let U be an open set in R^{n}, and let K be a compact set in R^{n} that is contained in U. It is well known and not too difficult to show that there is a continuous real-valued function f on R^{n} with compact support such that

(7.1.6) 0 ≤ f ≤ 1 on R^{n}, f ≡ 1 on K, and f ≡ 0 on R^{n} \ U.

A version of Urysohn’s lemma states that this works on any locally compact Hausdorff topological space. One can do this more directly on R^{n}, and there are related results with additional conditions on f.

198 CHAPTER 7. WEAK^{∗} LIMITS AND FOURIER TRANSFORMS

7.1.4 Nonnegative linear functionals on C_{com}(R^{n})

Note that C_{com}(R^{n}) and C_{com}(R^{n}, C) may be considered as vector spaces over the real and complex numbers, with respect to pointwise addition and scalar multiplication. Let μ be a locally finite Borel measure on R^{n}, as in Section 2.10. If f is a continuous real or complex-valued function on R^{n} with compact support, then f is integrable with respect to μ. Thus

f ↦ ∫_{R^{n}} f dμ (7.1.7)

defines a linear functional on each of C_{com}(R^{n}) and C_{com}(R^{n}, C), i.e., a linear mapping from each of these spaces into the real or complex numbers, as appropriate, considered as one-dimensional vector spaces over themselves. This linear functional is nonnegative, in the sense that

∫_{R^{n}} f dμ ≥ 0 (7.1.8)

when f is a continuous nonnegative real-valued function on R^{n} with compact support.

It is well known and not too difficult to show that μ is uniquely determined by this linear functional on C_{com}(R^{n}). If U is an open set in R^{n}, then U is an F_{σ} set, as in Subsection 2.2.2, and thus σ-compact, as in Subsection 2.10.2. One can use this to get that μ(U) is determined by the linear functional on C_{com}(R^{n}) associated to μ, by considering elements of C_{com}(R^{n}) as in (7.1.6). More precisely, if K and f are as in (7.1.6), then

μ(K) ≤ ∫_{R^{n}} f dμ ≤ μ(U). (7.1.9)

One can use this to approximate μ(U) by the integrals of suitable elements of C_{com}(R^{n}) with respect to μ on R^{n}.

We also have that μ is outer regular on R^{n}, because R^{n} can be expressed as the union of a sequence of bounded open sets, as in Subsection 2.12.1. This implies that μ is uniquely determined on the σ-algebra of Borel sets in R^{n} by its restriction to the collection of open sets in R^{n}, and so by the corresponding linear functional on C_{com}(R^{n}).

Conversely, a well-known version of the Riesz representation theorem states that every nonnegative linear functional on C_{com}(R^{n}) corresponds to a locally finite Borel measure on R^{n} in this way. In fact, this works on any locally compact Hausdorff topological space X, where the corresponding Borel measure is outer regular, and open sets in X satisfy the inner regularity condition with respect to μ and compact subsets of X. See 7.2 on p212 of [118], 12.36 on p177 of [159], and Theorem 2.14 on p42 of [298].

7.2 The complex exponential function

If z is a complex number, then the exponential of z is defined as a complex number by

exp z = ∑_{j=0}^{∞} \frac{z^{j}}{j!} . (7.2.1)

It is easy to see that the series on the right converges absolutely, using the ratio test, or more directly by comparison with a suitable convergent geometric series. It follows that this defines a continuous complex-valued function on the complex plane, by standard results. Of course, if z is a real number, then this the Taylor series for the usual exponential function on the real line.

If w is another complex number, then it is well known that

(7.2.2) exp(z + w) = (exp z) (exp w).

More precisely, the right side may be expressed as the Cauchy product

∑_{n=0}^{∞} ( ∑_{j=0}^{n} \frac{z^{j}}{j!} \frac{w^{n−j}}{(n − j)!}) (7.2.3)

of (7.2.1) and the analogous series for exp w, because of the absolute convergence of these series. This is the same as the series expansion for exp(z + w), by the binomial theorem. In particular,

(7.2.4) (exp z) (exp(−z)) = exp(z − z) = 1,

so that exp z ≠ 0, with (exp z)^{−1} = exp(−z).

Observe that \overline{(exp z)} = exp(\overline{z}). (7.2.5)

It follows that

(7.2.6)  | exp z|^{2} = (exp z) \overline{(exp z)} = (exp z) (exp \overline{z})
= exp(z + \overline{z}) = exp(2 Re z).

Euler’s identity states that

(7.2.7) exp(i y) = cos y + i sin y

for every y ∈ R.

7.2.1 The Fourier transform of an integrable function on R

Let f be a real or complex-valued function on the real line that is integrable with respect to Lebesgue measure. The Fourier transform of f is the complex-valued function defined on R by

f̂(ξ) = ∫_{−∞}^{+∞} f(x) exp(i x ξ) dx. (7.2.8)

One can check that this is a continuous function on R, and in fact it is uniformly continuous. Note that

|f̂(ξ)| ≤ ∫_{−∞}^{+∞} |f(x)| dx (7.2.9)

for every ξ ∈ R.

The Riemann–Lebesgue Lemma states that

f̂(ξ) → 0 as ξ → ±∞. (7.2.10)

This can be verified directly when f is a step function, and otherwise it can be obtained by approximating f by step functions. If f is real-valued, then

\overline{f̂(ξ)} = f̂(−ξ) (7.2.11)

for every ξ ∈ R. If f is complex-valued, then \overline{f̂(ξ)} is the same as the Fourier transform of \overline{f} at −ξ.

7.2.2 The Fourier transform of a finite Borel measure on R

Let ν be a finite Borel measure on R. The Fourier transform of ν is the complex-valued function defined on R by

ν̂(ξ) = ∫_{−∞}^{+∞} exp(i x ξ) dν(x). (7.2.12)

Suppose for the moment that ν is of the form

ν(E) = ∫_{E} f(x) dx (7.2.13)

for some nonnegative real-valued function f on R that is integrable with respect to Lebesgue measure. In this case, the Fourier transform of ν is the same as the Fourier transform of f, as in Subsection 3.15.1.

It is not too difficult to show that ν̂ is uniformly continuous on R. This is somewhat easier when ν(R \ [−r, r]) = 0 for some r ≥ 0, and otherwise one can approximate ν by finite Borel measures on R with this property. It is easy to see that |ν̂(ξ)| ≤ ν(R) (7.2.14)

and \overline{ν̂(ξ)} = ν̂(−ξ) (7.2.15)

for every ξ ∈ R, as before. If a ∈ R and δ_{a} is the measure defined on R as in Subsection 1.7.3, then δ̂_{a}(ξ) = exp(i a ξ) (7.2.16)

for every ξ ∈ R, as in Subsection 3.9.2.

7.2.3 The Fourier transform of a finite Borel measure on R^{n}

Let n be a positive integer, and let ν be a finite Borel measure on R^{n}. The Fourier transform of ν is the complex-valued function defined on R^{n} by

ν̂(ξ) = ∫_{R^{n}} exp(i x · ξ) dν(x), (7.2.17)

where x · ξ = ∑_{j=1}^{n} x_{j} ξ_{j} is the usual dot product on R^{n}. It is well known that ν is uniquely determined by its Fourier transform. This is related to the fact that ν is uniquely determined by the associated linear functional on C_{com}(R^{n}), as in Subsection 7.1.4.

Let ν_{1}, …, ν_{n} be finite Borel measures on R, and suppose for the moment that ν = ν_{1} × ⋯ × ν_{n} (7.2.18)

as measures on R^{n}. One can check that

ν̂(ξ) = ∏_{j=1}^{n} ν̂_{j}(ξ_{j}) (7.2.19)

for each ξ ∈ R^{n}, where ν̂_{j} is the Fourier transform of ν_{j} on R. This uses Fubini’s theorem, and the fact that

exp(i x · ξ) = ∏_{j=1}^{n} exp(i x_{j} ξ_{j}) (7.2.20)

for all x, ξ ∈ R^{n}.

7.3 Characteristic functions of random variables

Let (X, A, μ) be a probability space, and let f be a measurable real-valued function on X. The characteristic function of f as a random variable on X with respect to μ may be defined as the complex-valued function on the real line with ϕ_{f}(ξ) = E(exp(i ξ f)) = ∫_{X} exp(i ξ f)dμ (7.3.1)

for each ξ ∈ R, as on p297 of [20]. If ν_{f} is the probability distribution on R associated to μ and f as in Subsection 4.11.1, then

ϕ_{f} = ν̂_{f}, (7.3.2)

as in [20]. If a and b are real numbers, then it is easy to see that

ϕ_{a f}(ξ) = ϕ_{f}(a ξ) (7.3.3) and ϕ_{f+b}(ξ) = (exp(i b ξ)) ϕ_{f}(ξ) (7.3.4)

for every ξ ∈ R, as on p298 of [20].

Let f_{1}, …, f_{n} be finitely many measurable real-valued functions on X that are independent as random variables, and suppose for the moment that f = ∑_{j=1}^{n} f_{j}. Under these conditions, we have that

ϕ_{f}(ξ) = ∏_{j=1}^{n} ϕ_{f_{j}}(ξ) (7.3.5)

for each ξ ∈ R, as in Proposition 21.38 on p298 of [20]. More precisely,

(7.3.6)  ϕ_{f}(ξ) = ∫_{X} ( ∏_{j=1}^{n} exp(i ξ f_{j})) dμ

= ∏_{j=1}^{n} ( ∫_{X} exp(i ξ f_{j}) dμ) = ∏_{j=1}^{n} ϕ_{f_{j}}(ξ),

using the multiplication theorem in the second step, as in Section 6.10.

It is well known that a finite Borel measure on R is uniquely determined by its Fourier transform, as mentioned in the previous section. This implies that

(7.3.7) ν_{f} is uniquely determined by ϕ_{f},

as in Corollary 21.40 on p300 of [20].

7.3.1 Joint characteristic functions

Let n be a positive integer, let f_{1}, …, f_{n} be measurable real-valued functions on X, and let F be the corresponding mapping from X into R^{n}, whose lth component is f_{l} for l = 1, …, n. The joint characteristic function of f_{1}, …, f_{n} or equivalently F with respect to μ on X is the complex-valued function on R^{n} defined by

ϕ_{F} (ξ) = E(exp(i ξ · F)) = ∫_{X} exp(i ξ · F) dμ, (7.3.8)

as on p303 of [20]. Here ξ · F = ∑_{j=1}^{n} ξ_{j} f_{j}, as before.

Let ν_{F} be the joint distribution of f_{1}, …, f_{n} with respect to μ, as in Section 6.6.1. Observe that ϕ_{F} = ν̂_{F}, (7.3.9)

as in Subsection 4.11.3.

Let ν_{f_{l}} be the probability distribution on R associated to μ and f_{l} for each l = 1, …, n, as in Subsection 4.11.1. Remember that f_{1}, …, f_{n} are independent as random variables on X if and only if ν_{F} is equal to the product measure

ν_{f_{1}} × ⋯ × ν_{f_{n}}, (7.3.10) as Borel measures on R^{n}, as in Subsection 6.6.1. The Fourier transform of (7.3.10) is equal to

∏_{l=1}^{n} ν̂_{f_{l}}(ξ_{l}), (7.3.11)

as in (7.2.19). It follows that ν_{F} is equal to (7.3.10) if and only if ν̂_{F}(ξ) is equal to (7.3.11) for every ξ ∈ R^{n}, because a finite Borel measure on R^{n} is uniquely determined by its Fourier transform, as mentioned in the previous section.

Thus f_{1}, …, f_{n} are independent as random variables on X if and only if ν̂_{F}(ξ) is equal to (7.3.11) for every ξ ∈ R^{n}. This is the same as saying that

ϕ_{F} (ξ) = ∏_{l=1}^{n} ϕ_{f_{l}}(ξ_{l}) (7.3.12)

for every ξ ∈ R^{n}. This corresponds to Proposition 21.45 on p303 of [20].

7.4 Convolutions of finite Borel measures on R

Let n be a positive integer, and let ν_{1}, …, ν_{n} be finite Borel measures on the real line. Also let a_{n} be the mapping from R^{n} into R defined by addition, so that

a_{n}(x) = ∑_{j=1}^{n} x_{j} (7.4.1)

for every x ∈ R^{n}. Of course, it is well known that this mapping is continuous. Remember that the product measure

ν_{1} × ⋯ × ν_{n} (7.4.2)

may be defined as a Borel measure on R^{n} as in Section 5.12.

The convolution of ν_{1}, …, ν_{n} is the Borel measure defined on R by

ν_{1} ∗ ⋯ ∗ ν_{n} = (a_{n})_{∗}(ν_{1} × ⋯ × ν_{n}), (7.4.3)

as in Section 4.11. This means that

(ν_{1} ∗ ⋯ ∗ ν_{n})(E) = (ν_{1} × ⋯ × ν_{n})(a_{n}^{−1}(E)) (7.4.4)

for all Borel sets E ⊆ R, as before. Note that

a_{n}^{−1}(E) = {x ∈ R^{n} : a_{n}(x) ∈ E} = {x ∈ R^{n} : ∑_{j=1}^{n} x_{j} ∈ E}. (7.4.5)

In particular,

(7.4.6)  (ν_{1} ∗ ⋯ ∗ ν_{n})(R) = (ν_{1} × ⋯ × ν_{n})(a_{n}^{−1}(R))
= (ν_{1} × ⋯ × ν_{n})(R^{n}) = ∏_{l=1}^{n} ν_{l}(R).

Often one defines convolutions initially for two measures on R, as on p270 of [118], and Exercise 5 on p158 of [298]. It is easy to see that the convolution of two measures is commutative, and one can check that it is associative as well. More precisely, if n = 3, then

(ν_{1} ∗ ν_{2}) ∗ ν_{3} = ν_{1} ∗ (ν_{2} ∗ ν_{3}) = ν_{1} ∗ ν_{2} ∗ ν_{3}, (7.4.7)

as on p271 of [118]. The convolution of n measures may be obtained using convolutions of two measures repeatedly, as in (10.3) on p317 of [118].

If f is a nonnegative Borel measurable function on R, then f ∘ a_{n} is Borel measurable on R^{n}, and

∫_{R} f d(ν_{1} ∗ ⋯ ∗ ν_{n}) = ∫_{R^{n}}(f ∘ a_{n}) d(ν_{1} × ⋯ × ν_{n}), (7.4.8)

as in Subsection 4.11.3. This also works for real or complex-valued Borel measurable functions with suitable integrability properties, as before. In particular, this works for bounded real and complex-valued Borel measurable functions on R, because the measures are finite, by hypothesis. This corresponds to part b of Proposition 8.48 on p270 of [118], and it is related to part (b) of Exercise 5 on p158 of [298].

One can use this to check that the Fourier transform of the convolution is

(ν_{1} ∗ ⋯ ∗ ν_{n})(ξ) = ∏_{l=1}^{n} ν̂_{l}(ξ), (7.4.9)

as mentioned on p272 of [118].

Suppose for the moment that n = 2, and that for j = 1, 2,

ν_{j}(E) = ∫_{E} ρ_{j}(x) dx (7.4.10)

for all Borel sets E ⊆ R^{n}, where ρ_{j} is a nonnegative real-valued function on R^{n} that is integrable with respect to Lebesgue measure. It is well known and not difficult to show that

(ν_{1} ∗ ν_{2})(E) = ∫_{E}(ρ_{1} ∗ ρ_{2})(x) dx (7.4.11)

for each Borel set E ⊆ R^{n}, where

(ρ_{1} ∗ ρ_{2})(x) = ∫_{R} ρ_{1}(x − y) ρ_{2}(y) dy. (7.4.12)

This corresponds to part (d) of Proposition 8.48 on p271 of [118], and part (f) of Exercise 5 on p159 of [298].

If a ∈ R, then let δ_{a} be the Borel measure on R defined as in Subsection 1.7.3. One can check that δ_{a} ∗ δ_{b} = δ_{a+b} (7.4.13)

for all a, b ∈ R.

7.5. WEAK^{∗} CONVERGENCE FOR BOREL MEASURES ON R^{N} 205

7.4.1 Sums of independent random variables

Let (X, A, μ) be a probability space, let f_{1}, …, f_{n} be measurable real-valued functions on X, and let F be the corresponding mapping into R^{n}. Put

f = ∑_{l=1}^{n} f_{l} = a_{n} ∘ F. (7.4.14)

Let ν_{f_{1}}, …, ν_{f_{n}} and ν_{f} be the probability distributions on the real line associated to f_{1}, …, f_{n} and f, respectively, as in Subsection 4.11.1, and let ν_{F} be the corrresponding joint distribution on R^{n}, as in Subsection 6.6.1. Observe that

ν_{f} = f_{∗}(μ) = (a_{n} ∘ F)_{∗}(μ) = (a_{n})_{∗}(F_{∗}(μ)) = (a_{n})_{∗}(ν_{F}), (7.4.15)

as in Subsection 4.11.2.

Suppose that f_{1}, …, f_{n} are independent as random variables on X, so that ν_{F} is the same as the product measure ν_{f_{1}} × ⋯ × ν_{f_{n}} on R^{n}, as in Subsection 6.6.1. In this case, we get that

ν_{f} = ν_{f_{1}} ∗ ⋯ ∗ ν_{f_{n}}. (7.4.16)

This corresponds to Proposition 10.4 on p317 of [118]. One can take the Fourier transform of both sides of the equation, to get that the characteristic function of f is equal to the product of the characteristic functions of f_{1}, …, f_{n}, as in the previous section.

7.5 Weak^{∗} convergence for Borel measures on R^{n}

Let n be a positive integer, let {μ_{j}}_{j=1}^{∞} be a sequence of locally finite Borel measures on R^{n}, and let μ be another locally finite Borel measure on R^{n}. Let us say that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense if

lim_{j→∞} ∫_{R^{n}} f dμ_{j} = ∫_{R^{n}} f dμ (7.5.1)

for every f ∈ C_{com}(R^{n}). To be more precise, we may say that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}) under these conditions. Note that this implies that (7.5.1) holds for every f ∈ C_{com}(R^{n}, C) as well.

Of course, the limit is unique when it exists, because a locally finite Borel measure on R^{n} is uniquely determined by the corresponding linear functional on C_{com}(R^{n}), as in Subsection 7.1.4. The existence of the limit implies in particular that the sequence of integrals ∫_{R^{n}} f dμ_{j} is bounded for each f ∈ C_{com}(R^{n}). If K is a compact subset of R^{n}, then one can use this to get that

sup_{j≥1} μ_{j}(K) < +∞, (7.5.2)

by considering f ≥ 0 with f ≡ 1 on K.

If y ∈ R^{n}, then let δ_{y} be the probability measure on R^{n} associated to y as in Subsection 1.7.3. If {y_{j}}_{j=1}^{∞} is a sequence of elements of R^{n} that converges to y, then it is easy to see that {δ_{y_{j}}}_{j=1}^{∞} converges to δ_{y} in the weak^{∗} sense with respect to C_{com}(R^{n}). If |y_{j}| → +∞ as j → ∞, then {δ_{y_{j}}}_{j=1}^{∞} converges to 0 in the weak^{∗} sense with respect to C_{com}(R^{n}). In this case, {a_{j} δ_{y_{j}}}_{j=1}^{∞} converges to 0 in the weak^{∗} sense with respect to C_{com}(R^{n}) for every sequence {a_{j}}_{j=1}^{∞} of nonnegative real numbers.

Suppose for the moment that there is a sequence {ρ_{j}}_{j=1}^{∞} of nonnegative Borel measurable real-valued functions on R^{n} such that

μ_{j}(E) = ∫_{E} ρ_{j}(x) dx (7.5.3)

for each Borel set E ⊆ R^{n} and j ≥ 1, and similarly that there is a nonnegative Borel measurable real-valued function ρ on R^{n} such that

μ(E) = ∫_{E} ρ(x) dx (7.5.4)

for all Borel sets E ⊆ R^{n}. More precisely, the ρ_{j}’s and ρ should be locally integrable on R^{n} with respect to Lebesgue measure, in the sense that they are integrable on bounded Borel sets in R^{n}, in order for the μ_{j}’s and μ so be locally finite on R^{n}. If

lim_{j→∞} ∫_{B} |ρ_{j}(x) − ρ(x)| dx = 0 (7.5.5)

for every bounded Borel set B ⊆ R^{n}, then it is easy to see that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}). In this case, we also have that (7.5.3) tends to (7.5.4) as j → ∞ when E is a bounded Borel set in R^{n}.

7.5.1 Weak^{∗} convergence and measures of open sets in R^{n}

Suppose again that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}), and let U be an open set in R^{n}. Also let f be a continuous real-valued function on R^{n} with compact support such that

(7.5.6) 0 ≤ f ≤ 1 on R^{n} and f ≡ 0 on R^{n} \ U.

Of course, this implies that

∫_{R^{n}} f dμ_{j} ≤ μ_{j}(U) (7.5.7)

for each j. It follows that

∫_{R^{n}} f dμ ≤ lim inf_{j→∞} μ_{j}(U), (7.5.8)

because of (7.5.1).

If K is a compact set in R^{n} that is contained in U, then there are continuous functions f as in the preceding paragraph with f ≡ 1 on K, as in Subsection 7.1.3. This means that

μ(K) ≤ lim inf_{j→∞} μ_{j}(U). (7.5.9)

One can use this to get that

μ(U) ≤ lim inf_{j→∞} μ_{j}(U), (7.5.10)

as in Subsection 2.10.3. This uses the fact that U is an F_{σ} set, as in Subsection 2.2.2, so that U is σ-compact, because R^{n} is σ-compact, as in Subsection 2.10.2.

7.5.2 Weak^{∗} convergence and measures of compact sets in R^{n}

Now let K be any compact set in R^{n}, and let f be a continuous real-valued function on R^{n} with compact support such that

(7.5.11) 0 ≤ f ≤ 1 on R^{n} and f ≡ 1 on K.

This implies that

μ_{j}(K) ≤ ∫_{R^{n}} f dμ_{j} (7.5.12)

for each j. It follows that

lim sup_{j→∞} μ_{j}(K) ≤ ∫_{R^{n}} f dμ, (7.5.13)

because of (7.5.1) again.

If U is an open set in R^{n} that contains K, then there are continuous functions f as in the previous paragraph with f ≡ 0 on R^{n} \ U, as in Subsection 7.1.3 again. In this case, we can use (7.5.13) to get that

lim sup_{j→∞} μ_{j}(K) ≤ μ(U). (7.5.14)

This implies that

lim sup_{j→∞} μ_{j}(K) ≤ μ(K), (7.5.15)

because of outer regularity, as in Subsection 2.12.1. In fact, we can get this outer regularity property more directly here, as follows.

Remember that K can be expressed as the intersection of a sequence of open sets, as in Subsection 2.2.2. More precisely, K can be expressed as the intersection of a decreasing sequence of bounded open sets using this construction, although these additional properties could easily be arranged anyway. It follows that μ(K) is the limit of the measures of these open sets with respect to μ, as in Subsection 1.9.2. We can use this to get (7.5.15) from (7.5.14), as before.

7.6 Weak^{∗} convergence for Borel measures on R

Let {μ_{j}}_{j=1}^{∞} be a sequence of locally finite Borel measures on the real line that converges to a locally finite Borel measure μ on R in the weak^{∗} sense with respect to C_{com}(R), as in the previous section. If a, b ∈ R and a ≤ b, then

sup_{j≥1} μ_{j}([a, b]) < +∞, (7.6.1)

as in (7.5.2). This is a bit simpler here, using continuous nonnegative real-valued functions on R with compact support that are piecewise-linear.

Similarly, if a and b are real numbers with a < b, then

μ((a, b)) ≤ lim inf_{j→∞} μ_{j}((a, b)), (7.6.2)

as in (7.5.10). We also have that

lim sup_{j→∞} μ_{j}([a, b]) ≤ μ([a, b]), (7.6.3)

even when a = b, as in (7.5.15). In fact, (7.6.2) works when a = −∞ or b = +∞ as well. Let us see how the previous arguments can be simplified in these cases.

If a < a_{1} ≤ b_{1} < b, then there is a continuous nonnegative real-valued function f on R with compact support such that f(x) = 1 when a_{1} ≤ x ≤ b_{1}, f(x) ≤ 1 for every x ∈ R, and f(x) = 0 when x ≤ a and when x ≥ b. One can take f to be piecewise-linear, as before. It follows that

μ([a_{1}, b_{1}]) ≤ ∫_{R} f dμ ≤ lim inf_{j→∞} μ_{j}((a, b)), (7.6.4)

because of (7.5.1), as in the previous section. One can use this to get (7.6.2), as in Section 1.9.

Similarly, if a_{0} < a ≤ b < b_{0}, then there is a continuous nonnegative realvalued function f on R with compact support such that f(x) = 1 on [a, b], f(x) ≤ 1 for every x ∈ R, and f(x) = 0 when x ≤ a_{0} and when x ≥ b_{0}, and one can take f to be piecewise-linear, as usual. This implies that

lim sup_{j→∞} μ_{j}([a, b]) ≤ ∫_{R} f dμ ≤ μ((a_{0}, b_{0})), (7.6.5)

as before. One can obtain (7.6.3) from this as in Subsection 1.9.2.

7.6.1 When μ({a}) = μ({b}) = 0

Suppose now that a, b ∈ R, a < b, and

μ({a}) = μ({b}) = 0. (7.6.6)

This implies that

lim_{j→∞} μ_{j}({a}) = lim_{j→∞} μ_{j}({b}) = 0, (7.6.7)

7.6. WEAK^{∗} CONVERGENCE FOR BOREL MEASURES ON R 209

as in (7.6.3). It follows that

lim_{j→∞} μ_{j}((a, b)) = μ((a, b)) (7.6.8)

and

lim_{j→∞} μ_{j}([a, b]) = μ([a, b]) (7.6.9)

under these conditions, because of (7.6.2) and (7.6.3). Of course, the right sides of these two equations are the same in this case.

7.6.2 The set of x ∈ R with μ({x}) > 0 is at most countable

Let μ be a locally finite Borel measure on R. If ϵ and r are positive real numbers, then it is easy to see that μ({x}) ≥ ϵ (7.6.10)

for at most μ([−r, r])/ϵ (7.6.11)

elements x of [−r, r]. This implies that

μ({x}) > 0 (7.6.12)

for at most finitely or countably many x ∈ [−r, r]. This means that (7.6.12) holds for at most finitely or coutably many x ∈ R.

7.6.3 Some remarks about step functions

A step function on the real line may be expressed as a linear combination of indicator functions of finitely many intervals in R. Let E be a dense set in R. If f is a continuous real or complex-valued function on R with compact support, then one can check that f can be uniformly approximated by step functions that are linear combinations of indicator functions of intervals with endpoints in E, and whose supports are contained in a single closed interval. This uses the fact that f is uniformly continuous, as mentioned in Subsection 7.1.1.

7.6.4 A criterion for weak^{∗} convergence on R

Let {μ_{j}}_{j=1}^{∞} be a sequence of locally finite Borel measures on R again, and let μ be another locally finite Borel measure on R. Also let E be a dense set in the real line such that μ({x}) = 0 (7.6.13)

for every x ∈ E. Suppose that (7.6.9) holds for every a, b ∈ E with a ≤ b. In particular, this means that

lim_{j→∞} μ_{j}({x}) = 0 (7.6.14)

for every x ∈ E. Thus (7.6.8) holds for every a, b ∈ E with a < b too.

Note that (7.6.1) holds for all a, b ∈ E with a ≤ b. This implies the analogous statement for all a, b ∈ R with a ≤ b.

Let f be a step function on R with compact support that is a linear combination of indicator functions of intervals with endpoints in E. It is easy to see that

lim_{j→∞} ∫_{R} f dμ_{j} = ∫_{R} f dμ, (7.6.15)

by hypothesis.

One can use this to get that (7.6.15) holds for all continuous functions f on R with compact support, by approximating f by step functions as in the previous subsection. Thus {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R) under these conditions. Note that one can simply take E to be the set of all x ∈ R such that (7.6.13) holds.

7.7 Other types of weak^{∗} convergence on R^{n}

Let n be a positive integer again, let {μ_{j}}_{j=1}^{∞} be a sequence of finite Borel measures on R^{n}, and let μ be another finite Borel measure on R^{n}. Let us say that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{0}(R^{n}) if (7.5.1) holds for all continuous real-valued functions f on R^{n} that vanish at infinity, as in Subsection 7.1.2. Note that this implies the analogous condition for continuous complex-valued functions f on R^{n} that vanish at infinity. This corresponds to vague convergence on p223 of [118]. Clearly this implies that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}), as in Section 7.5.

Suppose that

sup_{j≥1} μ_{j}(R^{n}) < +∞. (7.7.1)

In this case, one can check that weak^{∗} convergence with respect to C_{com}(R^{n}) implies weak^{∗} convergence with respect to C_{0}(R^{n}). This uses the fact that C_{com}(R^{n}) is dense in C_{0}(R^{n}) with respect to the supremum metric, as mentioned in Subsection 7.1.2.

If y ∈ R^{n}, then let δ_{y} be the probability measure on R^{n} associated to y as in Subsection 1.7.3 again. If {y_{j}}_{j=1}^{∞} is a sequence of elements of R^{n} with |y_{j}| → +∞ as j → ∞, then it is easy to see that {δ_{y_{j}}}_{j=1}^{∞} converges to 0 in the weak^{∗} sense with respect to C_{0}(R^{n}).

7.7.1 The uniform boundedness principle

It is well known that (7.7.1) is a necessary condition for {μ_{j}}_{j=1}^{∞} to converge in the weak^{∗} sense with respect to C_{0}(R^{n}). More precisely, (7.7.1) is a necessary condition to have that

sup_{j≥1} |∫_{R^{n}} f dμ_{j}| < +∞ (7.7.2)

for every f ∈ C_{0}(R^{n}). This uses the uniform boundedness principle for bounded linear mappings on Banach spaces. Remember that C_{0}(R^{n}) is complete with respect to the supremum metric, as in Subsection 7.1.2, so that C_{0}(R^{n}) is a Banach space with respect to the supremum norm. This also uses the wellknown fact that f ↦ ∫_{R^{n}} f dμ_{j} (7.7.3)

is a bounded linear functional on C_{0}(R^{n}) with respect to the supremum norm for each j, with dual norm equal to μ_{j}(R^{n}).

7.7.2 Convergence with respect to C_{b}(R^{n})

Let us say that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}) if (7.5.1) holds for all bounded continuous real-valued functions f on R^{n}. This is the same as saying that (7.5.1) holds for all bounded continuous complex-valued functions f on R, as usual. Of course, this implies that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{0}(R^{n}).

If {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}), then one can take f ≡ 1 on R^{n} in (7.5.1) to get that

lim_{j→∞} μ_{j}(R^{n}) = μ(R^{n}). (7.7.4)

If {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R), then

(7.7.5) μ(R^{n}) ≤ lim inf_{j→∞} μ_{j}(R^{n}),

as in (7.5.10).

If {y_{j}}_{j=1}^{∞} is a sequence of elements of R^{n} such that |y_{j}| → +∞ as j → ∞, then {δ_{y_{j}}}_{j=1}^{∞} does not converge in the weak^{∗} sense with respect to C_{b}(R^{n}). This is because {δ_{y_{j}}}_{j=1}^{∞} converges to 0 in the weak^{∗} sense with respect to C_{0}(R^{n}), and the inequality in (7.7.5) is strict in this case.

Suppose for the moment that there is a sequence {ρ_{j}}_{j=1}^{∞} of nonnegative integrable real-valued functions on R^{n} with respect to Lebesgue measure, and another nonnegative integrable real-valued function ρ with respect to Lebesgue measure on R^{n}, such that μ_{j} and μ are as in (7.5.3) and (7.5.4), respectively. If {ρ_{j}}_{j=1}^{∞} converges to ρ in L^{1} on R^{n} with respect to Lebesgue measure, then it is easy to see that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}). However, this type of convergence is more restrictive that convergence in the weak^{∗} sense with respect to C_{b}(R^{n}).

7.8 A related condition on sequences of measures

Let n be a positive integer, and let \overline{B}(x, r) be the closed ball in R^{n} centered at x ∈ R^{n} with radius r ≥ 0 with respect to the standard Euclidean metric, as in Subsection 2.2.2. If μ is a finite Borel measure on R^{n}, then

lim_{l→∞} μ(R^{n} \ \overline{B}(0, l)) = 0, (7.8.1)

as in Subsection 1.9.2.

Let {μ_{j}}_{j=1}^{∞} be a sequence of finite Borel measures on R^{n}. Consider the condition that for each ϵ > 0 there be an r(ϵ) ≥ 0 such that

μ_{j}(R^{n} \ \overline{B}(0, r(ϵ))) ≤ ϵ (7.8.2)

for each j. A sequence of probability measures with this property may be said to be tight. This corresponds to the definition of a tight sequence of distribution functions of probability measures on the real line on p295 of [20]. Note that (7.8.2) is the same as saying that

μ_{j}(R^{n}) ≤ μ_{j}(\overline{B}(0, r(ϵ))) + ϵ (7.8.3)

for each j.

7.8.1 A reformulation using continuous functions

Consider the condition that for each ϵ > 0 there be a continuous real-valued function θ_{ϵ} on R^{n} with compact support such that 0 ≤ θ_{ϵ} ≤ 1 on R^{n} and

μ_{j}(R^{n}) ≤ ∫_{R^{n}} θ_{ϵ} dμ_{j} + ϵ (7.8.4)

for each j. This is the same as saying that

∫_{R^{n}}(1 − θ_{ϵ}) dμ_{j} ≤ ϵ (7.8.5)

for each j.

If (7.8.3) holds, then (7.8.4) holds when θ_{ϵ} ≡ 1 on \overline{B}(0, r(ϵ)). Conversely, if (7.8.4) holds, then (7.8.3) holds when the support of θ_{ϵ} is contained in \overline{B}(0, r(ϵ)).

7.8.2 A connection with weak^{∗} convergence

Let μ be a finite Borel measure on R^{n} again, and suppose that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}). Let ϵ > 0 be given, and suppose that θ_{ϵ} ∈ C_{com}(R^{n}) is as in the previous subsection. Observe that

lim_{j→∞} ∫_{R^{n}} θ_{ϵ} dμ_{j} = ∫_{R^{n}} θ_{ϵ} dμ ≤ μ(R^{n}). (7.8.6)

This implies that

(7.8.7)  lim sup_{j→∞} μ_{j}(R^{n}) − ϵ ≤ μ(R^{n}),

because of (7.8.4). If this holds for every ϵ > 0, then we get that

(7.8.8)  lim sup_{j→∞} μ_{j}(R^{n}) ≤ μ(R^{n}).

Combining this with (7.7.5), we get that (7.7.4) holds when {μ_{j}}_{j=1}^{∞} also satisfies the previous condition. Conversely, suppose that (7.7.4) holds, and let us check that {μ_{j}}_{j=1}^{∞} satisfies this condition.

7.8.3 Necessity of this condition

If f is a continuous real-valued function on R^{n} with compact support, then we can use weak^{∗} convergence with respect to C_{com}(R^{n}) and (7.7.4) to get that

lim_{j→∞} ∫_{R^{n}}(1 − f) dμ_{j} = ∫_{R^{n}}(1 − f) dμ. (7.8.9)

Let ϵ > 0 be given, and let f_{ϵ} be a continuous real-valued function on R^{n} with compact support such that 0 ≤ f_{ϵ} ≤ 1 on R^{n} and

∫_{R^{n}}(1 − f_{ϵ}) dμ < ϵ/2. (7.8.10)

This holds when f_{ϵ} ≡ 1 on a closed ball in R centered at 0 with large enough radius, as in (7.8.1). This implies that

∫_{R^{n}}(1 − f_{ϵ}) dμ_{j} < ϵ (7.8.11)

when j is large enough, because of (7.8.9).

If the support of f_{ϵ} is contained in \overline{B}(0, r(ϵ)), then (7.8.11) implies (7.8.3), as before. This means that (7.8.3) holds when j is large enough and r(ϵ) has this property. It follows that (7.8.3) holds for all j when r(ϵ) is large enough, as in (7.8.1).

7.8.4 More on convergence with respect to C_{b}(R^{n})

Let us continue to suppose that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}), and suppose again now that {μ_{j}}_{j=1}^{∞} satisfies the condition mentioned earlier. We would like to show that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}). This is related to Exercise 26 on p225 of [118], with some simplifications. This is related to Proposition 21.31 on p293 of [20] when n = 1 as well.

Let f ∈ C_{b}(R^{n}) and ϵ > 0 be given, and let θ_{ϵ} be as before. Note that

∫_{R^{n}}(1 − θ_{ϵ}) dμ ≤ ϵ, (7.8.12)

because of (7.8.5), as in (7.8.9). We also have that

lim_{j→∞} ∫_{R^{n}} f θ_{ϵ} dμ_{j} = ∫_{R^{n}} f θ_{ϵ} dμ, (7.8.13)

because f θ_{ϵ} ∈ C_{com}(R^{n}). Of course,

∫_{R^{n}} f dμ_{j} − ∫_{R^{n}} f dμ = ∫_{R^{n}} f θ_{ϵ} dμ_{j} − ∫_{R^{n}} f θ_{ϵ} dμ

+ ∫_{R^{n}} f (1 − θ_{ϵ}) dμ_{j} − ∫_{R^{n}} f (1 − θ_{ϵ}) dμ. (7.8.14)

Observe that

|∫_{R^{n}} f (1 − θ_{ϵ}) dμ_{j}| ≤ ∫_{R^{n}} |f| (1 − θ_{ϵ}) dμ_{j} ≤ ϵ ∥f∥_{sup}, (7.8.15)

using (7.8.5) in the second step, as well as the definition of the supremum norm of f, as in Section 7.1. Similarly,

|∫_{R^{n}} f (1 − θ_{ϵ}) dμ| ≤ ∫_{R^{n}} |f| (1 − θ_{ϵ}) dμ ≤ ϵ ∥f∥_{sup}, (7.8.16)

using (7.8.12) in the second step. One can use this to get that the left side of (7.8.14) is arbitrarily small when j is large enough, because of (7.8.13).

7.9 Multi-indices and the Schwartz class

Let n be a positive integer. A multi-index is an n-tuple α = (α_{1}, …, α_{n}) of nonnegative integers. The order of α is defined by

|α| = ∑_{j=1}^{n} α_{j}. (7.9.1)

Of course, this is not necessarily the same as the standard Euclidean norm of α, and this should not normally cause any confusion.

If x ∈ R^{n}, then put x^{α} = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}. (7.9.2)

Here x_{j}^{α_{j}} is interpeted as being equal to 1 when α_{j} = 0, even when x_{j} = 0. This defines the monomial on R^{n} associated to α.

Let f be a continuous real or complex-valued function on R^{n} that is continuously differentiable of all orders. The derivative of f of order |α| corresponding to α may be denoted

∂^{α}f = \frac{∂^{|α|}f}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} . (7.9.3)

7.9.1 The Schwartz class

Let S(R^{n}) be the space of complex-valued functions f on R^{n} that are continuously differentiable of all orders such that

(1 + |x|^{2})^{k} |(∂^{β}f)(x)| (7.9.4) is bounded on R^{n} for every nonnegative integer k and multi-index β. One can check that this is equivalent to asking that

x^{α} (∂^{β}f)(x) (7.9.5)

be bounded on R^{n} for all multi-indices α and β. This is called the Schwartz class, and it is easy to see that this is a linear subspace of the space C_{0}(R^{n}, C) of continuous complex-valued function on R^{n} that vanish at infinity, as a vector space over the complex numbers.

If f ∈ S(R^{n}) and α is a multi-index, then one can check that

∂^{α}f ∈ S(R^{n}). (7.9.6)

One can also verify that x^{α} f(x) ∈ S(R^{n}), (7.9.7)

using the product rule for partial derivatives.

If f is continuously differentiable on R^{n} of all orders and f has compact support, then f ∈ S(R^{n}). One can use this to get that S(R^{n}) is dense in C_{0}(R^{n}, C) with respect to the supremum metric.

7.9.2 The Fourier transform on S(R^{n})

If f is a real or complex-valued function on R^{n} that is integrable with respect to Lebesgue measure, then the Fourier transform of f is the complex-valued function defined on R^{n} by

f̂(ξ) = ∫_{R^{n}} f(x) exp(i x · ξ) dx. (7.9.8)

If f is real-valued and nonnegative, then this is the same as the Fourier transform of the measure on R^{n} associated to f in the usual way, as in Subsection 7.2.3. It is easy to see that f̂ is bounded on R^{n}, with

|f̂(ξ)| ≤ ∫_{R^{n}} |f(x)| dx (7.9.9)

for every ξ ∈ R^{n}, as in Subsection 7.2.1 when n = 1. Another version of the Riemann–Lebesgue lemma states that f̂ vanishes at infinity on R^{n}.

If f ∈ S(R^{n}), then it is easy to see that f is integrable on R^{n} with respect to Lebesgue measure. In fact, it is well known and not too difficult to show that

f̂ ∈ S(R^{n}) (7.9.10)

in this case. This corresponds to part of Theorem 26.17 on p411 of [20], Corollary 8.23 on p250 of [118], part (d) of Theorem 7.4 on p168 of [297], Theorem 1.3 on p137 and Corollary 2.2 on p181 of [334], and Theorem 3.2 on p20 of [337].

Indeed, if f ∈ S(R^{n}) and α is a multi-index, then one can check that the Fourier transform of ∂^{α}f is

ˆ (∂^{α}f)(ξ) = (−i)^{|α|} ξ^{α} f̂(ξ), (7.9.11) using integration by parts. This is related to the remarks in Subsection 7.12.3 when n = 1. Similarly, one can verify that

(7.9.12) ∂^{α} f̂ is equal to the Fourier transform of i^{|α|} x^{α} f(x),

by differentiating under the integral sign. This is related to the remarks in Subsections 7.12.1 and 7.12.2 when n = 1.

7.9.3 More on the Fourier transform on S(R^{n})

More precisely, it is well known that the Fourier transform defines a one-to-one mapping from S(R^{n}) onto itself. This is related to the inverse Fourier transform of the Fourier transform of an integrable function whose Fourier transform is also integrable. This corresponds to part of Corollary 8.28 on p252 of [118], part (b) of Theorem 7.7 on p170 of [297], Corollary 1.10 on p142 and a remark on p184 of [334], and a remark after Theorem 3.2 on p20 of [337]. See also [117, 199, 219, 236, 252, 340, 386, 387] for more on these and related topics.

7.9.4 The multiplication formula

If μ and ν are finite Borel measures on R^{n}, then it is easy to see that

∫_{R^{n}} μ̂ dν = ∫_{R^{n}} ν̂ dμ, (7.9.13)

using Fubini’s theorem. Similarly, if f is a real or complex-valued function on R^{n} that is integrable with respect to Lebesgue measure, then

∫_{R^{n}} f̂dμ = ∫_{R^{n}} f(y) μ̂(y) dy. (7.9.14)

In particular, one can use this to get that μ is uniquely determined by μ̂. This also uses the fact that S(R^{n}) is dense in C_{0}(R^{n}, C) with respect to the supremum metric, as mentioned earlier.

7.10 More on sequences of finite measures on R^{n}

Let n be a positive integer, and let {μ_{j}}_{j=1}^{∞} be a sequence of finite Borel measures on R^{n}. Suppose for the moment that {μ_{j}(R^{n})}_{j=1}^{∞} is bounded, and that {μ_{j}}_{j=1}^{∞} satisfies the condition discussed near the beginning of Section 7.8. It is not too difficult to check that

(7.10.1) {μ̂_{j}}_{j=1}^{∞} is uniformly equicontinuous on R^{n}.

This means that for every ϵ > 0 there is a δ > 0 such that

|μ̂_{j}(ξ) − μ̂_{j}(η)| < ϵ (7.10.2)

7.10. MORE ON SEQUENCES OF FINITE MEASURES ON R^{N} 217

for all j and ξ, η ∈ R^{n} with |ξ − η| < δ.

More precisely, if there is an r ≥ 0 such that

μ_{j}(R^{n} \ \overline{B}(0, r)) = 0 (7.10.3)

for each j, then one can verify that μ̂_{j} is Lipschitz of order 1 on R^{n}, as in Section A.11, with bounded constant. Otherwise, one can use the condition in Section 7.8 to get that the μ̂_{j}’s can be uniformly approximated by Lipschitz functions of order 1 with constants that are uniformly bounded in j. This implies the uniform equicontinuity condition mentioned in the preceding paragraph, by standard arguments.

7.10.1 Getting pointwise convergence of {μ̂_{j}}_{j=1}^{∞}

Let μ be another finite Borel measure on R^{n}, and suppose that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}), in addition to the condition from Section 7.8. Remember that weak^{∗} convergence of {μ_{j}}_{j=1}^{∞} with respect to C_{com}(R^{n}) implies that the μ_{j}’s are uniformly bounded on compact subsets of R^{n}, as in Section 7.5. This implies that the μ_{j}’s are uniformly bounded on R^{n}, because of the condition from Section 7.8.

Remember that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}), as in Subsection 7.8.4. Conversely, if {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}), then μ_{j} converges to μ in the weak^{∗} sense with respect to C_{com}(R^{n}), and {μ_{j}(R^{n})}_{j=1}^{∞} converges to μ(R^{n}). This implies the condition mentioned near the beginning of Section 7.8, as in Subsection 7.8.3.

Observe that

lim_{j→∞} μ̂_{j}(ξ) = μ(ξ) (7.10.4)

for every ξ ∈ R^{n} when {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{b}(R^{n}). This is related to a remark after Proposition 8.50 on p272 of [118]. In fact, one can get uniform convergence on compact subsets of R^{n}, because of (7.10.1).

7.10.2 Using pointwise convergence of {μ̂_{j}}_{j=1}^{∞}

Suppose now that {μ_{j}}_{j=1}^{∞} is a sequence of finite Borel measures on R^{n}, μ is another finite Borel measure on R^{n}, and that (7.10.4) holds for every ξ ∈ R^{n}. In particular, we can take ξ = 0, to get that

(7.10.5) {μ_{j}(R^{n})}_{j=1}^{∞} converges to μ(R^{n}).

Note that this implies that {μ_{j}(R^{n})}_{j=1}^{∞} is bounded, so that {μ̂_{j}}_{j=1}^{∞} is uniformly bounded on R^{n}.

Let f be a real or complex-valued function on R^{n} that is integrable with respect to Lebesgue measure. The dominated convergence theorem implies that

lim_{j→∞} ∫_{R^{n}} f(y) μ̂_{j}(y) dy = ∫_{R^{n}} f(y) μ̂(y) dy. (7.10.6)

This means that

lim_{j→∞} ∫_{R^{n}} f̂dμ_{j} = ∫_{R^{n}} f̂dμ, (7.10.7)

as in (7.9.14).

Remember that f̂ can be any element of the Schwartz class S(R^{n}), as in Subsection 7.9.2. One can use this and (7.10.7) to get that {μ_{j}}_{j=1}^{∞} converges to μ in the weak^{∗} sense with respect to C_{0}(R^{n}, C). This also uses the fact that S(R^{n}) is dense in C_{0}(R^{n}, C) with respect to the supremum metric, as mentioned in Subsection 7.9.1, and the boundedness of {μ_{j}(R)}_{j=1}^{∞}. This corresponds to Proposition 8.50 on p272 of [118], with some simplifications. This is related to Theorem 21.44 on p302 of [20] as well.

7.11 Weak convergence of random variables

Let (X_{j}, A_{j}, μ_{j}) be a probability space for each positive integer j, and let (X, A, μ) be another probability space. Often one simply takes (X_{j}, A_{j}, μ_{j}) = (X, A, μ) for each j, although this is not really needed here.

Let f_{j} be a measurable real-valued function on X_{j} for each j, and let f be a measurable real-valued function on X. Also let ν_{f_{j}} be the Borel probability measure on R corresponding to f_{j} for each j, and let ν_{f} be the Borel probability measure on R corresponding to f, as in Subsection 4.11.1.

Let g be a bounded continuous real or complex-valued function on the real line. Note that ∫_{X_{j}}(g ∘ f_{j}) dμ_{j} = ∫_{R} g dν_{f_{j}} (7.11.1)

for each j, and

∫_{X}(g ∘ f) dμ = ∫_{R} g dν_{f}, (7.11.2)

as in Subsection 4.11.3. This means that

lim_{j→∞} ∫_{X_{j}}(g ∘ f_{j}) dμ_{j} = ∫_{X}(g ∘ f) dμ (7.11.3)

is the same as saying that

lim_{j→∞} ∫_{R} g dν_{f_{j}} = ∫_{R} g dν_{f}. (7.11.4)

The condition that (7.11.3) hold for all bounded continuous functions g on R characterizes a type of weak convergence of {f_{j}}_{j=1}^{∞} to f, as in Proposition 21.31 on p293 of [20]. This type of weak convergence is defined on p292 of [20], and is also known as convergence in distribution. This is the same as saying that {ν_{f_{j}}}_{j=1}^{∞} converges to ν_{f} in the weak^{∗} sense with respect to C_{b}(R), or equivalently C_{b}(R, C), as in the preceding paragraph.

The convergence of {ν_{f_{j}}}_{j=1}^{∞} to ν_{f} in the weak^{∗} sense with respect to C_{b}(R) is equivalent to convergence in the weak^{∗} sense with respect to C_{com}(R), because ν_{f_{j}}(R) = μ_{j}(X_{j}) = 1 for each j, and ν_{f}(R) = μ(X) = 1. This uses the remarks in Subsections 7.8.3 and 7.8.4. This is also equivalent to the convergence of {ν_{f_{j}}}_{j=1}^{∞} to ν_{f} in the weak^{∗} sense with respect to C_{0}(R), as in Section 7.7.

The definition of convergence in distribution on p292 of [20] uses the distribution functions associated to these random variables, as in Section 4.13. More precisely, it is defined to mean that the distribution functions associated to the f_{j}’s converge pointwise to the distribution function associated to f at points where the distribution function associated to f is continuous. This is related to Proposition 7.19 on p223 of [118], and to the remarks in Section 7.6.

Let us emphasize that while the probability distribution ν_{f} associated to f is uniquely determined by convergence in distribution, the random variable f is not. This is related to Example 21.33 on p295 of [20]. One can also look at this in terms of isomorphisms between probability spaces, as in Subsection 4.11.2.

7.11.1 When f is constant

Let c be a real number, and suppose for the moment that f is the constant function equal to c to X. This means that ν_{f} is the Borel probability measure δ_{c} defined on R as in Subsection 1.7.3. Suppose that {ν_{f_{j}}}_{j=1}^{∞} converges to ν in the weak^{∗} sense with respect to C_{com}(R), and let ϵ > 0 be given. Observe that

1 = ν_{f}((c − ϵ, c + ϵ)) ≤ lim inf_{j→∞} ν_{f_{j}}((c − ϵ, c + ϵ)), (7.11.5)

where the second step is as in Subsection 7.5.1, with U = (c − ϵ, c + ϵ). In fact, the previous argument can be simplified in this case, by taking K = {c}.

This is the same as saying that

lim_{j→∞} ν_{f_{j}}((c − ϵ, c + ϵ)) = 1, (7.11.6)

because the ν_{f_{j}}’s are probability measures on R. This means that

lim_{j→∞} ν_{f_{j}}(R \ (c − ϵ, c + ϵ)) = 0. (7.11.7)

Equivalently,

lim_{j→∞} μ_{j}(f_{j}^{−1}(R \ (c − ϵ, c + ϵ))) = 0, (7.11.8)

by the definition of ν_{f_{j}}. If (X_{j}, A_{j}, μ_{j}) = (X, A, μ) for each j, then it follows that {f_{j}}_{j=1}^{∞} converges to f in measure on X. This corresponds to part (2) of Proposition 21.32 on p294 of [20].

7.11.2 When {f_{j}}_{j=1}^{∞} converges pointwise

Let (X, A, μ) be a probability space again, let {f_{j}}_{j=1}^{∞} be a sequence of measurable real-valued functions on X, and let f be another measurable real-valued function on X. Also let ν_{f_{j}}, j ≥ 1, and ν_{f} be the Borel probability measures on R associated to the f_{j}’s and f, as before. Suppose for the moment that {f_{j}}_{j=1}^{∞} converges to f pointwise almost everywhere on X with respect to μ. If g is a continuous real or complex-valued function on the real line, then it follows that

(7.11.9) {g ∘ f_{j}}_{j=1}^{∞} converges to g ∘ f pointwise almost everywhere on X

with respect to μ.

If g is also bounded on R, then {g ∘ f_{j}}_{j=1}^{∞} is uniformly bounded on X. In this case, the bounded convergence theorem implies that

lim_{j→∞} ∫_{X}(g ∘ f_{j}) dμ = ∫_{X}(g ∘ f) dμ. (7.11.10)

This means that {ν_{f_{j}}}_{j=1}^{∞} converges to ν_{f} in the weak^{∗} sense with respect to C_{b}(R), as before. This corresponds to part of the proof of part (1) of Proposition 21.32 on p294 of [20].

7.11.3 When {f_{j}}_{j=1}^{∞} converges in measure

Suppose now that {f_{j}}_{j=1}^{∞} converges to f in measure on X with respect to μ. If g is a uniformly continuous real or complex-valued function on the real line, then one can check that

(7.11.11) {g ∘ f_{j}}_{j=1}^{∞} converges to g ∘ f in measure on X

with respect to μ.

If g is bounded on R as well, then {g ∘ f_{j}}_{j=1}^{∞} is uniformly bounded on X, as before, and the bounded convergence theorem for convergence in measure implies that (7.11.10) holds, as in Section 4.8. In particular, this works when g is continuous on R and vanishes at infinity, as in Subsection 7.1.2. This means that {ν_{f_{j}}}_{j=1}^{∞} converges to ν_{f} in the weak^{∗} sense with respect to C_{0}(R), as before. It follows that {ν_{f_{j}}}_{j=1}^{∞} converges to ν_{f} in the weak^{∗} sense with respect to C_{b}(R), as mentioned in the first part of this section. This corresponds to part (1) of Proposition 21.32 on p294 of [20].

Alternatively, let {f_{j_{l}}}_{l=1}^{∞} be a subsequence of {f_{j}}_{j=1}^{∞}. This subsequence converges to f in measure on X with respect to μ, and so there is a subsequence {f_{j_{lr}} }_{r=1}^{∞} of {f_{j_{l}}}_{l=1}^{∞} that converges to f pointwise almost everywhere on X with respect to μ, as in Subsection 4.7.3. If g is a bounded continuous function on R, then it follows that

lim_{r→∞} ∫_{X}(g ∘ f_{j_{lr}}) dμ = ∫_{X}(g ∘ f) dμ, (7.11.12)

as in the preceding subsection. One can use this to get (7.11.10), as in [20].

7.12 The derivative of exp(a t)

It is well known that the complex exponential function exp z is a complex analytic or holomorphic function on the complex plane, whose complex derivative is itself. This can be seen by differentiating the series expansion for exp z termwise.

7.12. THE DERIVATIVE OF exp(A T) 221

Similarly, if a is a complex number, then exp(a t) is differentiable as a function of t ∈ R, with

\frac{d}{dt} exp(a t) = a exp(a t). (7.12.1)

If a is imaginary, then it follows that

|\frac{d}{dt} exp(a t)| = |a| (7.12.2)

for each t ∈ R. One can use this to get that

| exp(a r_{1}) − exp(a r_{2})| ≤ |a| |r_{1} − r_{2}| (7.12.3)

for all r_{1}, r_{2} ∈ R in this case, using the fundamental theorem of calculus, for instance.

7.12.1 Differentiating the Fourier transform

Let ν be a finite Borel measure on the real line, and suppose that

∫_{−∞}^{+∞} |x| dν(x) < +∞. (7.12.4)

Under these conditions, it is well known that the Fourier transform of ν is differentiable on R, with

ν′(ξ) = ∫_{−∞}^{+∞} i x exp(i x ξ) dν(x). (7.12.5)

This can be obtained using the dominated convergence theorem and (7.12.3). One can also check that ν′ is continuous on R, and in fact it is uniformly continuous on R.

7.12.2 Differentiating ν̂ some more

Let k be a positive integer, and suppose that

∫_{−∞}^{+∞} |x|^{k} dν(x) < +∞. (7.12.6)

This implies that ν̂ is k-times continuously differentiable on R, with

ν^{(k)}(ξ) = ∫_{−∞}^{+∞}(i x)^{k} exp(i x ξ) dν(x). (7.12.7)

This is formulated for characteristic functions of random variables on probability spaces in Proposition 21.43 on p302 of [20]. This can be obtained by repeating the same type of arguments as for k = 1.

7.12.3 More on Fourier transforms of integrable functions on R

Let f be a real or complex-valued function on the real line that is integrable with respect to Lebesgue measure, and suppose that x f(x) is integrable with respect to Lebesgue measure on R as well. This implies that the Fourier transform of f is continuously differentiable on the real line, with the derivative of f̂ equal to the Fourier transform of (i x) f(x), as before. This type of result is often stated this way, as in Proposition 16.2 on p163 of [20]. See also part (i) of Theorem 3 on p8 of [45] and Theorem 1.7 on p4 of [337], for instance.

If f is integrable on R and the derivative of f is integrable on R with respect to Lebesgue measure too, then one would expect to have that

ˆ(f′)(ξ) = −i ξ f̂(ξ), (7.12.8)

using integration by parts. This corresponds to Proposition 16.3 on p163 of [20] when f is absolutely continuous on every closed interval in R, and this is essentially the same as in part (ii) of Theorem 3 on p8 of [45]. This also corresponds to Theorem 1.8 on p4 of [337] when f is differentiable in the L^{1} norm, as in Subsection 4.5.4. Results like these are often stated with slightly stronger hypotheses, for convenience, and of course the main principles are the same.

7.13 Gaussian distributions

It a is a positive real number, then it is well known that

∫_{−∞}^{+∞} exp(−ax^{2}) dx = \sqrt{π/a}, (7.13.1)

as in Proposition 2.53 on p79 of [118]. More precisely, the square of this integral may be considered as an integral over R^{2} that can be evaluated using polar coordinates.

If σ is a positive real number, then it follows that

\frac{1}{σ \sqrt{2 π}} ∫_{−∞}^{+∞} exp(−t^{2}/(2 σ^{2})) dt = 1. (7.13.2)

One can check that

\frac{1}{σ \sqrt{2 π}} ∫_{−∞}^{+∞} t exp(−t^{2}/(2 σ^{2})) dt = 0 (7.13.3)

and

\frac{1}{σ \sqrt{2 π}} ∫_{−∞}^{+∞} t^{2} exp(−t^{2}/(2 σ^{2})) dt = σ^{2}, (7.13.4)

as on p325 of [118].

The Borel probability measure defined on the real line by

ν_{0}^{σ^{2}}(E) = \frac{1}{σ \sqrt{2 π}} ∫_{E} exp(−t^{2}/(2 σ^{2})) dt (7.13.5)

is called the Gaussian or normal distribution with mean 0 and variance σ^{2}, as on p325 of [118]. If σ = 1, then this is called the standard normal distribution, as in [118]. This corresponds to the standard normal density on p395 of [14], and the standard normal distribution on p270 of [20]. A random variable on a probability space with a Gaussian distribution may be called a Gaussian or normal random variable, as on p300 of [20]. Of course, one may consider Gaussian distributions and random variables with arbitrary means as well, instead of mean 0.

It is well known that

ν̂_{0}^{σ^{2}}(ξ) = exp(−σ^{2} ξ^{2}/2). (7.13.6)

This means that if f is a Gaussian random variable on a probability space with mean 0 and variance σ^{2}, then its characteristic function is

ϕ_{f}(ξ) = exp(−σ^{2} ξ^{2}/2), (7.13.7)

as on p301 of [20]. If σ = 1, then (7.13.6) corresponds to part (1) of Proposition 16.5 on p164 of [20], and it is easy to reduce to that case. Of course, the Fourier transform of a Gaussian is discussed in many texts, with perhaps slightly different conventions.

7.14 A convergence theorem

Let λ be a Borel probability measure on the real line such that

∫_{−∞}^{+∞} t^{2} dλ(t) = 1 (7.14.1)

and

∫_{−∞}^{+∞} t λ(t) = 0. (7.14.2)

Note that the finiteness of the integral on the left side of (7.14.1) implies that

∫_{−∞}^{+∞} |t| dλ(t) < +∞, (7.14.3)

so that the integral on the left side of (7.14.2) is defined.

If n is a positive integer, then put

λ^{∗n} = λ ∗ ⋯ ∗ λ (n factors), (7.14.4)

as in Section 7.4. Let λ_{n} be the Borel measure defined on R by

λ_{n}(E) = λ^{∗n}(\sqrt{n} E), (7.14.5) where \sqrt{n} E = \{\sqrt{n} t : t ∈ E\}.

Under these conditions, Theorem 10.14 on p326 of [118] states that {λ_{n}}_{n=1}^{∞} converges to ν_{0}^{1} in the weak^{∗} sense with respect to C_{0}(R, C), where ν_{0}^{1} is the standard normal distribution on R, as in the previous section. More precisely, one can show that {λ̂_{n}}_{n=1}^{∞} converges to ν̂_{0}^{1} pointwise on R, as in [118]. This implies weak^{∗} convergence of the corresponding Borel measures with respect to C_{0}(R, C), as in Subsection 7.10.2.

7.14.1 The central limit theorem

Let (X, A, μ) be a probability space, and suppose that {f_{j}}_{j=1}^{∞} is a sequence of independent identically distributed square-integrable random variables on X with mean 0 and variance equal to 1. Note that

Var (n^{−1/2} ∑_{j=1}^{n} f_{j}) = n^{−1} ∑_{j=1}^{n} Var(f_{j}) = 1 (7.14.6)

for every positive integer n, as in Section 6.12. Under these conditions, the central limit theorem states that the sequence of probability distributions on the real line associated to the random variables

n^{−1/2} ∑_{j=1}^{n} f_{j} (7.14.7)

converges to the standard normal distribution ν_{0}^{1} in the weak^{∗} sense with respect to C_{0}(R, C), as in Theorem 21.46 on p306 of [20], and Theorem 10.15 on p326 of [118]. The means that this sequence of probability distributions converges to ν_{0}^{1} in the weak^{∗} sense with respect to C_{b}(R, C), as in Section 7.8.

This can be obtained from the convergence theorem discussed at the beginning of the section. If λ is the probability distribution on R associated to the f_{j}’s, then λ^{∗n} is the probability distribution associated to ∑_{j=1}^{n} f_{j}, as in Subsection 7.4.1. Using this, one can check that the probability distribution associated to (7.14.7) is λ_{n}.

7.14.2 Some remarks about logarithms

If {c_{n}}_{n=1}^{∞} is a sequence of complex numbers that converges to c ∈ C, then it is mentioned on p305 of [20] that

lim_{n→∞}(1 + (c_{n}/n))^{n} = exp c, (7.14.8)

and that this is used in the proof of the central limit theorem. The proof is left to the reader, with the additional remark that one should be careful about proofs using logarithms, because the complex logarithm is multi-valued.

If x and y are real numbers, then

(7.14.9) exp(x + i y) = (exp x) (exp(i y)) = (exp x) (cos y + i sin y), as in Section 7.2. Note that

| exp(x + i y)| = exp x. (7.14.10)

This implies that x is uniquely determined by exp(x + i y), because the exponential function is one-to-one on the real line.

If w is a complex number with |w| < 1, then the principal branch of log(1+w) may be defined in a standard way, so that

exp(log(1 + w)) = 1 + w (7.14.11)

and

lim_{w→0} w^{−1} (log(1 + w) − w) = 0. (7.14.12)

It follows that exp(n log(1 + w)) = (1 + w)^{n} (7.14.13)

for every positive integer n. One can use this to get (7.14.8).

Let λ be a Borel probability measure on the real line as at the beginning of the section again, and suppose for the moment that λ is symmetric about 0, in the sense that λ(−E) = λ(E) (7.14.14)

for every Borel set E ⊆ R. One can check that λ̂ is real-valued on R. This means that one can use the ordinary logarithm on the set of positive real numbers in the proof of Theorem 10.14 on p326 of [118]. Similarly, one can use the ordinary logarithm to get (7.14.8) when {c_{n}}_{n=1}^{∞} is a sequence of real numbers. This is the version that is used in the proof of the central limit theorem when the probability distribution associated to the random variables has this property.

Otherwise, in the proof of the central limit theorem in [20], or the convergence theorem in [118], one really only needs (7.14.8) in the case where c is a real number, even though the c_{n}’s may be complex. Of course, this means that the imaginary parts of the c_{n}’s converges to 0 as n → ∞. This can be used to get slightly better properties of the complex logarithm of 1 + c_{n}/n.

Appendix A

Metrics, ultrametrics, and product spaces

A.1 Some remarks related to compactness

Let X be a metric space, or a topological space. As usual, a subset K of X is said to be compact if every open covering of K can be reduced to a finite subcovering. Let us review some related conditions in this section.

A.1.1 The limit point property

Let A be a subset of X, and let x be an element of X. If (X, d) is a metric space, then x is said to be a limit point of A if for every r > 0 there is an y ∈ A such that x ≠ y and d(x, y) < r. In this case, it is well known and not difficult to show that there are infinitely many elements of A in B(x, r).

If X is any topological space, then x is said to be a limit point of A in X if for every open set U ⊆ X with x ∈ U, there is a y ∈ A such that x ≠ y and y ∈ U. It is easy to see that this is equivalent to the previous definition when the topology on X is determined by a metric.

Let us say that x is a strong limit point of A in X if for every open set U ⊆ X with x ∈ U, there are infinitely many elements of A in U. Clearly this implies that x is a limit point of A, and the converse holds in metric spaces, as in the preceding paragraph. More precisely, one can check that a limit point of A is a strong limit point when X satisfies the first separation condition.

A subset E of X is said to have the limit point property if every infinite subset of E has a limit point in E. Similarly, let us say that E has the strong limit point property if every infinite subset of E has a strong limit point in E. This implies that E has the limit point property, and the converse holds in metric spaces, or topological spaces that satisfy the first separation condition.

If E is compact, then it is well known that E has the strong limit point property. Otherwise, if there is an infinite subset A of E without a strong limit point in E, then E can be covered by open sets U such that A ∩ U has only finitely many elements. Compactness implies that E can be covered by finitely many such open sets, contradicting the hypothesis that A has infinitely many elements.

A.1.2 Sequential compactness

A subset E of X is said to be sequentially compact if every sequence of elements of E has a subsequence that converges to an element of E. It is easy to see that this implies that E has the strong limit point property, in any topological space.

It is well known and not too difficult to show that the limit point property implies sequential compactness in metric spaces. More precisely, the strong limit point property implies sequential compactness in a topological space X that satisfies the first countability condition, so that there is a countable local base for the topology of X at each point in X.

A.1.3 Countable compactness

Remember that a subset E of X is said to have the Lindelöf property if every open covering of E in X can be reduced to a subcovering with only finitely or countably many elements, as in Subsection 2.14.1. If every countable open covering of E can be reduced to a finite subcovering, then E is said to be countably compact. Thus E is compact if and only if E has the Lindelöf property and E is countably compact.

If E has the strong limit point property, then it is well known that E is countably compact. Indeed, suppose that U_{1}, U_{2}, U_{3}, … is a sequence of open subsets of X such that

E ⊆ ⋃_{j=1}^{n} U_{j} (A.1.1)

and

E ⊈ ⋃_{j=1}^{n} U_{j} (A.1.2)

for each positive integer n. Let x_{n} be an element of

E \ ( ⋃_{j=1}^{n} U_{j}) (A.1.3)

for each n. One can check that the set of x_{n}’s, n ∈ Z_{+}, has infinitely many elements, and thus a strong limit point y in E, by hypothesis. One can get a contradiction using the fact that y ∈ U_{l} for some positive integer l, so that x_{n} ∈ U_{l} for infinitely many n.

It is well known that countably compact sets have the strong limit point property in any topological space.

The limit point property and sequential compactness in metric spaces will be discussed further in the next section.

228APPENDIX A. METRICS, ULTRAMETRICS, AND PRODUCT SPACES

A.2 Totally bounded sets

Let (X, d) be a metric space. A subset E of X is said to be totally bounded if for every r > 0, E can be covered by finitely many open balls of radius r. One might also ask that these open balls have centers in E, and one can show that this leads to an equivalent definition. Another equivalent condition is that for every ϵ > 0, E can be covered by finitely many sets of diameter less than ϵ.

It is easy to see that compact sets are totally bounded.

If E is not totally bounded, then there is an r > 0 such that E cannot be covered by finitely many open balls of radius r. One can use this to get a sequence {x_{j}}_{j=1}^{∞} of elements of E such that

d(x_{j}, x_{l}) ≥ r (A.2.1)

when j ≠ l. It is not difficult to see that this implies that E does not have the limit point property, or equivalently that E is not sequentially compact. This means that E is totally bounded when E has the limit point property, or equivalently is sequentially compact.

A subset of X is said to be bounded if it is contained in a ball. It is easy to see that totally bounded sets are bounded. One can check that bounded sets in R^{n} are totally bounded, using the standard Euclidean metric on R^{n} for any positive integer n. If d is the discrete metric on X, and E ⊆ X is totally bounded, then one can verify that E has only finitely many elements.

A.2.1 Totally bounded metric spaces

If X is totally bounded, then there is a base for the topology with only finitely or countably many elements, as in Subsection 2.14.1. Equivalently, this means that X is a separable metric space. It follows that X has the Lindelöf property, by Lindelöf’s theorem, as before.

If X has the limit point property, or equivalently is sequentially compact, then X is totally bounded, and countably compact, as in the previous section. This implies that X is compact, because X has the Lindelöf property.

One can use this to show that if E ⊆ X has the limit point property, or equivalently is sequentially compact, then E is compact. Indeed, if E has the limit point property or is sequentially compact as a subset of X, then it is easy to see that E satisfies the same condition as a subset of itself, considered as a metric space, using the restriction of d to E. This implies that E is compact as a subset of itself, as before, and it is well known that this means that E is compact as a subset of X.

A.2.2 Another characterization of compactness

Suppose now that (X, d) is complete as a metric space. If E ⊆ X is a closed set that is also totally bounded, then it is well known that E is compact. This is part of Theorem A 4 on p369 of [297]. This corresponds to Theorem 5.1 on p20 of [124] when E = X, and it is not difficult to reduce to that case.

Compactness of a metric space is defined on p145 of [135] to mean that the metric space is complete and totally bounded. Theorem 6.5 B on p145 of [135] says that this is equivalent to sequential compactness of the metric space. Equivalence with the definition of compactness in terms of open coverings is discussed after that.

Compactness of a metric space is defined on p99 of [185] in terms of the limit point property, or equivalently sequential compactness. The equivalence of this with the definition of compactness in terms of open coverings is part of Theorem 72 on p101 of [185]. Compactness of complete metric spaces that are totally bounded is part of Theorem 74 on p101 of [185].

A.3 Some basic examples

Let X be a metric space, or a topological space, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of X that converges to a point x ∈ X. Under these conditions, one can check that {x_{j} : j ∈ Z_{+}} ∪ {x} (A.3.1) is compact in X.

Note that (A.3.1) may not be a closed set in X when X is a topological space that is not Hausdorff. This is related to the fact that the limit of a convergent sequence may not be unique in a topological space that is not Hausdorff.

If X is a metric space, and {x_{j}}_{j=1}^{∞} is a Cauchy sequence in X, then one can check that {x_{j} : j ∈ Z_{+}} (A.3.2) is totally bounded in X.

A.3.1 Some examples of ultrametrics

Let X = {x_{j} : j ∈ Z_{+}} ∪ {x^{∗}} (A.3.3) be a countably-infinite set, where {x_{j}}_{j=1}^{∞} is a sequence of distinct elements of X, and x^{∗} is another element of X, distinct from the terms of the previous sequence. Also let {a_{j}}_{j=1}^{∞} be a monotonically decreasing sequence of positive real numbers. We would like to define an ultrametric d on X, using this sequence.

Of course, we put d(x_{j}, x_{j}) = 0 for each j, and d(x^{∗}, x^{∗}) = 0. Also put

d(x_{j}, x^{∗}) = d(x^{∗}, x_{j}) = a_{j} (A.3.4)

for each j. If j < l, then put

d(x_{j}, x_{l}) = d(x_{l}, x_{j}) = a_{j}. (A.3.5)

One can check that this defines an ultrametric on X.

If {a_{j}}_{j=1}^{∞} converges to 0 in the real line, then {x_{j}}_{j=1}^{∞} converges to x^{∗} with respect to d. This implies that X is compact with respect to d, as before.

Note that the one-element set {x_{j}} is an open set in X for each j. If the infimum of the set of a_{j}’s is positive, then {x^{∗}} is an open set in X as well.

A.4 Products of n metric spaces

Let n be a positive integer, (X_{j}, d_{X_{j}}) be a metric space for each j = 1, …, n, and let

X = X_{1} × X_{2} × ⋯ × X_{n} = ∏_{j=1}^{n} X_{j} (A.4.1)

be their Cartesian product, as in Subsection 1.6.2. We would like to define suitable metrics on X using the d_{X_{j}}’s, and there are various ways to do this. One way is to take

d_{X,1}(x, y) = ∑_{j=1}^{n} d_{X_{j}}(x_{j}, y_{j}) (A.4.2)

for each x, y ∈ X.

Another way is to take

d_{X,2}(x, y) = ( ∑_{j=1}^{n} d_{X_{j}}(x_{j}, y_{j})^{2})^{1/2} (A.4.3)

for each x, y ∈ X, using the nonnegative square root on the right side, as usual. If X_{j} = R with the standard Euclidean metric for each j, then this is the same as the standard Euclidean metric on R^{n}. One can use the triangle inequality for the standard Euclidean norm on R^{n} to get that d_{X_{2}}(x, y) satisfies the triangle inequality for any metric spaces (X_{j}, d_{X_{j}}).

One can also take

d_{X,∞}(x, y) = max_{1≤j≤n} d_{X_{j}}(x_{j}, y_{j}) (A.4.4)

for each x, y ∈ X. In particular, one can check directly that this satisfies the triangle inequality on X. If d_{X_{j}} is an ultrametric on X_{j} for each j, as in Section 2.15, then it is easy to see that d_{X} is an ultrametric on X.

Clearly d_{X,∞}(x, y) ≤ d_{X_{1}}(x, y), d_{X_{2}}(x, y) (A.4.5)

for every x, y ∈ X. We also have that

d_{X,1}(x, y) ≤ n d_{X,∞}(x, y) (A.4.6)

and d_{X,2}(x, y) ≤ n^{1/2} d_{X,∞}(x, y) (A.4.7)

for every x, y ∈ X.

A.4.1 Some remarks about open and closed balls in X

Let x ∈ X and r > 0 be given, and let

B_{X_{j},d_{X_{j}}} (x_{j}, r), \overline{B}_{X_{j},d_{X_{j}}} (x_{j}, r) (A.4.8)

A.4. PRODUCTS OF N METRIC SPACES 231

be the open and closed balls in X_{j} centered at x_{j} with radius r with respect to d_{X_{j}} for each j, respectively, as in Subsection 2.2.2. Similarly, let

B_{X,d_{X,1}}(x, r), B_{X,d_{X,2}}(x, r), B_{X,d_{X,∞}}(x, r) (A.4.9)

and \overline{B}_{X,d_{X,1}}(x, r), \overline{B}_{X,d_{X,2}}(x, r), \overline{B}_{X,d_{X,∞}}(x, r) (A.4.10)

be the open and closed balls in X centered at x with radius r with respect to d_{X,1}, d_{X,2}, and d_{X,∞}, respectively. One can check that

B_{X,d_{X,∞}}(x, r) = ∏_{j=1}^{n} B_{X_{j},d_{X_{j}}} (x_{j}, r) (A.4.11)

and

\overline{B}_{X,d_{X,∞}}(x, r) = ∏_{j=1}^{n} \overline{B}_{X_{j},d_{X_{j}}} (x_{j}, r), (A.4.12)

using (A.4.4). The open and closed balls in X centered at x with radius r with respect to d_{X,1} and d_{X,2} may be compared with these using (A.4.5), (A.4.6), and (A.4.7).

A.4.2 Some remarks about open sets in X

If U_{j} is an open set in X_{j} with respect to d_{X_{j}} for each j, then one can check that

U = ∏_{j=1}^{n} U_{j} (A.4.13)

is an open set in X with respect to d_{X,∞}, using (A.4.11). This implies that U is an open set in X with respect to d_{X,1} and d_{X,2}, because of (A.4.5).

In fact, one can check that d_{X,1}, d_{X,2}, and d_{X,∞} determine the same collection of open sets in X, using (A.4.5), (A.4.6), (A.4.7). This is the same as the product topology on X associated to the topologies on the X_{j}’s determined by the d_{X_{j}}’s.

A.4.3 Some remarks about closures and closed sets in X

Let A_{j} be a subset of X_{j} for each j = 1, …, n, and put A = ∏_{j=1}^{n} A_{j}. One can check that the closure of A in X with respect to d_{X,∞} is

∏_{j=1}^{n} \overline{A_{j}}, (A.4.14)

where \overline{A_{j}} is the closure of A_{j} in X_{j} with respect to d_{X_{j}} for each j. This is the same as the closure of A in X with respect to d_{X,1} and d_{X,2}.

In particular, if A_{j} is a closed set in X_{j} with respect to d_{X_{j}} for each j, then A is a closed set in X with respect to d_{X,1}, d_{X,2}, and d_{X,∞}. Note that d_{X,1}, d_{X,2}, and d_{X,∞} determine the same collection of closed sets in X.

A.4.4 Sequences in X

Let {x(l)}_{l=1}^{∞} be a sequence of elements of X, so that

x(l) = (x_{1}(l), …, x_{n}(l)) (A.4.15)

for each l ≥ 1. This means that {x_{j}(l)}_{l=1}^{∞} is a sequence of elements of X_{j}, 1 ≤ j ≤ n.

One can check that

(A.4.16) {x(l)}_{l=1}^{∞} converges to x ∈ X

with respect to d_{X,∞} if and only if {x_{j}(l)}_{l=1}^{∞} converges to x_{j} in X_{j} with respect to d_{X_{j}} for each j. Convergence in X with respect to d_{X,∞} is equivalent to convergence with respect to d_{X,1} and d_{X,2}, because of (A.4.5), (A.4.6), and (A.4.7).

Similarly, (A.4.17) {x(l)}_{l=1}^{∞} is a Cauchy sequence in X

with respect to d_{X,∞} if and only if {x_{j}(l)}_{l=1}^{∞} is a Cauchy sequence in X_{j} with respect to d_{X_{j}} for each j. This is also equivalent to {x(l)}_{l=1}^{∞} being a Cauchy sequence in X with respect to d_{X,1} and d_{X,2}.

If the X_{j}’s are complete with respect to the d_{X_{j}}’s for each j, then it follows that X is complete with respect to d_{X,1}, d_{X,2}, and d_{X,∞}. The completeness of X with respect to d_{X,1}, d_{X,2}, and d_{X,∞} are equivalent. If X is complete with respect to any of these three metrics, and X_{j} ≠ ∅ for each j, then it is easy to see that X_{j} is complete for each j.

A.5 Diameters in metric spaces

Let (Y, d_{Y}) be a metric space, and let E be a subset of Y. If E is nonempty and bounded, then the diameter of E may be defined as a nonnegative real number by (A.5.1) diam E = sup{d(w, z) : w, z ∈ E}.

More precisely, one can check that the boundedness of E in Y is equivalent to having an upper bound for the set of distances d(w, z) with w, z ∈ E. It is sometimes convenient to define the diameter of the empty set to be 0, and to interpret the diameter of an unbounded set in Y as being +∞. The diameter of E may also be denoted diam_{Y} E or diam_{Y,d_{Y}} E, to indicate the metric space and possibly the metric, when needed.

It is well known and not too difficult to show that

(A.5.2) diam \overline{E} = diam E,

where \overline{E} is the closure of E in Y, as usual. If E_{1} ⊆ E_{2} ⊆ Y, then

(A.5.3) diam E_{1} ≤ diam E_{2}.

Let \overline{B}_{Y} (y, r) be the closed ball in Y centered at y ∈ Y with radius r ≥ 0, as in Subsection 2.2.2. It is easy to see that

diam \overline{B}_{Y} (y, r) ≤ 2 r, (A.5.4)

using the triangle inequality. If d_{Y} is an ultrametric on Y, then

diam \overline{B}_{Y} (y, r) ≤ r. (A.5.5)

If E is a bounded set in Y and y ∈ E, then

E ⊆ \overline{B}_{Y} (y, diam E). (A.5.6)

If α is a positive real number, then one can use Carathéodory’s construction with ζ_{α}(A) = (diam A)^{α}, (A.5.7)

as in Subsection 1.11.3, to define α-dimensional Hausdorff measure on Y. See Chapter 9 of [25], p61 of [110], p171 of [113], p350 of [118], p145 of [159], p55 of [247], and p327 of [335], for instance.

A.5.1 Diameters in finite products

Let us return now to the same notation and hypotheses as in the previous section. If A is a subset of X, then let us use diam_{X,d_{X,1}} A, diam_{X,d_{X,2}} A, and diam_{X,d_{X,∞}} A to denote the diameters of A with respect to the metrics d_{X,1}, d_{X,2}, and d_{X,∞}, respectively, as before. It is easy to see that if A is bounded with respect to any of these three metrics, then it is bounded with respect to the other two metrics. In this case, we have that

(A.5.8)  diam_{X,d_{X,∞}}A ≤ diam_{X,d_{X,1}}A, diam_{X,d_{X,2}}A,
(A.5.9)  diam_{X,d_{X,1}}A ≤ n diam_{X,d_{X,∞}}A,
(A.5.10)  diam_{X,d_{X,2}}A ≤ n^{1/2} diam_{X,d_{X,∞}}A.

Let E_{j} be a nonempty bounded set in X_{j} with respect to d_{X_{j}} for each j, and put

E = ∏_{j=1}^{n} E_{j}. (A.5.11)

One can verify that E is bounded in X with respect to d_{X,1}, d_{X,2}, and d_{X,∞}. In fact, we have that

(A.5.12)  diam_{X,d_{X,1}}E = ∑_{j=1}^{n} diam_{X_{j},d_{X_{j}}} E_{j},

(A.5.13)  diam_{X,d_{X,2}}E = ( ∑_{j=1}^{n}(diam_{X_{j},d_{X_{j}}} E_{j})^{2})^{1/2},
(A.5.14)  diam_{X,d_{X,∞}}E = max_{1≤j≤n}(diam_{X_{j},d_{X_{j}}} E_{j}).

A.5.2 Total boundedness in finite products

If A ⊆ X is totally bounded with respect to any of the metrics d_{X,1}, d_{X,2}, or d_{X,∞}, then it is easy to see that A is totally bounded with respect to the other two metrics.

If E_{j} is totally bounded in X_{j} with respect to d_{X_{j}} for each j, then E is totally bounded in X with respect to d_{X,∞}, and thus with respect to d_{X,1} and d_{X,2}. More precisely, let r > 0 be given, and suppose that E_{j} can be covered by L_{j}(r) open balls in X_{j} of radius r. One can check that E can be covered by

L(r) = ∏_{j=1}^{n} L_{j}(r) (A.5.15)

open balls in X of radius r with respect to d_{X,∞}. There are analogous statements using closed balls of radius r, or sets of diameter less than r, or less than or equal to r.

A.6 Compactness in finite products

Let us continue with the same notation and hypotheses as in Section A.4. If a subset of X is compact with respect to any of d_{X,1}, d_{X,2}, or d_{X,∞}, then it is compact with respect to the other two metrics, because all three metrics determine the same collection of open sets in X. Similarly, if a subset of X is sequentially compact with respect to d_{X,1}, d_{X,2}, or d_{X,∞}, then one can check directly that it is sequentially compact with respect to the other two metrics.

If K_{j} is a compact subset of X_{j} with respect to d_{X_{j}} for each j = 1, …, n, then it is well known that

K = ∏_{j=1}^{n} K_{j} (A.6.1)

is compact in X with respect to the metrics d_{X,1}, d_{X,2}, and d_{X,∞}. This may be considered as a particular case of Tychonoff’s theorem for topological spaces. Let us mention a couple of other ways to look at this for products of finitely many metric spaces.

One can show more directly that K is sequentially compact using the sequential compactness of K_{j} for each j, and this works as well for products of finitely many topological spaces, using the product topology. Let {x(l)}_{l=1}^{∞} be a sequence of elements of K, so that {x_{j}(l)}_{l=1}^{∞} is a sequence of elements of K_{j} for each j. There is a subsequence {x_{1}(l_{m})}_{m=1}^{∞} of {x_{1}(l)}_{l=1}^{∞} that converges to an element x_{1} of K_{1} in X_{1}, because K_{1} is sequentially compact in X_{1}. Note that we can use the same sequence of indices {l_{m}}_{m=1}^{∞} to get a subsequence {x(l_{m})}_{m=1}^{∞} of {x(l)}_{l=1}^{∞}.

There is a subsequence {x_{2}(l_{m_{r}})}_{r=1}^{∞} of {x_{2}(l_{m})}_{m=1}^{∞} that converges to an element x_{2} of K_{2}, because K_{2} is sequentially compact. We can use the same sequence {l_{m_{r}}}_{r=1}^{∞} of indices to get a subsequence {x(l_{m_{r}})}_{r=1}^{∞} of {x(l_{m})}_{m=1}^{∞}, as before. We also have that {x_{1}(l_{m_{r}})}_{r=1}^{∞} converges to x_{1} in X_{1}, because {x_{1}(l_{m_{r}})}_{r=1}^{∞} is a subsequence of {x_{1}(l_{m})}_{m=1}^{∞}.

We can repeat the process to get a subsequence of {x(l)}_{l=1}^{∞} such that the corresponding sequence of jth coordinates converges to an element x_{j} of K_{j} in X_{j} for each j. It follows that x = (x_{1}, …, x_{n}) ∈ K, and that this subsequence of {x(l)}_{l=1}^{∞} converges to x in X, as in Subsection A.4.4.

A.6.1 Using total boundedness and completeness

Alternatively, if K_{j} is compact in X_{j} for each j, then K_{j} is closed and totally bounded for each j. This implies that K is closed and totally bounded in X with respect to each of d_{X_{1}}, d_{X_{2}}, and d_{X∞}, as in Subsections A.4.3 and A.5.2.

Suppose for the moment that X_{j} is complete with respect to d_{X_{j}} for each j. This implies that X is complete with respect to d_{X,1}, d_{X,2}, and d_{X,∞}, as in Subsection A.4.4. It follows that K is compact in X, as in Subsection A.2.2.

Otherwise, we may consider K_{j} as a metric space for each j, using the restriction of d_{X_{j}} to K_{j}. It is well known that K_{j} is complete for each j, because it is compact. Indeed, one can use sequential compactness to get that a Cauchy sequence of elements of K_{j} has a subsequence that converges to an element of K_{j}, and one can check that the Cauchy sequence converges to the same element of K_{j} under these conditions.

Thus the previous argument implies that K is compact as a subset of itself with respect to the restriction of any of d_{X,1}, d_{X,2}, or d_{X,∞} to K. This means that K is compact as a subset of X, by a well-known result.

A.7 Some remarks about truncating metrics

Let t be a positive real number. It is easy to see that

(A.7.1) min(a + b, t) ≤ min(a, t) + min(b, t)

for all nonnegative real numbers a and b. We also have that

min(a_{1}, t) ≤ min(a_{2}, t) (A.7.2)

when a_{1} and a_{2} are nonnegative real numbers such that a_{1} ≤ a_{2}. This implies that (A.7.3) min(max(a, b), t) = max(min(a, t), min(b, t)) for all a, b ≥ 0.

A.7.1 Some new metrics on a metric space

Let (Y, d_{Y}) be a metric space, and put

d_{Y,t}(y, z) = min(d_{Y} (y, z), t) (A.7.4)

for all y, z ∈ Y. One can check that this defines a metric on Y too, using (A.7.1) and (A.7.2). If d_{Y} is an ultrametric on Y, then d_{Y,t} is an ultrametric on Y as well, because of (A.7.2) and (A.7.3).

A.7.2 Another new metric on Y

Observe that

\frac{a + b}{1 + a + b} = \frac{a}{1 + a + b} + \frac{b}{1 + a + b} ≤ \frac{a}{1 + a} + \frac{b}{1 + b} (A.7.5)

for all a, b ≥ 0. If 0 ≤ a_{1} ≤ a_{2}, then

\frac{a_{1}}{1 + a_{1}} = 1 − \frac{1}{1 + a_{1}} ≤ 1 − \frac{1}{1 + a_{2}} = \frac{a_{2}}{1 + a_{2}} . (A.7.6)

It follows that

(A.7.7) \frac{max(a, b)}{1 + max(a, b)} = max (\frac{a}{1 + a}, \frac{b}{1 + b})

for all a, b ≥ 0, as before.

One can use this to get that

d̃_{Y} (y, z) = \frac{d_{Y} (y, z)}{1 + d_{Y} (y, z)} (A.7.8)

is a metric on Y, and an ultrametric on Y when d_{Y} is an ultrametric on Y. This is another commonly used way to get a bounded metric from any other metric, as in Exercise 11 on p44 of [299] when Y is the real line equipped with the standard metric.

A.7.3 Some more easy inequalities

If a is a nonnegative real number, then

\frac{a}{1 + a} ≤ min(a, 1). (A.7.9)

If a ≤ 1, then

\frac{a}{1 + a} ≥ a/2. (A.7.10)

If a ≥ 1, then

\frac{a}{1 + a} = 1 − \frac{1}{1 + a} ≥ 1 − 1/2 = 1/2. (A.7.11)

This means that

min(a, 1)/2 ≤ \frac{a}{1 + a} (A.7.12)

for all a ≥ 0.

Thus d_{Y,1}(y, z)/2 ≤ d̃_{Y} (y, z) ≤ d_{Y,1}(y, z) (A.7.13)

for every y, z ∈ Y.

A.7.4 Open and closed balls with respect to d_{Y,t}

If y ∈ Y and r is a positive real number, then let B_{Y,d_{Y}} (y, r) and \overline{B}_{Y,d_{Y}} (y, r) be the open and closed balls in Y centered at y with radius r with respect to d_{Y}, as in Subsection 2.2.2, and let B_{Y,d_{Y,t}}(y, r) and \overline{B}_{Y,d_{Y,t}}(y, r) be the corresponding balls with respect to d_{Y,t}. Observe that

(A.7.14)  B_{Y,d_{Y,t}}(y, r) = B_{Y,d_{Y}} (y, r) when r ≤ t
= Y  when r > t

and

(A.7.15)  \overline{B}_{Y,d_{Y,t}}(y, r) = \overline{B}_{Y,d_{Y}} (y, r) when r < t
= Y  when r ≥ t.

A.7.5 Compatibility of d_{Y} and d_{Y,t}

It is easy to see that d_{Y} and d_{Y,t} determine the same collections of open sets, closed sets, compact sets, and connected sets in Y. Similarly, d_{Y} and d_{Y,t} determine the same convergent sequences in Y, with the same limits, and they determine the same Cauchy sequences in Y. In particular, Y is complete with respect to d_{Y} if and only if it is complete with respect to d_{Y,t}.

We also have that d_{Y} and d_{Y,t} determine the same collections of totally bounded sets in Y. Of course, Y is automatically bounded with respect to d_{Y,t}.

A.7.6 Sequences of mappings into Y

Let E be a set, let {f_{j}}_{j=1}^{∞} be a sequence of mappings from E into Y, and let f be another mapping from E into Y. Note that {f_{j}}_{j=1}^{∞} converges to f pointwise on E with respect to d_{Y} on Y if and only if {f_{j}}_{j=1}^{∞} converges to f pointwise on E with respect to d_{Y,t} on Y. One can check that {f_{j}}_{j=1}^{∞} converges to f uniformly on E with respect to d_{Y} on Y if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly on E with respect to d_{Y,t} on Y.

A.7.7 Mappings to and from other metric spaces

Let (Z, d_{Z}) be another metric space. A mapping from Y into Z is continuous with respect to d_{Y} on Y if and only if it is continuous with respect to d_{Y,t} on Y. Similarly, a mapping from Y into Z is uniformly continuous with respect to d_{Y} on Y if and only if it is uniformly continuous with respect to d_{Y,t} on Y.

One can verify that a mapping from Z into Y is continuous with respect to d_{Y} on Y if and only if it is continuous with respect to d_{Y,t} on Y. Similarly, a mapping from Z into Y is uniformly continuous with respect to d_{Y} on Y if and only if it is uniformly continuous with respect to d_{Y,t}.

In particular, the identity mapping on Y is uniformly continuous as a mapping from Y equipped with d_{Y} into Y equipped with d_{Y,t}. The identity mapping on Y is also uniformly continuous as a mapping from Y equipped with d_{Y,t} into 238APPENDIX A. METRICS, ULTRAMETRICS, AND PRODUCT SPACES Y equipped with d_{Y}. Most of the previous statements could be considered as consequences of these two.

A.8 Sequences of metric spaces

If A_{j} is a set for each positive integer j, then the Cartesian product of the A_{j}’s is the set ∏_{j=1}^{∞} A_{j} of sequences a = {a_{j}}_{j=1}^{∞} with a_{j} ∈ A_{j} for each j.

A.8.1 Some metrics on countable products

Let (X_{j}, d_{X_{j}}) be a metric space for each positive integer j. Also let {b_{j}}_{j=1}^{∞} be a sequence of positive real numbers, and suppose that d_{X_{j}}(x_{j}, w_{j}) ≤ b_{j} (A.8.1) for each j and x_{j}, w_{j} ∈ X_{j}. Suppose from now on this section that (A.8.2) {b_{j}}_{j=1}^{∞} is a bounded sequence in R.

Put

d_{X}(x, w) = sup_{j≥1} d_{X_{j}}(x_{j}, w_{j}) (A.8.3)

for every x, w ∈ X. One can check that this defines a metric on X. More precisely, if x, w, z ∈ X, then d_{X_{j}}(x_{j}, z_{j}) ≤ d_{X_{j}}(x_{j}, w_{j}) + d_{X_{j}}(w_{j}, z_{j}) ≤ d_{X}(x, w) + d_{X}(w, z). (A.8.4) This implies that d_{X}(x, z) ≤ d_{X}(x, w) + d_{X}(w, z). (A.8.5)

A.8.2 When d_{X_{j}} is an ultrametric on X_{j} for each j

Suppose for the moment that d_{X_{j}} is an ultrametric on X_{j} for each j. If x, w, and z are elements of X, then

(A.8.6)  d_{X_{j}}(x_{j}, z_{j}) ≤ max(d_{X_{j}}(x_{j}, w_{j}), d_{X_{j}}(w_{j}, z_{j}))
≤ max(d_{X}(x, w), d_{X}(w, z))

for each j. This implies that d_{X}(x, z) ≤ max(d_{X}(x, w), d_{X}(w, z)), (A.8.7) so that d_{X} is an ultrametric on X.

If d_{X_{j}} is the discrete metric on X_{j} for each j, then one can check that d_{X} is the discrete metric on X.

A.8.3 Closed balls in X are products of closed balls

Let x ∈ X and a positive real number r be given, and let B_{X_{j},d_{X_{j}}} (x_{j}, r), \overline{B}_{X_{j},d_{X_{j}}} (x_{j}, r) be the open and closed balls in X_{j} centered at x_{j} with radius r with respect to d_{X_{j}} for each j, respectively, as in Subsection 2.2.2. Similarly, let B_{X,d_{X}} (x, r), \overline{B}_{X,d_{X}} (x, r) be the open and closed balls in X centered at x with radius r with respect to d_{X}, respectively. It is easy to see that

B_{X,d_{X}} (x, r) ⊆ ∏_{j=1}^{∞} B_{X_{j},d_{X_{j}}} (x_{j}, r) (A.8.8)

and

\overline{B}_{X,d_{X}} (x, r) = ∏_{j=1}^{∞} \overline{B}_{X_{j},d_{X_{j}}} (x_{j}, r). (A.8.9)

A.8.4 Some open sets in X

Let U_{j} be an open set in X_{j} with respect to d_{X_{j}} for each j. If U_{j} = X_{j} for all but finitely many j, then one can verify that

U = ∏_{j=1}^{∞} U_{j} (A.8.10)

is an open set in X with respect to d_{X}, using (A.8.8). This means that the topology determined on X by d_{X} is at least as strong as the product topology corresponding to the topology determined on X_{j} by d_{X_{j}} for each j.

A.8.5 Products of closed sets are closed sets in X

Let A_{j} be a subset of X_{j} for each j, and put A = ∏_{j=1}^{∞} A_{j}. One can check that the closure of A in X with respect to d_{X} is

∏_{j=1}^{∞} \overline{A_{j}}, (A.8.11)

where \overline{A_{j}} is the closure of A_{j} in X_{j} with respect to d_{X_{j}} for each j. In particular, if A_{j} is a closed set in X_{j} with respect to d_{X_{j}} for each j, then A is a closed set in X with respect to d_{X}.

A.8.6 Convergence of sequences in X

Let {x(l)}_{l=1}^{∞} be a sequence of elements of X. Thus x(l) = {x_{j}(l)}_{j=1}^{∞} for each l, and {x_{j}(l)}_{l=1}^{∞} is a sequence of elements of X_{j} for each j. If {x(l)}_{l=1}^{∞} converges to an element x = {x_{j}}_{j=1}^{∞} of X with respect to d_{X}, then it is easy to see that {x_{j}(l)}_{l=1}^{∞} converges to x_{j} in X_{j} with respect to d_{X_{j}} for each j.

Suppose now that {x(l)}_{l=1}^{∞} is a Cauchy sequence in X with respect to d_{X}, so that for every ϵ > 0 there is a positive integer L(ϵ) such that

d_{X}(x(l), x(m)) < ϵ (A.8.12)

for all l, m ≥ L(ϵ). This implies that

d_{X_{j}}(x_{j}(l), x_{j}(m)) < ϵ (A.8.13)

for all j, and all l, m ≥ L(ϵ). In particular, this means that {x_{j}(l)}_{l=1}^{∞} is a Cauchy sequence in X_{j} with respect to d_{X_{j}} for each j.

Suppose that {x_{j}(l)}_{l=1}^{∞} converges to x_{j} ∈ X_{j} with respect to d_{X_{j}} for each j, and put x = {x_{j}}_{j=1}^{∞}. We would like to check that {x(l)}_{l=1}^{∞} converges to x in X with respect to d_{X} under these conditions. One can use (A.8.13) to get that d_{X_{j}}(x_{j}(l), x_{j}) ≤ ϵ (A.8.14)

for all j, and all l ≥ L(ϵ). This means that

d_{X}(x(l), x) ≤ ϵ (A.8.15)

for all l ≥ L(ϵ).

If X_{j} is complete as a metric space with respect to d_{X_{j}} for each j, then it follows that X is complete with respect to d_{X}.

A.8.7 Diameters of products in X

Note that X_{j} is bounded with respect to d_{X_{j}} for each j, and that X is bounded with respect to d_{X}, by hypothesis. Let E_{j} be a nonempty subset of X_{j} for each j, and put E = ∏_{j=1}^{∞} E_{j}. One can verify that

(A.8.16) diam_{X,d_{X}} E = sup_{j≥1} (diam_{X_{j},d_{X_{j}}} E_{j}).

A.9 When {b_{j}}_{j=1}^{∞} converges to 0

Let us continue with the same notation and hypotheses as at the beginning of the previous section, and suppose now that

lim_{j→∞} b_{j} = 0. (A.9.1)

This implies that the supremum on the right side of (A.8.3) is attained for all x, w ∈ X, so that

d_{X}(x, w) = max_{j≥1} d_{X_{j}}(x_{j}, w_{j}). (A.9.2)

This is clear when x = w, and otherwise one can check that the maximum on the right reduces to the maximum of finitely many terms.

A.9. WHEN {B_{J}}_{J=1}^{∞} CONVERGES TO 0 241

A.9.1 Open balls in X are products of open balls

If x ∈ X and r is a positive real number, then

B_{X,d_{X}} (x, r) = ∏_{j=1}^{∞} B_{X_{j},d_{X_{j}}} (x_{j}, r), (A.9.3)

because of (A.9.2). We also have that

B_{X_{j},d_{X_{j}}} (x_{j}, r) = X_{j} (A.9.4)

when b_{j} < r, because of (A.8.1). This means that (A.9.4) holds for all but finitely many j, because of (A.9.1).

One can use this to get that every open set in X with respect to d_{X} is an open set with respect to the product topology corresponding to the topology determined on X_{j} by d_{X_{j}} for each j. It follows that this product topology is the same as the topology determined on X by d_{X}, because every open set with respect to the product topology is an open set with respect to d_{X}, as mentioned in Subsection A.8.4.

Note that \overline{B}_{X_{j},d_{X_{j}}} (x_{j}, r) = X_{j} (A.9.5)

when b_{j} ≤ r. This holds for all but finitely many j, because of (A.9.1), as before.

A.9.2 More on convergence of sequences in X

Let {x(l)}_{l=1}^{∞} be a sequence of elements of X, and let x be an element of X. If {x_{j}(l)}_{j=1}^{∞} converges to x_{j} in X_{j} with respect to d_{X_{j}} for each j, then one can check that {x(l)}_{l=1}^{∞} converges to x in X with respect to d_{X}. This uses the fact that d_{X_{j}}(x_{j}(l), x_{j}) ≤ b_{j} (A.9.6)

for all j and l, as in (A.8.1). More precisely, the right side is as small as we want for all but finitely many j, because of (A.9.1). In order to get that d_{X}(x(l), x) is as small as we want when l is sufficiently large, it suffices to have that d_{X_{j}}(x_{j}(l), x_{j}) is as small as we want when l is sufficiently large for fnitely many j.

A.9.3 Total boundedness in infinite products

Suppose that A_{j} is a totally bounded subset of X_{j} with respect to d_{X_{j}} for each j, and consider A = ∏_{j=1}^{∞} A_{j}. It is not too difficult to see that A is totally bounded in X with respect to d_{X}, because of (A.9.1).

Indeed, put

X^{n} = ∏_{j=1}^{n} X_{j} and A^{n} = ∏_{j=1}^{n} A_{j} (A.9.7)

for each positive integer n, and let d_{X^{n},∞} be the metric defined on X^{n} as in Section A.4. Note that A^{n} is totally bounded in X^{n} with respect to d_{X^{n},∞}, as in Subsection A.5.2.

In order to show that A is totally bounded, one would like to be able to cover A by finitely many balls of arbitrarily small radius, or finitely many sets of arbitrarily small diameter. To do this, one can use an analogous covering of A^{n} for n sufficiently large. This can be obtained from (A.9.4) or (A.9.5), using coverings by open or closed balls, as appropriate, and the previous descriptions of open and closed balls in X and X^{n} with respect to d_{X} and d_{X^{n},∞}, respectively.

A.9.4 Compactness in infinite products

If K_{j} is a compact set in X_{j} with respect to d_{X_{j}} for each j, then it is well known that

K = ∏_{j=1}^{∞} K_{j} (A.9.8)

is compact in X with respect to d_{X}. This may be considered as another case of Tychonoff’s theorem for topological spaces. This can also be obtained using the remarks about total boundedness in X in the previous subsection, as in Subsection A.6.1.

Alternatively, one can show that K is sequentially compact using the sequential compactness of K_{j} for each j, and this works as well for countable products of topological spaces, using the product topology. More precisely, let {x(l)}_{l=1}^{∞} be a sequence of elements of K, so that {x_{j}(l)}_{l=1}^{∞} is a sequence of elements of K_{j} for each j. One can use the sequential compactness of the K_{j}’s to get a sequence of subsequences of {x(l)}_{l=1}^{∞} with the following properties, as in Section A.6.

The first property is that if r ≥ 2 is an integer, then the rth subsequence is a subsequence of the (r−1)th subsequence. This implies that the rth subsequence is a subsequence of all of the previous subsequences.

Let {x(m, r)}_{m=1}^{∞} be the rth subsequence of {x(l)}_{l=1}^{∞} mentioned before. The second property is that for each r ≥ 1 there be an element x_{r} of K_{r} such that (A.9.9) {x_{r}(m, r)}_{m=1}^{∞} converges to x_{r}

in X_{r} with respect to d_{X_{r}}. It follows that

(A.9.10) {x_{j}(m, r)}_{m=1}^{∞} converges to x_{j}

in X_{j} with respect to d_{X_{j}} when j ≤ r, because {x(m, r)}_{m=1}^{∞} is a subsequence of the jth subsequence, as in the preceding paragraph.

Put y(n) = x(n, n) (A.9.11)

for each positive integer n. One can check that {y(n)}_{n=1}^{∞} is a subsequence of {x(l)}_{l=1}^{∞}, by construction. Similarly, one can verify that {y(n)}_{n=r}^{∞} is a subsequence of {x(m, r)}_{m=1}^{∞} for each r. This implies that

(A.9.12) {y_{j}(n)}_{n=r}^{∞} converges to x_{j}

in X_{j} with respect to d_{X_{j}} when j ≤ r. This means that {y_{j}(n)}_{n=1}^{∞} converges to x_{j} in X_{j} with respect to d_{X_{j}} for each j, so that {y(n)}_{n=1}^{∞} converges to x = {x_{j}}_{j=1}^{∞} in X with respect to d_{X}, as before.

A.10 Snowflake metrics

Let (X, d_{X}) be a metric space, and let a be a positive real number. It is easy to see that d_{X}(x, w)^{a} (A.10.1)

satisfies the first two conditions in the definition of a metric space, i.e., it is equal to 0 if and only if x = w and it is symmetric in x and w. Thus, if (A.10.1) satisfies the triangle inequality on X, then it also defines a metric on X.

One can check directly that (A.10.1) is a metric on X when a = 1/2. In fact, this holds when 0 < a < 1, because of (A.10.2). If d_{X} is an ultrametric on X, then (A.10.1) is an ultrametric for every a > 0. If X is the real line and d_{X} is the standard Euclidean metric on R, then (A.10.1) is not a metric on R when a > 1.

A.10.1 A nice inequality

If a ≤ 1, then it is well known that

(r + t)^{a} ≤ r^{a} + t^{a} (A.10.2)

for all nonnegative real numbers r, t. Indeed, observe that

max(r, t) ≤ (r^{a} + t^{a})^{1/a}. (A.10.3)

It follows that

r + t ≤ (r^{a} + t^{a}) max(r, t)^{1−a} ≤ (r^{a} + t^{a})^{1+(1−a)/a} = (r^{a} + t^{a})^{1/a}, (A.10.4)

using (A.10.3) in the second step. This is equivalent to (A.10.2).

A.10.2 Some properties of d_{X}^{a}

Suppose that (A.10.1) is a metric on X for some a > 0, and let B_{X,d_{X}^{a}} (x, r) and \overline{B}_{X,d_{X}^{a}} (x, r) be the open and closed balls in X centered at x ∈ X with radius r > 0 with respect to (A.10.1), as in Subsection 2.2.2. Observe that

B_{X,d_{X}^{a}} (x, r^{a}) = B_{X,d_{X}} (x, r) (A.10.5)

and \overline{B}_{X,d_{X}^{a}} (x, r^{a}) = \overline{B}_{X,d_{X}} (x, r) (A.10.6) for every x ∈ X and r > 0. It is easy to see that (A.10.1) determines the same collections of open sets, closed sets, compact sets, and connected sets in X as d_{X}. Similarly, (A.10.1) determines the same convergent sequences in X, with the same limits, and the same Cauchy sequences in X as d_{X}, so that X is complete with respect to (A.10.1) if and only if X is complete with respect to d_{X}.

A subset A of X is bounded with respect to (A.10.1) if and only if it is bounded with respect to d_{X}, with

(A.10.7) diam_{X,d_{X}^{a}} A = (diam_{X,d_{X}} A)^{a}.

Similarly, A is totally bounded with respect to (A.10.1) if and only if it is totally bounded with respect to d_{X}.

Let {f_{j}}_{j=1}^{∞} be a sequence of mappings from a set E into X, and let f be another mapping from E into X. Clearly {f_{j}}_{j=1}^{∞} converges to f pointwise on E with respect to (A.10.1) on X if and only if {f_{j}}_{j=1}^{∞} converges to f pointwise on E with respect to d_{X} on X, as before. We also have that {f_{j}}_{j=1}^{∞} converges to f uniformly on E with respect to (A.10.1) on X if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly on E with respect to d_{X} on X.

If (Z, d_{Z}) is another metric space, then continuity and uniform continuity of mappings from X into Z with respect to (A.10.1) on X are equivalent to the same properties with respect to d_{X} on X. Similarly, continuity and uniform continuity of mappings from Z into X with respect to (A.10.1) on X are equivalent to the same properties with respect to d_{X} on X. The identity mapping on X is uniformly continuous as a mapping from X equipped with (A.10.1) into X equipped with d_{X}, and vice-versa.

A.11 Lipschitz mappings

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces, let α be a positive real number, and let f be a mapping from X into Y. We say that f is Lipschitz or Hölder continuous of order α if d_{Y} (f(x), f(w)) ≤ C d_{X}(x, w)^{α} (A.11.1)

for some nonnegative real number C and all x, w ∈ X. This implies that f is uniformly continuous, and this holds with C = 0 if and only if f is constant on X. Sometimes one simply says that f is Lipschitz when this holds with α = 1.

Suppose for the moment that X is bounded with respect to d_{X}, and let α_{0} be a positive real number less than α. In this case, (A.11.1) implies that

d_{Y} (f(x), f(w)) ≤ C (diam_{X,d_{X}} X)^{α−α_{0}} d_{X}(x, w)^{α_{0}} (A.11.2)

for all x, w ∈ X, so that f is Lipschitz of order α_{0} on X too.

Suppose for the moment again that X = [0, 1], d_{X} is the restriction of the standard Euclidean metric on the real line to [0, 1], and that Y = R with the standard Euclidean metric d_{Y}. If f is Lipschitz of order α > 1 on X, then one can check that f is constant on X. More precisely, the derivative of f is equal to 0 at every point in [0, 1] under these conditions, although one can also verify that f is constant more directly.

A.11.1 Using snowflake metrics

Suppose that (A.10.1) is a metric on X for some a > 0, and similarly that

d_{Y} (y, z)^{b} (A.11.3)

is a metric on Y for some b > 0. Observe that (A.11.1) is equivalent to saying that

d_{Y} (f(x), f(w))^{b} ≤ C^{b} (d_{X}(x, w)^{a})^{α b/a} (A.11.4)

for all x, w ∈ X. This condition is the same as saying that f is Lipschitz or Hölder continuous of order α b/a with respect to (A.10.1) on X and (A.11.3) on Y, with constant C^{b}.

A.11.2 Real-valued Lipschitz functions

Let us take Y = R with the standard Euclidean metric again, so that (A.11.1) is the same as saying that

|f(x) − f(w)| ≤ C d_{X}(x, w)^{α} (A.11.5)

for all x, w ∈ X. This implies that

f(x) ≤ f(w) + C d_{X}(x, w)^{α} (A.11.6)

for all x, w ∈ X. Conversely, if (A.11.6) holds for all x, w ∈ X, then one can check that (A.11.5) holds for all x, w ∈ X, by interchanging the roles of x and w.

If p ∈ X, then it follows that

f_{p}(x) = d_{X}(x, p) (A.11.7)

is Lipschitz of order 1 with constant 1 with respect to d_{X} on X.

Suppose that (A.10.1) is a metric on X for some a > 0 again, and consider

f_{p,a}(x) = d_{X}(x, p)^{a} (A.11.8)

as a real-valued function on X. This function is Lipschitz of order 1 with respect to (A.10.1) on X, with constant equal to 1, as in the preceding paragraph. Equivalently, this means that (A.11.8) is Lipschitz of order a with respect to d_{X} on X, with constant equal to 1, as in the previous subsection.

A.12 The set of binary sequences

Let B be the set of all binary sequences, i.e., sequences x = {x_{j}}_{j=1}^{∞} with x_{j} ∈ {0, 1} for each j. This is the same as the Cartesian product ∏_{j=1}^{∞} B_{j}, with B_{j} = {0, 1} for each j, as in Section A.8.

Let θ be a real number with 0 < θ ≤ 1, and put

(A.12.1)  d_{B_{j},θ}(x_{j}, w_{j}) = θ^{j−1} when x_{j} ≠ w_{j}
= 0  when x_{j} = w_{j}

for each j and x_{j}, w_{j} ∈ B_{j}. This is θ^{j−1} times the discrete metric on B_{j}, and an ultrametric on B_{j} in particular. Note that

d_{B_{j},θ}(x_{j}, w_{j})^{a} = d_{B_{j},θ^{a}}(x_{j}, w_{j}) (A.12.2)

for every positive real number a.

Put

d_{B,θ}(x, w) = sup_{j≥1} d_{B_{j},θ}(x_{j}, w_{j}) (A.12.3)

for every x, w ∈ B, as in Subsection A.8.1. This is an ultrametric on B, as in Subsection A.8.2. In fact, this is the discrete metric on B when θ = 1, and we shall primarily be concerned with θ < 1. We also have that

d_{B,θ}(x, w)^{a} = d_{B,θ^{a}}(x, w) (A.12.4)

for every a > 0, as in (A.12.2).

Of course, d_{B_{j},θ}(x_{j}, w_{j}) ≤ θ^{j−1} (A.12.5)

for each j, as in Subsection A.8.1, with

b_{j} = θ^{j−1}. (A.12.6)

If θ < 1, then b_{j} = θ^{j−1} → 0 as j → ∞, as in Section A.9. This implies that

d_{B,θ}(x, w) = max_{j≥1} d_{B_{j},θ}(x_{j}, w_{j}) (A.12.7)

for all x, w ∈ B, as before, and this works in this case when θ = 1 as well.

A.12.1 Another description of d_{B,θ}(x, w)

If x, w ∈ B and x ≠ w, then let l(x, w) be the smallest positive integer such that

x_{l(x,w)} ≠ w_{l(x,w)}. (A.12.8)

Observe that d_{B,θ}(x, w) = θ^{l(x,w)−1}. (A.12.9)

It is sometimes convenient to interpret l(x, w) as being equal to +∞ when x = w, so that the right side of (A.12.9) may be interpreted as being equal to 0 when θ < 1.

A.12.2 The usual mapping from B onto [0, 1]

If x ∈ B, then put

ϕ(x) = ∑_{j=1}^{∞} x_{j} 2^{−j}, (A.12.10)

where the sum on the right converges by comparison with the geometric series ∑_{j=1}^{∞} 2^{−j} = 1. This is the usual way of associating a real number to a binary sequence, and it is well known that

ϕ(B) = [0, 1]. (A.12.11)

If x, w ∈ B and x ≠ w, then one can check that

|ϕ(x) − ϕ(w)| ≤ ∑_{j=l(x,w)}^{∞} 2^{−j} = 2^{1−l(x,w)}. (A.12.12)

This means that |ϕ(x) − ϕ(w)| ≤ d_{B,1/2}(x, w), (A.12.13)

because of (A.12.9).

Suppose for the moment that

ϕ(x) = ϕ(w), (A.12.14)

and that x_{l(x,w)} = 0, w_{l(x,w)} = 1. (A.12.15)

Otherwise, we can interchange the roles of x and w. Under these conditions, it is well known and not too difficult to see that

(A.12.16) x_{j} = 1 and w_{j} = 0 for each j ≤ l(x, w) + 1.

A.12.3 A mapping from B onto the Cantor set

If x ∈ B, then put

ψ(x) = ∑_{j=1}^{∞} 2 x_{j} 3^{−j}, (A.12.17)

where the sum on the right converges by comparison with

2 ∑_{j=1}^{∞} 3^{−j} = (2/3) ∑_{j=0}^{∞} 3^{−j} = (2/3) (1 − (1/3))^{−1} = 1. (A.12.18)

Observe that (A.12.19) ψ(B) is the middle-thirds Cantor set,

as in Example 3.4 on p13 of [159]. This uses the definition of the Cantor set on p19 of [135], or equivalently the characterization in Exercise 19 on p81 of [299].

This corresponds to 2.75 on p57 (= 2010 base 3) of [14], the definition of the Cantor set on p38 of [118], Theorem 6.64 on p71 of [159], and Exercise 2 (a) on p38 of [335].

If x, w ∈ B and x ≠ w, then one can verify that

(A.12.20) |ψ(x) − ψ(w)| ≤ 2 ∑_{j=l(x,w)}^{∞} 3^{−j}
= 2 · 3^{−l(x,w)} (1 − (1/3))^{−1} = 3^{1−l(x,w)}.

Thus |ψ(x) − ψ(y)| ≤ d_{B,1/3}(x, w), (A.12.21)

as in (A.12.9).

One can also check that

|ψ(x) − ψ(w)| ≥ 2 · 3^{−l(x,w)} − 2 ∑_{j=l(x,w)+1}^{∞} 3^{−j}. (A.12.22)

This implies that

|ψ(x) − ψ(w)| ≥ 2 · 3^{−l(x,w)} − 2 · 3^{−l(x,w)−1} ∑_{j=0}^{∞} 3^{−j}
= 2 · 3^{−l(x,w)} − 2 · 3^{−l(x,w)−1} (1 − (1/3))^{−1}
= 2 · 3^{−l(x,w)} − 3^{−l(x,w)} = 3^{−l(x,w)}.

This is the same as saying that

|ψ(x) − ψ(w)| ≥ d_{B,1/3}(x, w)/3, (A.12.23)

because of (A.12.9).

A.12.4 Mapping the Cantor set onto the unit interval

It follows that ϕ ∘ ψ^{−1} (A.12.24)

maps the Cantor set onto [0, 1]. One can check that this mapping is monotonically increasing on the Cantor set as well. This mapping can be extended to a monotonically increasing continuous mapping from [0, 1] onto itself that is constant on each of the open intervals in the complement of the Cantor set in [0, 1], as in Definition 2.77 on p58 of [14], beginning on p32 of [20], on p39 of [118], Exercise 8.28 of [159], Example 8.20 (b) on p179 of [298], parts (b), (c), and (d) of Exercise 2 on p38 of [335], and discussed further beginning on p125 of [335].

In fact, this function is Lipschitz of order

α = log 2/ log 3, (A.12.25)

A.13. MORE ON B AND ϕ 249

as in Lemma 2.3 on p331 of [335]. Note that

d_{B,1/3}(x, w)^{α} = d_{B,1/2}(x, w) (A.12.26)

for all x, w ∈ B, as in (A.12.4). This implies that

|ϕ(x) − ψ(x)| ≤ d_{B,1/2}(x, w) = d_{B,1/3}(x, w)^{α}
(A.12.27)  ≤ 3^{α} |ψ(x) − ψ(w)|^{α} = 2 |ψ(x) − ψ(w)|^{α},

using (A.12.13) in the first step, and (A.12.23) in the third step. This is equivalent to a Lipschitz condition of order α for (A.12.24) as a mapping from the Cantor set onto [0, 1], using the restriction of the standard metric on the real line to both sets. One can use this to get the same Lipschitz condition for the extension of this mapping to a monotonically increasing function on the unit interval mentioned in the previous paragraph.

A.13 More on B and ϕ

Let us continue with the same notation and hypotheses as in the previous section. Put A_{0,k} = {x ∈ B : x_{j} = 0 for all j ≥ k} (A.13.1)

and A_{1,k} = {x ∈ B : x_{j} = 1 for all j ≥ k} (A.13.2)

for each positive integer k. These are finite sets with exactly 2^{k−1} elements, and

A_{0,k} ⊆ A_{0,k+1}, A_{1,k} ⊆ A_{1,k+1} (A.13.3)

for each k. Thus

A_{0} = ⋃_{k=1}^{∞} A_{0,k}, A_{1} = ⋃_{k=1}^{∞} A_{1,k}, A = A_{0} ∪ A_{1} (A.13.4)

are countably infinite subsets of B. It is easy to see that A is dense in B with respect to d_{B,θ} when θ < 1.

Similarly, E_{k} = {m · 2^{1−k} : m = 0, 1, …, 2^{k−1}} (A.13.5)

is a subset of [0, 1] with exactly 2^{k−1} + 1 elements for every positive integer k. Note that E_{k} ⊆ E_{k+1} (A.13.6)

for each k, and put

E = ⋃_{k=1}^{∞} E_{k}. (A.13.7)

This is a countable dense set in [0, 1].

250APPENDIX A. METRICS, ULTRAMETRICS, AND PRODUCT SPACES

A.13.1 The restrictions of ϕ to A_{0}, A_{1}, and B \ A

One can check that ϕ(A_{0,k}) = E_{k} \ {1}, ϕ(A_{1,k}) = E_{k} \ {0} (A.13.8) for each k, and that the restrictions of ϕ to A_{0,k} and A_{1,k} are one-to-one. It follows that ϕ(A_{0}) = E \ {1}, ϕ(A_{1}) = E \ {0}, (A.13.9) and that the restrictions of ϕ to A_{0} and A_{1} are one-to-one as well. Using (A.13.9), we obtain that ϕ(A) = ϕ(A_{0}) ∪ ϕ(A_{0}) = E. (A.13.10)

More precisely, one can verify that ϕ^{−1}(E) = A. (A.13.11) We also have that ϕ(B \ A) = [0, 1] \ E, (A.13.12) and that the restriction of ϕ to B \ A is one-to-one.

A.13.2 Continuity of ϕ^{−1} on [0, 1] \ E

Let x ∈ B \ A and w ∈ B be given, with x ≠ w, and remember that l(x, w) is as in Subsection A.12.1. If x_{l(x,w)} = 0, then w_{l(x,w)} = 1, and

ϕ(w) − ϕ(x) = 2^{−l(x,w)} + ∑_{j=l(x,w)+1} w_{j} 2^{−1} − ∑_{j=l(x,w)+1}^{∞} x_{j} 2^{−j}
(A.13.13)  ≥ 2^{−l(x,w)} − ∑_{j=l(x,w)+1}^{∞} x_{j} 2^{−1} = ∑_{j=l(x,w)+1}^{∞}(1 − x_{j}) 2^{−j}.

Similarly, if x_{l(x,w)} = 1, then w_{l(x,w)} = 0, and

ϕ(x) − ϕ(w) = 2^{−l(x,w)} + ∑_{j=l(x,w)+1} x_{j} 2^{−j} − ∑_{j=l(x,w)+1}^{∞} w_{j} 2^{−j}
(A.13.14)  ≥ ∑_{j=l(x,w)+1}^{∞} x_{j} 2^{−j}.

Note that

∑_{j=k}^{∞} x_{j} 2^{−j}, ∑_{j=k}^{∞}(1 − x_{j}) 2^{−j} > 0 (A.13.15)

for every positive integer k, because x ∉ A_{0,k}, A_{1,k}, by hypothesis. If |ϕ(x) − ϕ(w)| (A.13.16) is sufficiently small, then we get that l(x, w) has to be as large as we like, because of (A.13.13) and (A.13.14). This means that d_{B,1/2}(x, w) is as small as we like when (A.13.16) is sufficiently small.

Let us use ϕ^{−1} for the inverse of the restriction of ϕ to B \ A, as a mapping defined on [0, 1] \ E. The previous argument shows that

(A.13.17) ϕ^{−1} is continuous at ϕ(x),

with respect to the restriction of d_{B,1/2} to B \ A, and the restriction of the standard Euclidean metric on the real line to [0, 1] \ E. Remember that ϕ is continuous as a mapping from B onto [0, 1], as in Subsection A.12.2.

A.14 More on product spaces

Let X_{1}, …, X_{n} be finitely many metric spaces, or topological spaces, and consider their Cartesian product X = ∏_{j=1}^{n} X_{j}, equipped with the product topology, or one of the three metrics discussed in Section A.4, as appropriate. If E_{j} is a dense set in X_{j} for each j, then one can check that E = ∏_{j=1}^{n} E_{j} is dense in X. This follows from a remark about the closure in X of a product of subsets of the X_{j}’s, as in Subsection A.4.3.

If E_{j} has only finitely or countably many elements for each j, then it is well known that E has only finitely or countably many elements too. It follows that X is separable when X_{1}, …, X_{n} are separable.

A.14.1 Some bases for the topology of X

Let B_{X_{j}} be a base for the topology of X_{j} for each j, as in Section 2.3. One can check that

(A.14.1) B_{X} = { ∏_{j=1}^{n} U_{j} : U_{j} ∈ B_{X_{j}} for each j}

is a base for the topology of X.

If B_{X_{j}} has only finitely or countably many elements for each j, then it is not too difficult to see that B_{X} has only finitely or countably many elements as well. More precisely,

∏_{j=1}^{n} B_{X_{j}} (A.14.2)

has only finitely or countably many elements in this case, as before. Observe that

(U_{1}, …, U_{n}) ↦ ∏_{j=1}^{n} U_{j} (A.14.3)

defines a mapping from (A.14.2) onto (A.14.1). One can use this to get that (A.14.1) has only finitely or countably many elements, because (A.14.2) has only fnitely or countably many elements.

If X_{1}, …, X_{n} are metric spaces, then one can use separability to get a base for the topology of X with only finitely or countably many elements, as in Section 2.3. More precisely, this base consists of open balls in X, as before. If one uses the metric d_{X,∞} defined on X as in Section A.4, then open balls in X are products of open balls, as before.

A.14.2 Dense sets in countable products

Now let X_{1}, X_{2}, X_{3}, … be an infinite sequence of nonempty topological or metric spaces, and consider their Cartesian product X = ∏_{j=1}^{∞} X_{j}, equipped with the product topology. If the X_{j}’s are metric spaces, then we ask that the corresponding metrics be bounded, with a sequence of upper bounds that converges to 0, as in Section A.9. This implies that the topology determined on X by the metric defined previously is the same as the product topology, as before.

Let u_{j} be an element of X_{j} for each j, and suppose that E_{j} is a dense set in X_{j} for each j. If n is a positive integer, then put

(A.14.4)  E(n) = {x ∈ X : x_{j} ∈ E_{j} for j = 1, …, n,
and x_{j} = u_{j} when j > n}.

This is the same as the Cartesian product of the sequence of sets E_{1}, …, E_{n} followed by {u_{j}} for j > n.

One can check that

⋃_{n=1}^{∞} E(n) (A.14.5)

is dense in X under these conditions. More precisely, if x ∈ X and n is a positive integer, then one can approximate x by elements of E(n), by approximating x_{j} by an element of E_{j} when j ≤ n, because E_{j} is dense in X_{j}, by hypothesis, and simply replacing x_{j} with u_{j} when j > n. One can get approximations to x like these that are as close to x as we want when n is sufficiently large, because of the hypothesis at the beginning of Section A.9, or by definition of the product topology on X.

If E_{j} has only finitely or countably many elements for each j, then E(n) has only finitely or countably many elements for each n, as usual. This implies that (A.14.5) has only finitely or countably many elements, so that X is separable.

A.14.3 Some bases for the topology of X again

Let B_{X_{j}} be a base for the topology of X_{j} for each j. If n is a positive integer, then put

(A.14.6)  B_{X}(n) = { ∏_{j=1}^{∞} U_{j} : U_{j} ∈ B_{X_{j}} for j = 1, …, n,

and U_{j} = X_{j} when j > n}.

This is a collection of open sets in X, by definition of the product topology, or because the topology determined on X by the metric defined previously is the corresponding product topology when the X_{j}’s are metric spaces.

One can check that

⋃_{n=1}^{∞} B_{X}(n) (A.14.7)

is a base for the topology of X. This follows from the definitions when one uses the product topology on X. If the X_{j}’s are metric spaces, then this uses the description of open balls in X with respect to the usual metric in Section A.9. Of course, this description of the open balls in X implies that the topology determined by the metric is the same as the product topology, as before.

A.14.4 Countability of these bases for the topology of X

Suppose that B_{X_{j}} has only finitely or countably many elements for each j. This implies that ∏_{j=1}^{n} B_{X_{j}} has only finitely or countably many elements for each n, as before. One can use this to get that (A.14.6) has only finitely or countably many elements for each n. It follows that (A.14.7) has only finitely or countably many elements as well.

If the X_{j}’s are metric spaces, then one can use separability of X to get a countable base for the topology of X, as in Section 2.3. This base consists of open balls in X with respect to the metric d_{X} in Section A.9. These open balls in X are products of open balls in the X_{j}’s, where all but finitely many factors are equal to X_{j}, as before.

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Index

absolute convergence in L^{1}, 113 with respect to d, d̃, 119

absolute convergence in L^{p}, 134 absolute value, 94 absolutely continuous functions, 106 absolutely continuous measures, 102 algebras of sets, 10 almost every, 85 almost surely, 163 almost uniform convergence, 116 α-lengths of intervals, 47 automorphisms of measure spaces, 122

Banach spaces, 211 base for a topology, 42 Borel families, 11 Borel measures, 40 Borel outer measures, 41 Borel sets, 11, 40 Borel–Cantelli lemma, 169 bounded convergence theorem, 97, 117 bounded intervals

in R, 14 in R^{n}, 14

bounded sets, 228

C, 79 C_{0}(R^{n}) = C_{0}(R^{n}, R), 197 C_{0}(R^{n}, C), 197 C_{b}(R^{n}) = C_{b}(R^{n}, R), 196 C_{b}(R^{n}, C), 196 C_{com}(R^{n}) = C_{com}(R^{n}, C), 197 C_{com}(R^{n}, C), 197 Carathéodory’s construction, 30 Carathéodory’s criterion, 29 Cartesian products, 14, 238 Cauchy sequences in L^{1}, 113 Cauchy sequences in L^{∞}, 139 Cauchy sequences in L^{p}, 133 Cauchy sequences in measure, 116 Cauchy–Schwarz inequality, 136 central limit theorem, 224 characteristic functions, 9

of random variables, 201 Chebyshev’s inequality, 115, 133, 185 closed balls, 41 compact sets, 226 complete measure spaces, 39 completion of a measure space, 39 complex conjugates, 93 conjugate exponents, 136 connection with riemann integrals, 98 convergence in distribuation, 218 convergence in L^{1}, 113 convergence in L^{∞}, 139 convergence in L^{p}, 133 convergence in measure, 114 convergence in probability, 115 convergent sequences of sets, 10 convolutions, 203 countably compact sets, 227 countably measurable sets, 40, 142 counting measure, 17 cylinder sets, 155

density functions, 126 diameters of sets, 232 discrete metric, 71 distribution functions, 124 dominated convergence theorem, 96

Egorov’s theorem, 115 elementary families of sets, 142 elementary subsets

of R, 14 of R^{n}, 15

essentially bounded functions, 138 Euler’s identity, 199 events in probability theory, 163 expectation of a random variable, 163 expected value, 163 exponential function, 199 extended real numbers, 1

topology on, 3, 76, 77 extension theorem, 30, 31, 57 exterior measures, 27

metric, 28

F_{σ} sets, 41 Fatou’s lemma, 92 finite moments, 183 Fourier transform, 199–201, 215 Fubini’s theorem, 149

G_{δ} sets, 41 Gaussian distributions, 223 Gaussian random variables, 223

Hölder continuity, 244 Hölder’s inequality, 136 Hausdorff measures, 233

identically distributed r.v.’s, 187 image measures, 122 imaginary parts, 93 independent events, 163, 164 independent random variables, 171 indicator functions, 8 induced σ-algebras, 75 induced topologies, 75 infinite products, 165 inner regularity

with respect to closed sets, 31 with respect to compact sets, 31

integrable functions complex-valued, 94 real-valued, 93

intervals in R, 14

in R^{n}, 15

invariance under translations, 99 isomorphisms

between measurable spaces, 122 between measure spaces, 122

joint characteristic functions, 202 joint distributions, 175

Kolmogorov’s inequality, 190

Lebesgue decomposition theorem, 103 Lebesgue integrals, 84, 86, 87, 93, 94 Lebesgue measurable sets, 37 Lebesgue measure, 37 Lebesgue outer measure, 36 Lebesgue’s theorem, 104, 108 Lebesgue–Stieltjes measures, 48 Lebesgue–Stieltjes outer measures, 47 length of an interval, 14 limit inferior, 6, 9 limit point property, 226 limit points, 226 limit superior, 6, 9 limits of functions, 44 Lindelöf property, 69 Lindelöf’s theorem, 69 Lipschitz mappings, 244 locally compact, 60

at a point, 59 locally finite measures, 59 locally finite outer measures, 59 locally integrable functions, 108, 206 lower Lebesgue sums, 86 lower limits, 6, 9

Markov’s inequality, 115 mean of a random variable, 163 measurable functions, 73

extended real-valued, 77 measurable mappings, 74 measurable rectangles, 143 measurable sets, 17

and outer measures, 28 measurable spaces, 17 measure spaces, 17

finite, 17 measure-preserving mappings, 122 measures

as outer measures, 27 Borel, 40 countably additive, 16 finite, 17 finitely additive, 16 on image sets, 122 probability, 17

Minkowski’s inequality, 132 modulus, 94 monomials, 214 monotone classes, 50 monotone convergence theorem, 89

for sums, 22 monotone families, 50 multi-indices, 214 multiplication theorem, 183 mutually singular measures, 103

normal distributions, 223 normal random variables, 223

one-sided continuity, 45 one-sided limits, 45 open balls, 41 order of a multi-index, 214 orthogonal functions, 141 outer measures, 27

Borel, 41 Borel regular, 54 metric, 28 regular, 52

outer regularity, 31 of outer measures, 56 with respect to subsets of compact

sets, 34

p-summable series in L^{p}, 134 partitions of sets, 86 premeasures, 17 probability distributions, 122 probability spaces, 17 product measures, 154, 159 product outer measures, 142 product measures, 147 product σ-algebras, 143, 150, 155 product topology, 231 pth-power integrability, 131 pushing measures forward, 122

Q, 43

R, 1 R^{n}, 14 Rademacher functions, 173 Radon–Nikodym derivative, 103 Radon–Nikodym theorem, 102 random variables (r.v.’s), 122, 163 random vectors, 175 real parts, 93 relatively open sets, 75 Riemann–Lebesgue lemma, 200, 215 Riesz representation theorem, 198 rings of sets, 12

S(R^{n}), 214 Schwartz class, 215 separable spaces, 43 sequentially compact sets, 227 σ-algebras of sets, 11 σ-compact sets, 60 σ-finite measure spaces, 40 σ-finite sets, 39, 40, 54, 142 σ-nice families of sets, 61 σ-rings of sets, 13 simple functions, 83

canonical form, 83

standard representation, 83 square-integrable functions, 133 standard deviation, 184 standard normal distribution, 223 step functions, 209 strong law of large numbers, 190, 194,

195 strong limit point property, 226 strong limit points, 226 subadditivity

countable, 26 finite, 26

summable functions, 20 supports of functions, 196 supremum metric, 196 supremum norm, 196 symmetric differences of sets, 12

Tchebychev’s inequality, 115, 133 tight sequences of measures, 212 Tonelli’s theorem, 148 totally bounded sets, 228 Tychonoff’s theorem, 234, 242

ultrametrics, 71 uniform boundedness principle, 211 upper Lebesgue sums, 86 upper limits, 6, 9 Urysohn’s lemma, 197

vague covergence, 210 vanishing at infinity, 197 variance, 184 volume

of an elementary set, 34 of an interval, 14, 15

weak law of large numbers, 186 weak^{∗} convergence of measures

with respect to C_{0}(R^{n}), 210 with respect to C_{b}(R), 211 with respect to C_{com}(R^{n}), 205

Z_{+}, 13 zero-dimensional spaces, 70

at a point x, 70