MATH 426:

Topics in Real Analysis, Spring 2002


This course is a continuation of Math 425 which treated Lebesgue Integration Theory and Lebesgue Measure.  Math 426 will focus around Hausdorff measure.   Whereas Lebesgue (outer) measure   L   provides a notion of the     n   dimensional size for any subset  A  of Rn ,   the k dimensional Hausdorff measure  Hk (A) gives, for every nonnegative number  k ,  a precise notion of   k  dimensional size .   While H n (A) = L (A), one may use the other numbers Hk(A)   for   k < n    to   define the ( Hausdorff) dimension of  A as

 sup{ j : Hj(A) = infty }  or    inf { k : Hk(A) = 0 }. 

 Then countable sets have dimension 0 , smooth curves have dimension 1, and smooth surfaces have dimension 2.  A more striking example is the Cantor ternary set which has dimension  k = log2/ log3 and in fact finite positive Hausdorff measure in this dimension. In general, one expects a self-similar set , consisting of    m   essentially disjoint congruent pieces each similar to the whole set but scaled down by a factor 1/n ,   to have dimension   log m / log n .   Sets of non-integer dimension are examples of fractals.   The Lebesgue integral employing Hausdorff measure is an important and useful tool, but not all properties of Lebesgue measure have analogues with Hausdorff measure.   For example, Lebesgue’s beautiful theorem that the n dimensional density function  

lim r->0L(A ^ B(x,r))/ L(B(0,1))rn   of a Lebesgue measurable subset   A of Rn   coincides almost everywhere with the characteristic function of   A    is no longer true for the k dimensional density of a compact subset of finite k dimensional measure when   k < n . Nevertheless, there are some general density properties that are true for   H k  which we will discuss. For k being an integer there are even some results about tangency, the extent to which the set locally looks near some points at small scales like a   k dimensional plane.   These are important both historically and for applications in the special case of one dimensional sets in the plane R2.    After proving many of these, we will turn to the relatively recent application from image processing called image segmentation.  

Here the problem concerns improving the appearance of a noisy image which is given as a (grayscale intensity) function   g(x) defined on a planar region   U  or digitally on a discrete grid (pixels) in the plane.   One has two competing goals in improving the image:   1) to improve the sharpness of the edges (as in a profile curve) and  2) to improve the regularity in smooth regions (as in a facial area).   It is mathematically difficult to describe or characterize these processes and hence to find some kind of algorithmic procedure to effect these.   There have been many models proposed and used for image segmentation and related problems. We will discuss some of the variational theory and mathematics associated with the Mumford-Shah model where one considers minimizers of an energy   
E(u,K)  =   JU\K |Du|2dL + JU (u-g)2dL   +  H 1 (K)   corresponding to an image   u   with edge set   K .    Although there is now a considerable literature on this model, there are still many open questions. We will try to highlight some aspects that fit very well with our study of 1 dimensional subsets of   R 2 .   

Background for the course is a knowledge of Lebesgue integration as in Math 425.


TThurs 9:25-10:40 in Herman Brown 423


Robert Hardt  Office: Herman Brown 430; Office hours: 1-2 MWF (and others by appt.),

Email:, Telephone: ext 3280


Frank Jones, Lebesgue Integration on Euclidean Space First Course, Jones and Bartlett, 1993.
(See Frank for a good deal if you don't already have a copy.)

Other references:

K.J. Falconer, The Geometry of Fractal Sets, (paperback) Cambridge University Press, 1985.

J-M. Morel and S. Solimini,  Variational Methods in Image Segmentation, Birkhauser, 1995.


1.  It was stated in an early lecture that the projection onto the Y axis of the lifted-thirds Cantor set was countable.  This is not true.  It is a Cantor set.

2.  Exercise 14 was not stated correctly in class.  See the correct statement below.

3.  Exercise 13 (now corrected) should say Hausdorff measure rather than Lebesgue in the conclusion.

4. In class, I  followed Frank Jones' proof of the Theorem on P.539 about the primitive F of an L1 function f .

Haidee (Zheng Meng)  made the nice suggestion for the proof of the formula for  V_F(a,b).  After verifying that F is BV and proving the < or = , (page540 midpage),
one can use the Theorem on p.461 to say that F' = f   a.e. and then apply the corollary on p.538 with f = F to get the opposite inequality. This argument replaces p.541 and 1/2 of p.540.


1.  Show that if  E  is a Lebesgue measurable subset of  Rn , then  the union of  E  with some set  Z  having
Lebesgue measure L(Z) = 0  is the intersection of  a decreasing sequence 
U1 , U2  , U3 , ...  of open sets. Also  E  is the union of  set  Y  having  L(Y) = 0 and the union of an increasing sequence 
K1 , K2  , K3 , ...  of compact sets.

2. For the “lifted middle-thirds” set  K  constructed in class, verify that lower and upper densities,
liminf r -> 0 r -1 [H1 (K intersect B(0,r))] and limsup r -> 0 r -1 [H 1 (K intersect B(0,r))] ,
are different (where 0 = (0,0) is the lower left corner of the set).

3. Show that if  A  is a convex open subset of  Rn  and  f  is continuously differentiable on  A , then 
Lip f  =  sup { |grad f(x)| : x  is in  A } .  ( Note that this formula is correct for  f  being real-valued.  
For  f  being vector-valued one should replace  |grad f(x)|  by  sup { e . D f(x)  :  e in Rn , |e|=1  }. )

(#1-3 due Thursday, January 24)

4.  Let  S = [0,1]2  (the unit square). Using only the definition of Hausdorff measure (and the fact that
the Lebesgue measure of S is 1 ), show that   0 < H 2 (S) < 2 .

5.  Suppose that  0 < a < 1  and that  Ka  is the Cantor set obtained by recursively omitting from intervals  I
the middle interval of length  a [diam I ]. (So the usual Cantor set is  K 1/3 .) Find the (self-similarity) dimension of  K a

6. For each unit vector  e  in the plane  R2, let  pe (z) =  z - (z . e)e (so that  pe is the projection onto the line perpendicular to e ).  For the “lifted middle-thirds” set  K  constructed in class, verify that  H1[pe (K)] = 0 for every such  e  except (0,1) or (0,-1).  

(#4-6 due Thursday, January 31)

7. Show that Steiner symmetrization  S1 preserves Lebesgue measure, i.e.  L(S1(E)) = L(E)  for  E  in  Rn.
(Here  S1(E) = { (t,x2,x3 ,...,x n ) :  (x1,...,xn) in E , |t| is at most H 1 { s : (s,x2,x 3 ,...,x n ) in E }  } . )

8.  Show that Steiner symmetrization  S1 doesn't increase diameter, i.e. diam(S1(E)) is at most diam(E)  for  E  in  R n .

9. For any outer measure  m  on  X , show that the family  M = { m measurable subsets of  X } is a 
sigma algebra  and the triple  ( X, M, m restricted to M )  is a measure space.

(#7-9 due Thursday, February 7)

Homeworks 10,11,12 pdf file
 Homeworks 10,11,12 ps file

Homeworks 13,14,15 pdf file  Homeworks 13,14,15 ps file

16.  Exercise number 1 on Page 512 of Jones.

17. Exercise number 16 on Page 530 of Jones.

(#16-17 due Thursday, February 28)

18.  Exercise number 17 on Page 532 of Jones.

19. Exercise number 19 on Page 536 of Jones.

(#18-19 due Thursday, March14)

Note:  I have been accepting late homework. But I encourage you not to get
too far behind and, if you are, to try to catch up over the Rice break.

For exercises 20-22 , suppose that  E  is a  subset of  R N and recall that
dim E = inf {t : Ht(E) = 0 }. 

20. Show that  Hk+1( E x [0,1] ) = 0  if and only if  H k(E) = 0 .

21. Show that  dim (E x E)  is at most   2k .

22.  Show that  dim { x-y : x and y belong to  E } is at most  2k.

(#20-22 due Thursday, March 21)

23. Show that every sequence in a precompact metric space contains a Cauchy subsequence.
(Recall that a metric space is precompact (or totally bounded) if it admits, for each positive d, a finite cover by open balls of radius  d.)

24.  Show that a metric space  E  is compact if and only if  E  is precompact and complete.

(#23-24 due Thursday, March 28)

25. Show that if  K  is a compact subset of  Rn  with  H n-1( K ) = 0 , then the complement of  K  is connected.  
26. Show that for  W equalling the open unit square  (0,1) x (0,1)  in  R 2,   the Sobolev space  H(W)  is separable, that is, has a countable dense subset. Hint:  First approximate by smooth functions, then by
certain continuous piece-wise affine functions.

27. Show that   H(W)   has a countable Hilbert basis.

(#25-27 due Tuesday, April 16)

28. Show that if  U is a smoothly bounded domain,  uK  is a minimizer of
 IK(u)  =   J U\K  ( | grad u |2 + | u-g |2 ) dx    ,  and  uK  is  C2  in  U  and  C1  up to the boundary of  U , then the exterior normal derivative of  uK  vanishes at
every point of the boundary of  U .

29. Show that if  f = f(x,t)  is a C1  function on the plane, then
d/dt J01 f(x,t) dx =  J01 d/dt f(x,t) dx .

30.  Suppose that  F  is a family of closed squares in R2  which covers a set  A  of  finite
Lebesgue measure.  Suppose also that, for all  a  in  A ,  inf {diam(S) :  S in F and a in S } =  0  .
Show that some countable disjoint subfamily of  F  covers Lebesgue almost all of  S .

31.  Suppose that  G  is a (possibly uncountable) family of open rectangles in  R2 ,
and, for each rectangle  I  in  G , AI  is some set containing  I  and contained in  Clos I .
Prove that the union, over all  I  in  G , of the AI is Lebesgue measurable.

32.  Suppose that  f : [a,b] -> R  is absolutely continuous,  f(a) < -1 and
f(b) > 1.  Show that Jab ( f ' )2 dt  >  4/(b-a) .

33.  Let  U  be the open square  (-1,1) x (-1,1) ,
  g(x,y) = -3  for -1<x<0, -1 < y < 1 ,
  g(x,y) = +3  for 0<x<1, -1 < y < 1 ,

and suppose that (u,K)  is a minimizer for the Mumford-Shah functional

H1(K) + JU\K |grad u|2 + |u-g|2 dx .

(1)  Show that  H1(K) is at most 2 .

(2)  Show that  H1(K) is at least 1 .

Hint: Consider the set of points  y   such that  u(x,y) >-1  for all x in (-1,0)
 as well as the set of points  y   such that  u(x,y) < 1  for all x in (0,1) .

(3)  Give some ideas about how one would try to find the minimizer  (u,K) .

No Final Exam but all problems due May 8 !

There will be no class Thursday, April 25.  So April 23 will be the last class.
I will be out of town April 24-April 30. After that I will be available for any questions
about homework, the course, etc.


This page is maintained by Robert Hardt ( email )
Last edited 4/22/02.