A subset M of R^{n}
is a *k rectifiable set* if M is
contained in a
countable union M_{0} U
M_{1} U M_{2}
U ...

where M_{0} has k dimensional Hausdorff measure 0 and each M_{k}
for k=1,2,..., is a continuously differentiable submanifold
of R^{n} . Here the k
dimensional
Hausdorff (outer) measure may be defined
for any
subset A of a metric space and any nonnegative real
number
k by using sums of the kth powers of
diameters
of efficient countable covers of A by small sets. In
the case k=0 , H^{0} is
counting measure and
M is 0 rectifiable iff it is
countable.
In the case k=n, H^{n}
is n dimensional Lebesgue
measure and *all*
subsets are n rectifiable.
But the really interesting cases are where k
=
2,3,…, n-1 .

Though rectifiable sets may have possibly have
many
"holes", they enjoy, and are even characterized, by important *density*
and* tangential* properties. The k dimensional *density*
of A at a point a is the limit (if
it exists) as r ->0 of

H^{k}[A
intersect B_{r}(a)] / w_{k}
r^{k} for a suitable
constant w_{k}
(which equals the Lebesgue measure of the
unit ball in R^{k} in case k
is an integer). For a
rectifiable set M the density
function
coincides H^{k}^{ }a.e. with the characteristic function X_{M}
.
Also larger and larger homothetic expansions of M
about H^{k}^{ }
almost every point a
of M converge weakly or measure-theoretically to
a
$k$ dimensional affine plane (the approximate tangent space at a ). A few of the striking theorems
about a compact set A of finite
positive H^{k}^{ }
measure
that we will prove are:

(1) (Marstrand) If the k dimensional density
exists and
is positive and finite at H^{k}^{
}almost all a in A ,
then k
is an integer.

(2) (Besicovitch, Federer) If k is an integer , then A = M U N for some k rectifiable set M and a set N whose projection onto almost all k dimensional subspaces has k dimensional Lebesgue measure 0 .

(3) (Matilla, Preiss)
If k
is an integer and the k dimensional density =1 for H^{k}^{ }almost all a
in A , then A is k
rectifiable.

One may alternately characterize a rectifiable set
as an H^{k} almost subset of a
countable union of images *Lipschitz*
maps of
subsets of R^{k}.
This can be used to define a k rectifiable subset of a metric
space,
although some interesting metric spaces, like the set
N above may
be *totally unrectifiable*
in the sense that they do not contain *any*
rectifiable subset of positive H^{k
}measure. For example the square of a ½ dimensional
Cantor
set is 1 unrectifiable and the
Heisenberg
group is 2 unrectifiable. Nevertheless many of the basic properties of
rectifiable sets do carry over to metric spaces, following work of Ambrosio, Kirchheim,
and others
over the last 10 years.

Rectifiable sets have proven useful for various
geometric variational problems involving
area minimization or image processing. Rectifiable sets with additional
structures or properties enjoy compactness results. For example,
rectifiable
sets (possibly with integer multiplicities) of uniformly bounded
measures with
bounds on their first variations (weakly defined mean curvatures) have
weakly
convergent subsequences. Adding H^{k}^{
} measurable orientations of the
approximate tangent spaces
gives rectifiable currents and these become compact when there are
bounds on mass and
boundary mass. This allows for the
solvability of a k dimensional Plateau problem of
mass-minimization with a given boundary. Singular sets of Plateau
solutions
have been shown to be rectifiable in the appropriate dimensions by L.
Simon.

Also the solutions themselves, such as
“soap-bubbles” inside a very dirty medium enjoy an extra *uniform*

*rectifiability* property following works of
David and
Semmes.

In the course we will first review Hausdorff measures, basic density properties, and the calculus-type area and co-area formulas. Much of the course will involve only basic measure theory combined with numerous geometric constructions.

We will frequently refer to the recent book by F.Lin and X.Yang *Geometric Measure Theory*—*An Introduction*,
Science Press (*Geometric Measure Theory*, Springer,
1970) and various papers.

Tues.Thurs

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

Email: hardt@rice.edu, Telephone: ext 3280

http://math.rice.edu/~hardt/521F04/

This page is maintained by Robert Hardt ( email )