Rectifiable Sets

Lipschitz Functions

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\to Y$ is called Lipschitz if there is a constant $C$, called the Lipschitz constant, such that

$$d_Y(f(x),f(y))\le C{d}_X(x,y)$$

for all $x,y\in X$. In particular, $f:{\mathbb{R}}^m\to {\mathbb{R}}^n$ is Lipschitz if there is a constant $C$ such

$$\Vert f(x)-f(y)\Vert \le C\Vert x-y\Vert .$$

for all $x,y\in {\mathbb{R}}^m$.

Rademacher's Theorem

A Lipschitz function is differentiable almost everywhere.

One interesting result of Lipschitz functions is that they can be approximated by $C^1$ functions. Suppose $f:A\subset {\mathbb{R}}^m\to {\mathbb{R}}^n$ is Lipschitz. Then for every $ϵ>0$, there is a $C^1$ function $g:{\mathbb{R}}^m\to {\mathbb{R}}^n$ such that

$$m(\{x\in A:f(x)\ne g(x)\})\le ϵ$$

where $m$ is the Lebesgue measure.

Rectifiable Sets

A set $E\subset {\mathbb{R}}^n$ is called $d$-dimensional rectifiable if $H^d(E)<\infty$ and $H^d$-almost all of $E$ is contained in the union of the images of countably many Lipschitz functions from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$, i.e., there are a countable collection of Lipschitz functions $f_k:{\mathbb{R}}^m\to {\mathbb{R}}^n$ and a set $F$ such that

$$E\subset \left(\bigcup _{k=1}^{\infty }f({\mathbb{R}}^m)\right)\cup F\text{ where }{H}^d(F)=0.$$

For example, the collection of lines below is a 1-rectifiable set,

One may view rectifiable sets as a countable union of manifolds (of which their total area is finite) and arbitrary subsets of ${\mathbb{R}}^n$.