TUT CAL2 1

We will show that $S^1=\{(x,y)|x^2+y^2=1\}$ has measure zero in ${\mathbb{R}}^{{}^2}$.

[!Lemma]
Measure zero is preserved under continuously differentiable functions. That is, if $f:\overline{B}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n\in C^1(\overline{B})$ (where $B$ is an open ball), $A\subseteq B$ has measure $0$ $⟹$ $f(A)$ has measure zero.

[!Proof]-
${\mathbf{D}}_{p}f:{\mathbb{R}}^n\to {\mathbb{R}}^n\in \text{Hom}({\mathbb{R}}^n,{\mathbb{R}}^n)$.

for all $p\in B$, $\Vert D_{p}f\Vert <M$ for some $M>0$.

yadayada

[!Theorem] Sard
$f:U\to {\mathbb{R}}^m$ is a $C^{\infty }$ map where $U$ is an open subset of ${\mathbb{R}}^n$. $\{f(x)|x\in U,\text{ rank }{\mathbf{D}}_{x}f<m\}$ has $m$-dimensional measure zero.