TITLE: Lebesgue measure theory applications
QUESTION [20 upvotes]: I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example.
Theorem: Let $X$ be a differentiable submanifold of $\mathbb{R}^n$ with codimension $\geq 3$. Then $\mathbb{R}^n\setminus X$ is simply conected.
Proof. Let $\alpha:S^1\to \mathbb{R}^n\setminus X$ be a closed $C^1$ curve. We want to show  that there exists a point $p$ outside $X$ s.t. a linear homotopy between $\alpha$ and $p$ can be contructed. Well, define $F:\mathbb{R}\times S^1\times X\to \mathbb{R}^n $ by
    $$
F(t,s,x)= (1-t)\alpha(s)+tx.
 $$
Note that, 1) $F$ collects all the bad lines, i.e., the lines connecting $\alpha(s)$ and $x\in X$. 2) $F$ is $C^1$. 3) $\dim(\mathbb{R}\times S^1\times X)\leq n-1$. So, by Sard's theorem, the set $F(\mathbb{R} \times S^1\times X)$ has zero Lebesgue  measure, and therefore its complement is non-empty. This easily implies the result.
Also as an example we have the the weak form of the  Whitney immersion theorem, for which one can use the same kind of argument in the proof.
I want to know more "simple" applications of the type the above mentioned, in areas other than measure theory. But not too complicated ones!
Sorry if this question is too basic.

REPLY [7 votes]: I do not know what the protocol is when you give another answer; this has nothing to do with my previous example. The most spectacular applications of measure theory that I know come from Margulis' work. For example, suppose $\Gamma \subset SL_3(\mathbb R)$ is a discrete subgroup with compact quotient. Then Margulis shows that every non-trivial normal subgroup of $\Gamma $ has finite index in $\Gamma$. The proof uses measure theory ( and a lot else besides) in a serious way. His proof that such a $\Gamma$ is arithmetic also uses ergodic theory (and measure theory). These purely "algebraic" statements were proved by use of measure theory.