[Math] Lebesgue measure theory applications

Question

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.

Answer 1

This was American Mathematical Monthly Problem #11526, circa 2010:

Proposition. There is no function $f$ from $\mathbb R^3$ to $\mathbb R^2$ with the property that $|f(x)-f(y)| \ge |x-y|$ for all $x,y \in \mathbb R^3$.

Proof. (Mouse over below...)

Such $f$ would be injective, and its inverse $f^{-1}$ would be a surjective Lipschitz map from a subset of $\mathbb{R}^2$ onto $\mathbb{R}^3$. But Lipschitz maps don't increase Hausdorff dimension, so the image of $f^{-1}$ must have Lebesgue measure zero, contradicting surjectivity.