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.

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