If $f : S \to S$ is volume preserving and $S \subseteq \mathbb{R}^n$ has finite volume.

There is a very elegant proof of Koebe–Andreev–Thurston theorem (given in the book on combinatorial geometry by Agarwal and Pach) using this property.

If $f : S \to S$ is volume preserving and $S \subseteq \mathbb{R}^n$ has finite volume, then f is injective iff f is surjective.

There is a very elegant proof of Koebe–Andreev–Thurston theorem (given in the book on combinatorial geometry by Agarwal and Pach) using this property.