Diffeomorphism implies nullset-preserving

Statement

Let $U$ and $V$ be open subsets in $\mathbb {R} ^{n}$, and let $f:U\to V$ be a diffeomorphism (i.e. $f$ is a smooth map and the inverse of $f$ is also smooth). Then, $f$ is a nullset-preserving map, in the sense that a subset $A$ of $U$ has Lebesgue measure zero iff $f(A)$ has Lebesgue measure zero.

Proof

Proof idea

The idea behind the proof is the change-of-variables formula: namely, that for any $A$, the measure of $f(A)$ is given by:

$\int _{A}|\det(Df)|dm$

Because the map is invertible, the determinant is everywhere nonzero, so we are integrating a strictly positive function on $A$. The integral of this function is positive iff $m(A)>0$.

Note that the proof only requires the map to be $C^{1}$ and have a $C^{1}$ inverse; in fact, even more weakly, we only require the map to be Lipschitz and have a Lipschitz inverse.