Nullset-preserving map

Definition

Let $M$ and $N$ be differential manifolds. A map $f:M\to N$ (not necessarily smooth) is termed nullset-preserving if $f(A)$ has measure zero if and only if $A$ has measure zero.

Any diffeomorphism is nullset-preserving. In fact, it is because we have this fact for Euclidean space that we can define the notion of a nullset on a differential manifold to begin with. Further information: Diffeomorphism implies nullset-preserving