Nullset-preserving map

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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