Diffeomorphism implies nullset-preserving

From Diffgeom
Jump to: navigation, search

Statement

Let U and V be open subsets in \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.