Diffeomorphism implies nullset-preserving
Let and be open subsets in , and let be a diffeomorphism (i.e. is a smooth map and the inverse of is also smooth). Then, is a nullset-preserving map, in the sense that a subset of has Lebesgue measure zero iff has Lebesgue measure zero.
The idea behind the proof is the change-of-variables formula: namely, that for any , the measure of is given by:
Because the map is invertible, the determinant is everywhere nonzero, so we are integrating a strictly positive function on . The integral of this function is positive iff .
Note that the proof only requires the map to be and have a inverse; in fact, even more weakly, we only require the map to be Lipschitz and have a Lipschitz inverse.