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2008-05-18T19:38:07Z

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Vipul at 20:36, 13 January 2008

2008-01-13T20:36:57Z

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==Statement==

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

==Proof==

===Proof idea===

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

<math>\int_A |\det(Df)| dm</math>

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

Note that the proof only requires the map to be <math>C^1</math> and have a <math>C^1</math> inverse; in fact, even more weakly, we only require the map to be Lipschitz and have a Lipschitz inverse.

Diffeomorphism implies nullset-preserving - Revision history

Vipul: 1 revision

Vipul at 20:36, 13 January 2008