On the convergence in variation for the images of measures under differentiable mappings

Abstract

Let F j , F : ℝ n → ℝ n be measurable mappings such that F j,→ F and ∂x i F j → ∂x i F in measure on a measurable set E. We give conditions ensuring that the images of Lebesgue measure λ¦ E on E under the mappings F j converge in the variation norm to the image of λ¦ E under F. For example, a sufficient condition is that F j → F in the Sobolev space W p , 1 ( ℝ n , ℝ n ) with p ≥ n and E ⊂{detD F≠ 0. Analogous results are obtained for mappings between Riemannian manifolds and mappings from infinite-dimensional spaces.