Geodesic
On the convergence of induced measures in variation
Sbornik. Mathematics, Tome 190 (1999) no. 9, pp. 1229-1245 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $F_j$, $F\colon\mathbb R^n\to\mathbb R^n$ be measurable maps such that $F_j\to F$ and $\partial _{x_i}F_j\to\partial _{x_i}F$ in measure on a measurable set $E$. Conditions ensuring that the images of Lebesgue measure $\lambda \big|_E$ on $E$ under the maps $F_j$ converge in variation to the image of $\lambda \big |_E$ under $F$ are presented. For example, one sufficient condition is the convergence of the $F_j$ to $F$ in a Sobolev space $W^{p,1}(\mathbb R^n,\mathbb R^n)$ with $p\geqslant n$ and the inclusion $E\subset \{\det DF\ne 0\}$. Similar results are obtained for maps between Riemannian manifolds and maps from infinite dimensional spaces. 
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     title = {On the convergence of induced measures in variation},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_9_a0/}
}
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D. E. Aleksandrova; V. I. Bogachev; A. Yu. Pilipenko. On the convergence of induced measures in variation. Sbornik. Mathematics, Tome 190 (1999) no. 9, pp. 1229-1245. http://geodesic.mathdoc.fr/item/SM_1999_190_9_a0/

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