On the convergence of induced measures in variation

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.