$L_{loc}^1$-convergence of Jacobians of Sobolev homeomorphisms via area formula

We prove that given a sequence of homeomorphisms $f_k:\mathrm{\Omega }\to {\mathbb{R}}^n$ convergent in $W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^n)$, $p\ge 1$ for $n=2$ and $p>n-1$ for $n\ge 3$, to a homeomorphism $f$ which maps sets of measure zero onto sets of measure zero, Jacobians $J{f}_k$ converge to $Jf$ in $L_{loc}^1(\mathrm{\Omega })$. We prove it via Federer’s area formula and investigation of when $\mathrm{\mid }{f}_k(E)\mathrm{\mid }\to \mathrm{\mid }f(E)\mathrm{\mid }$ as $k\to \mathrm{\infty }$ for Borel subsets $E⋐\mathrm{\Omega }$.