Convergence in L^p for function multiplied for another function

Question

Problem: Let $\Omega \subset R^n$ and let $f_{\epsilon}, f \in W^{1,p}(\Omega)$ be functions such that $f_{\epsilon} \rightarrow f$ in $L^p(\Omega)$. Which condition should have another function $g$ so that $$\int_{\Omega}g(x)f_{\epsilon}(x) \ dx \rightarrow \int_{\Omega} g(x)f(x) \ dx \ \ \ ?$$