Weak* convergence in $L^\infty$ and the strong convergence in $L^2$ of a mollification?

Let $f=0$. Then we want to prove $$ \int_{\mathbb R^n} \left(\int_\Omega \rho(x-y)f_n(y)dy\right)^2 dx \to0. $$ Since $\Omega$ is bounded, $f_n\rightharpoonup0$ in $L^2(\Omega)$, so $\int_\Omega \rho(x-y)f_n(y)dy \to 0$ for all $x$. That is, the integrand in the integral above converges pointwise to zero. In addition, $$ |\int_\Omega \rho(x-y)f_n(y)dy| \le |\Omega|\|\rho\|_{L^\infty} \sup_n \|f\|_{L^\infty} $$ for all $x$, which is a square-integrable pointwise upper bound. Then convergence follows by dominated convergence.

This is some kind of compactness result for the simple integral operator $f\mapsto \rho *f$ from $L^2(\Omega)$ to $L^2(\Omega)$.