Assume that $u\in L^{p}(\Omega)$, $\Omega \subset R^n$ is bounded, and that $u_{\varepsilon}$ is the standard mollification. Does this imply that
$$ \chi\left\{ x\in\Omega:u_{\varepsilon}(x)>0\right\} \rightarrow\chi\left\{ x\in\Omega:u(x)>0\right\} \text{ in }L^{p}_{loc}(\Omega), $$
where $\chi$ is the charasteristic function?
By boundedness of $\Omega$, we may apply the dominated convergence theorem to the a.e. convergence $$ \chi_{\left\{ x\in\Omega:u_{\varepsilon}(x)>0\right\}} \rightarrow \chi_{\left\{ x\in\Omega:u(x)>0\right\}}, ~~~~\epsilon \to 0 $$ which comes from the fact that $$ \operatorname{ess sup} u_\epsilon \subseteq \operatorname{ess sup} u + B_\epsilon(0) $$ and the a.e. convergence $u_\epsilon \to u$; to prove this convergence, we pick a sequence of simple functions $s_n \to u$ (a.e. convergence) so that the convergences $s_n^- \to u^-$ and $s_n^+ \to u^+$ are monotone (so that we also obtain $L^p$ convergence by the monotone convergence theorem) and apply the triangle inequality twice, first choosing $n$ and then $\epsilon$.