I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is shown that:
If we have $f \in L^p(\mathbb{R}^n)$, then $(\rho_n *f) \to f $ in $L^p(\mathbb{R}^n)$, for a sequence $(\rho_n)$ of mollifiers. (*)
I would like to know if this result still holds for an open set $\Omega \subset \mathbb{R}^n$ and also how to prove it, i.e:
$f \in L^p(\Omega)$, then $(\rho_n *f) \to f $ in $L^p(\Omega)$, for a sequence $(\rho_n)$ of mollifiers and any open set $\Omega \subset \mathbb{R}^n$.
Note: Brezis relies on the density of $C_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ to show the result stated here for $\mathbb{R}^n$ (*), and then shows this result that, indeed, $C_c(\Omega)$ is dense in $L^p(\Omega)$, for any open set $\Omega \subset \mathbb{R}^n$.
Strictly speaking, the result you ask about cannot be true because if $f\in L^p(\Omega)$ then $\rho*f(x)$ is undefined for $x$ near the boundary (because the definition involves $f(x-t)$ where $x-t\notin\Omega$).
Given $f\in L^p(\Omega)$ you can make sense of $\rho*f$ by regarding $f$ as defined on all of $\Bbb R^n$ and vanishing off $\Omega$. But then you're not really talking about $L^p(\Omega)$, and in any case $\rho*f$ is not supported in $\Omega$.
Of course there's no problem using convolutions to show that $C_c(\Omega)$ is dense in $L^p(\Omega)$. First choose $g\in L^p(\Omega)$ so that $||f-g||_p<\epsilon$ and such that $g$ vanishes outside some compact set $K\subset\Omega$; now if the support of $\rho$ is small enough you will have $\rho*g\in C_c(\Omega)$.