Suppose $D$ is a bounded open set in $\mathbb{R}^{m}$ with $m\geq2$, and $u(x)$ is locally integrable on $D$. We know that if $\omega$ is an open set that is included with its closure to $D$, the mollifier $u_{n}$ of $u$ (https://en.wikipedia.org/wiki/Mollifier) exists on $\omega$ for $n$ sufficiently big. We know also that if $u$ is subharmonic on $D$, then the mollifier is subharmonic on $\omega$ for $n$ sufficiently big.
My question is: suppose $u$ is locally integrable on $D$ but subharmonic on $D\setminus F$, where $F$ is a closed set with empty interior. Can we say that $u_{n}$ is subharmonic on $\omega\setminus F$ for $n$ big enough?