Extension of plurisubharmonic functions with subharmonic singularities

Let $u$ be a subharmonic function which is furthermore plurisubharmonic outside a closed set $E$. Then $u$ is said to be a plurisubharmonic function with subharmonic singularities along $E$. We prove the following:
Theorem. Let $E\subseteq\Omega\subseteq \mathbb C^{n}$ be a closed subset of Lebesgue measure zero. Then any subharmonic function $u$ in $\Omega$ which is plurisubharmonic in $\Omega\setminus E$ is actually plurisubharmonic in the whole $\Omega$. This can be seen as a removable singularity theorem with special assumptions. In particular this solves a problem posed by Chirka.