Let $u_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi_\delta$ a family of standard mollifiers with compact support. Hörmander claims in The Analysis of Linear Partial Differential Operators Vol I, Theorem 4.1.9(b) that if $u_k$ converge as distributions to a subharmonic $u$ then $v_j * \psi_\delta(x) \to v * \psi_\delta$ "uniformly on compact sets if $\delta$ is small enough" (I assume this means that for every compact $K \subset X$ there exists such a $\delta$).
I can easily see why the convergence is true pointwise, but I don't know how to verify that it's uniform.
If you assume the conclusion of the theorem, namely that $u_k \to u$ in $L^1_\mathrm{loc}(X)$ then the result follows quickly from the contiuity of convolutions as a map $L^1 \times L^\infty \to C^0$, but this would of course be circular.
