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.