Suppose $D$ is a bounded domain of $\mathbb{R}^m$ for $m>1$ and $\{u_n\}_{n\geq1}$ is a sequence of subharmonic functions on $D$. Assume $u_n\to u_0$ pointwise on $D$ and $u_0$ is subharmonic on $D$. Let $\mu_n$ be the Riesz measure associted to each $u_n$ for $n\geq0$. Suppose also that for a compact set $K\subset D$ we have $$\mu_n(K)=0$$ for all $n>0$. It is well-known that the sequence of measures $\{\mu_n\}$ has a subsequence that is vaguely convergent, and so $$\int_Kf(x)d\mu_{n_k}(x)\to \int_Kf(x)d\nu(x),$$ as $k\to\infty$, for all continuous functions $f$ and for some measure $\nu$.

My question is: can we conclude that the restrictions of $\nu$ and $\mu_0$? to $K$ coincide? In particular, do we have also $\mu_0(K)=0$?