Convergence of measures and potential theory The following implication should hold: $\mu_{n}, \mu$ are positive measures whose supports are included in a compact set $K\subset \mathbb{C}$ and 
$$\lim_{n\to\infty}U^{\mu_n}(z)=U^{\mu}(z)$$
uniformly in compact subsets of $\mathbb{C}\setminus K$, where
$$ U^{\nu}(z)=\int_{\mathbb{C}}\log|z-x|d\nu(x)$$
is the logarithmic potential of the measure $\nu$. Then $\mu_{n}\to\mu$, for $n\to\infty$ in the weak$^\star$ topology.
I read the paper where this implication is used. The authors state only that it is a standard result of Potential theory. Unfortunately, I do not have solid background in Potential theory and I can not see why the statement is true by myself. 
Although I went through several books devoted entirely to the Potential theory (Saff & Totik, Ransford, Landkof), I did not find a place where this implication would be addressed directly (as a separate proposition, for example).
Can anyone, who is familiar with the subject, give some relevant reference? Many thanks!
 A: So, I'm not a potential theorist, so I'm sure someone else can give a better answer than this. For me, most of the basic potential theory stuff that comes up I can understand using Chapter I of the (free!) book Complex Analytic and Differential Geometry by Demailly.
But coming back to the statement you're trying to understand. I think the basic idea is this. The function $U^{\mu}$ and $U^{\mu_n}$ are potentials of their corresponding measures, which means that when you take their Laplacian (suitably scaled), you get their measure out: $$\Delta U^{\mu} = \mu$$ where this equality is in the sense of distributions. So, suppose that $U^{\mu_n}\to U^\mu$ uniformly on compacts (or even I think in $L^1_{loc}$). Then if $\phi$ is a test function (compactly supported smooth), you get that $$\int \phi\,d\mu_n = \int \phi\,d(\Delta U^{\mu_n}) := \int U^{\mu_n} \Delta\phi\,dx\,dy.$$ Since $U^{\mu_n}\to U^\mu$ uniformly on compacts (and hence in particular on the support of $\phi$, you obtain $$\int U^{\mu_n}\Delta\phi\,dx\,dy\to \int U^\mu \Delta\phi\,dx\,dy := \int\phi\,d(\Delta U^\mu) = \int \phi\,d\mu$$ This proves the convergence $$\int \phi\,d\mu_n\to \int \phi\,d\mu$$ for every compactly supported smooth function $\phi$, which should be equivalent to weak convergence $\mu_n\to \mu$. 
