Let $ \mu$ be a compactly supported borel probability measure on $ \mathbb C$ then it’s logaruthmic potential is,

$ P_{\mu}(z)= \int_{\mathbb C} log|z-w|d\mu(w)$

It’s well known that $ P_\mu$ is subharmonic on $ \mathbb C$ and harmonic on (Supp $ {\mu}$ )$ ^C$ .

In general $ P_\mu$ need not be continuous on $ \mathbb C$ . So under what conditions (necessary and sufficient or at least sufficient) on measure $ \mu$ , $ P_{\mu}$ is continuous?

Thanks for any reference or help.