Polar set in potential theory

Question

I'm studying potential theory in complex plane and I have studied polar set with the following definition:

A set $E$ in $\mathbb{C}$ is a polar set if for every $z \in E$, there is a neighborhood $U$ of $z$ and a non-constant subharmonic function $u$ on $U$ such that $u(z) = -\infty,\forall z \in E \cap U$.

However, in Ransford's books, they used that fact that boundary of bounded domain $U$ is non-polar without any explanation.

I have tried to prove it, but I got stuck for over a weeks. Can someone give me a hint about that fact?

Thanks for your help.

Answer 1

This follows for example from the maximum principle for subharmonic functions.

Other approaches, see Thm 3.6.3 in (my edition of) Ransford.