polar set

Definition.

Let $G\subset {ℝ}^n$ and let$f:G\to ℝ\cup \{-\mathrm{\infty }\}$ be a subharmonicfunction which is not identically $-\mathrm{\infty }$.The set $𝒫:=\{x\in G\mid f(x)=-\mathrm{\infty }\}$ iscalled a polar set.

Proposition.

Let $G$ and $\mathrm{P}$ be as above and suppose that $g$ is acontinuoussubharmonic function on $G\mathrm{\setminus }\mathrm{P}$. Then $g$ is subharmonicon the entire set $G$.

The requirement that $g$ is continuous cannot be relaxed.

Proposition.

Let $G$ and $\mathrm{P}$ be as above. Then the Lebesgue measure of$\mathrm{P}$ is 0.