# polar set

###### Definition.

Let $G\subset{\mathbb{R}}^{n}$ and let $f\colon G\to{\mathbb{R}}\cup\{-\infty\}$ be a subharmonic function which is not identically $-\infty$. The set ${\mathcal{P}}:=\{x\in G\mid f(x)=-\infty\}$ is called a polar set.

###### Proposition.

Let $G$ and ${\mathcal{P}}$ be as above and suppose that $g$ is a continuous subharmonic function on $G\setminus{\mathcal{P}}$. Then $g$ is subharmonic on the entire set $G$.

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

###### Proposition.

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

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title polar set PolarSet 2013-03-22 14:29:13 2013-03-22 14:29:13 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 31C05 msc 31B05 msc 31A05