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
Canonical name	PolarSet
Date of creation	2013-03-22 14:29:13
Last modified on	2013-03-22 14:29:13
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 31C05
Classification	msc 31B05
Classification	msc 31A05