polar set

Definition.

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

Proposition.

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

单词	PolarSet
释义	polar set Definition. Let $G\\subset{\\mathbb{R}}^{n}$ and let $f\\colon G\\to{\\mathbb{R}}\\cup\\{-\\infty\\}$ be a subharmonicfunction which is not identically $-\\infty$ .The set ${\\mathcal{P}}:=\\{x\\in G\\mid f(x)=-\\infty\\}$ iscalled a polar set. Proposition. Let $G$ and ${\\mathcal{P}}$ be as above and suppose that $g$ is acontinuoussubharmonic function on $G\\setminus{\\mathcal{P}}$ . Then $g$ is subharmonicon 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.
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