# polar set

###### Definition.

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

###### Proposition.

Let $G$ and $\mathrm{P}$ be as above and suppose that $g$ is a
continuous^{}
subharmonic function on $G\mathrm{\setminus}\mathrm{P}$. Then $g$ is subharmonic
on 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.

## 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 |