---
tag: C95QJ7YH5T
kind: concept
name: Polar set
msc: [31B15, 31A15, 31C15]
sources:
  - url: https://en.wikipedia.org/wiki/Polar_set_(potential_theory)
same_as:
  wikidata: Q7209102
  wikipedia: Polar set (potential theory)
---

## Summary

A polar set in $\R^n$, $n \ge 2$, is a set on which some subharmonic function, not identically $-\infty$, takes the value $-\infty$.
Polar sets are the negligible sets of [[potential theory|9PM8Q2E30Q]].
A Borel set is polar exactly when it has [[capacity|4YZVFGJVSZ]] zero, Newtonian capacity for $n \ge 3$ and [[logarithmic capacity|DBQXRTC0VN]] for $n = 2$.
Such sets are removable for bounded harmonic functions, and Brownian motion almost surely never hits them.

## Explanations

### L1 human

A [[subharmonic function|Z1MHMGFNFW]] may take the value $-\infty$ at some points, like $\log \abs{z}$ at $0$ in the plane.
A polar set is a set that is small enough to be sent to $-\infty$ by one subharmonic function at once.
Points and countable sets are polar.
A line segment in the plane is not polar, even though it has area zero.
So polar sets are much smaller than sets of measure zero.