Polar set (potential theory)


In mathematics, in the area of classical potential theory, polar sets are the "negligible sets", similar to the way in which sets of measure zero are the negligible sets in measure theory.

Definition

A set in is a polar set if there is a non-constant subharmonic function
such that
Note that there are other ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and by in the definition above.

Properties

The most important properties of polar sets are:
A property holds nearly everywhere in a set S if it holds on S−E where E is a Borel polar set. If P holds nearly everywhere then it holds almost everywhere.