A set in (where ) is a polar set if there is a non-constant subharmonic function
on
such that
Note that there are other (equivalent) 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 singleton set in is polar.
A countable set in is polar.
The union of a countable collection of polar sets is polar.
A polar set has Lebesgue measure zero in
Nearly everywhere
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.[1]