Polar set (potential theory)

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Template:Nofootnotes {{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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.


A set in (where ) is a polar set if there is a non-constant subharmonic function


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.


The most important properties of polar sets are:

Nearly everywhere

A property holds nearly everywhere in a set S if it holds on SE where E is a Borel polar set. If P holds nearly everywhere then it holds almost everywhere.[1]

See also


  1. Ransford (1995) p.56
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External links