Polar set (potential theory)

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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[edit]

A set Z in \R^n (where n\ge 2) is a polar set if there is a non-constant subharmonic function

u on \R^n

such that

Z \subseteq \{x: u(x) = -\infty\}.

Note that there are other (equivalent) ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and -\infty by \infty in the definition above.

Properties[edit]

The most important properties of polar sets are:

  • A singleton set in \R^n is polar.
  • A countable set in \R^n is polar.
  • The union of a countable collection of polar sets is polar.
  • A polar set has Lebesgue measure zero in \R^n.

Nearly everywhere[edit]

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[edit]

References[edit]

  1. ^ Ransford (1995) p.56

External links[edit]