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.

1 Definition
2 Properties
3 See also
4 References

[edit] Definition

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.

[edit] Properties

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 $\R^n.$

[edit] See also

Pluripolar set

[edit] References

J. L. Doob. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, Berlin Heidelberg New York, ISBN 3-540-41206-9.
L. L. Helms (1975). Introduction to potential theory. R. E. Krieger ISBN 0-88275-224-3.
Polar set on PlanetMath