Polar set (potential theory)

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.

Definition

A set $Z$ in $\mathbb {R} ^{n}$ (where $n\geq 2$) is a polar set if there is a non-constant subharmonic function

$u$ on $\mathbb {R} ^{n}$

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

The most important properties of polar sets are:

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

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.

See also

• Pluripolar set

References

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External links

• Template:Planetmath reference

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