Polar set (potential theory)

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

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

such that

${\displaystyle Z\subseteq \{x\in \mathbb {R} ^{n}:u(x)=-\infty \}.}$

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

Properties

The most important properties of polar sets are:

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

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.^[1]

See also

• Pluripolar set