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.

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.

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 ${\displaystyle \mathbb {R} ^{n}.}$

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]

• Pluripolar set

• Doob, Joseph L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften. Vol. 262. Berlin Heidelberg New York: Springer-Verlag. ISBN 3-540-41206-9. Zbl 0549.31001.
• Helms, L. L. (1975). Introduction to potential theory. R. E. Krieger. ISBN 0-88275-224-3.
• Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: Cambridge University Press. ISBN 0-521-46654-7. Zbl 0828.31001.

• "Polar set". PlanetMath.