Polar set (potential theory)

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

:$u$ on $\backslash R^n$

such that

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

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

The most important properties of polar sets are:

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

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.Ransford (1995) p.56

• Pluripolar set

{{reflist}}

• {{cite book | first=Joseph L. | last=Doob | authorlink=Joseph L. Doob | title=Classical Potential Theory and Its Probabilistic Counterpart | publisher=Springer-Verlag | location=Berlin Heidelberg New York | isbn=3-540-41206-9 | zbl=0549.31001 | series=Grundlehren der Mathematischen Wissenschaften | volume=262 | year=1984 }}
• {{cite book | first=L. L. | last=Helms | year=1975 | title=Introduction to potential theory | publisher=R. E. Krieger | isbn=0-88275-224-3 }}
• {{cite book | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=Cambridge University Press | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 | url-access=registration | url=https://archive.org/details/potentialtheoryi0000rans }}

• {{planetmath reference|urlname=PolarSet|title=Polar set}}

Category:Subharmonic functions