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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 set]]s in [[measure theory]].
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| == Definition ==
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| A set <math>Z</math> in <math>\R^n</math> (where <math>n\ge 2</math>) is a polar set if there is a non-constant [[subharmonic function]]
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| :<math>u</math> on <math>\R^n</math> | |
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| such that
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| :<math>Z \subseteq \{x: u(x) = -\infty\}.</math>
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| Note that there are other (equivalent) ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and <math>-\infty</math> by <math>\infty</math> in the definition above.
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| == Properties ==
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| The most important properties of polar sets are:
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| *A singleton set in <math>\R^n</math> is polar.
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| *A countable set in <math>\R^n</math> is polar.
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| *The union of a countable collection of polar sets is polar.
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| *A polar set has Lebesgue measure zero in <math>\R^n.</math>
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| ==Nearly everywhere==
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| 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]].<ref name=Ran56>Ransford (1995) p.56</ref>
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| ==See also==
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| * [[Pluripolar set]]
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| == References ==
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| {{reflist}}
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| * {{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 }}
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| * {{cite book | first=L. L. | last=Helms | year=1975 | title=Introduction to potential theory | publisher=R. E. Krieger | isbn=0-88275-224-3 }}
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| * {{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 }}
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| ==External links==
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| * {{planetmath reference|id=6020|title=Polar set}}
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| [[Category:Subharmonic functions]]
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| {{mathanalysis-stub}}
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