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 in (where ) is a polar set if there is a non-constant subharmonic function


such that

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


The most important properties of polar sets are:

  • A singleton set in is polar.
  • A countable set in is polar.
  • The union of a countable collection of polar sets is polar.
  • A polar set has Lebesgue measure zero in

Nearly everywhere[edit]

A property holds nearly everywhere in a set S if it holds on SE where E is a Borel polar set. If P holds nearly everywhere then it holds almost everywhere.[1]

See also[edit]


  1. ^ Ransford (1995) p.56
  • Doob, Joseph L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften. 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. 28. Cambridge: Cambridge University Press. ISBN 0-521-46654-7. Zbl 0828.31001.

External links[edit]