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Set function

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A mapping of a certain collection of subsets of a given set into another set, usually into the real numbers or the complex numbers . An important class of set functions are the additive set functions, for which

(*)

and the -additive set functions, which satisfy equation (*) for a countably infinite collection of sets also (replace by ). If takes only non-negative values, , and is a -algebra, then is called a measure.

References

[1] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)


Comments

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Set function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Set_function&oldid=15370
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article