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Signed measure

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generalized measure

An extended real-valued -additive set function that is defined on the -algebra of Borel subsets of a domain and that is finite on compact sets . The difference between two measures one of which is finite on is a charge; conversely, all charges may be obtained in this way: for any charge there exists a decomposition of into two disjoint Borel sets and such that for and for . The measures and are independent of the choice of and and are known respectively as the positive and negative variations of the charge ; the measure is called the total variation of . With this notation, the so-called Hahn–Jordan decomposition: holds, so that the properties of charges may be phrased in terms of measure theory.

References

[1] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[2] P.R. Halmos, "Measure theory" , v. Nostrand (1950)


Comments

A charge is also called a signed measure [a1], a real measure or a signed content. It can, more generally, be defined on a ring of subsets of a space , or, alternatively, on a Riesz space of functions on , see [a2].

Any pair as above is called a Hahn decomposition of with respect to . The pair , defined above, is also called the Jordan decomposition of .

References

[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
[a2] K. Jacobs, "Measure and integral" , Acad. Press (1978)
How to Cite This Entry:
Signed measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Signed_measure&oldid=19071
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article