# 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)