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Additive function

From Encyclopedia of Mathematics
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finitely-additive function (on sets, on domains)

A real-valued function defined on a system of sets and such that

(*)

for any finite number of pairwise-disjoint sets of whose union also belongs to . Countably-additive set functions are an important kind of additive functions (cf. Countably-additive set function).


Comments

Suppose that is a -algebra on a set . Then a non-negative function (taking, possibly, the value ) on is an additive (finitely-additive, countably-additive) measure if it satisfies (*) for an arbitrary (respectively, finite, countable) number of disjoint sets in .

Usually, a measure (sic) is a countably-additive measure.

References

[a1] H.L. Royden, "Real analysis", Macmillan (1968)
How to Cite This Entry:
Additive function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_function&oldid=19226
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article