# Vitali-Hahn-Saks theorem

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Let be a -algebra (cf. also Borel field of sets). Let be a non-negative set function and let , where is a normed space. One says that is absolutely continuous with respect to , denoted by , if for every there exists a such that whenever and (cf. also Absolute continuity). A sequence is uniformly absolutely continuous with respect to if for every there exists a such that whenever , and .

The Vitali–Hahn–Saks theorem [a7], [a2] says that for any sequence of signed measures which are absolutely continuous with respect to a measure and for which exists for each , the following is true:

i) the limit is also absolutely continuous with respect to this measure, i.e. ;

ii) is uniformly absolutely continuous with respect to . This theorem is closely related to integration theory [a8], [a3]. Namely, if is a sequence of functions from , where is the Lebesgue measure, and exists for each measurable set , then the sequence is uniformly absolutely -continuous and is absolutely -continuous, [a3].

R.S. Phillips [a5] and C.E. Rickart [a6] have extended the Vitali–Hahn–Saks theorem to measures with values in a locally convex topological vector space (cf. also Locally convex space).

There are also generalizations to functions defined on orthomodular lattices and with more general properties ([a1], [a4]).

How to Cite This Entry:
Vitali-Hahn-Saks theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vitali-Hahn-Saks_theorem&oldid=13281
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article