# Nikodým boundedness theorem

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A theorem [a5], [a4], saying that a family of countably additive signed measures (cf. Measure) defined on a -algebra and pointwise bounded, i.e. for each there exists a number such that is uniformly bounded, i.e. there exists a number such that As is well-known, the Nikodým boundedness theorem for measures fails in general for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness conditions are imposed on certain subfamilies of a given -algebra; those subfamilies need not be -algebras. The following definitions are useful [a2], [a7], [a8]:

SCP) An algebra has the sequential completeness property if each disjoint sequence from has a subsequence whose union is in SIP) An algebra has the subsequential interpolation property if for each subsequence of each disjoint sequence from there are a subsequence and a set such that and for .

The Nikodým boundedness theorem holds on algebras with SCP) and SIP).

A famous theorem of J. Dieudonné [a3] states that for compact metric spaces the pointwise boundedness of a family of regular Borel measures on open sets implies its uniform boundedness on all Borel sets. There are further generalizations of this theorem [a6].

How to Cite This Entry:
Nikodým boundedness theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikod%C3%BDm_boundedness_theorem&oldid=15213
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article