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Uniform boundedness principles for regular Borel vector measures

Part of: Classical measure theory Set functions, measures and integrals with values in abstract spaces Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  09 April 2009

Joseph Kupka
Joseph Kupka
Affiliation:
Department of Mathematics Monash UniversityClayton, Victoria 3168, Australia.
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The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

MSC classification

Secondary: 28A33: Spaces of measures, convergence of measures 28B05: Vector-valued set functions, measures and integrals 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 2 , March 1980 , pp. 206 - 218
Copyright
Copyright © Australian Mathematical Society 1980

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