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When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

Part of: Peculiar spaces Classical measure theory Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  09 April 2009

Thomas E. Armstrong
Thomas E. Armstrong
Affiliation:
Department of Mathematical Sciences Northern Illinois UniversityDe Kalb, Illinois 60115, U.S.A.
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Abstract

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It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

Keywords

Borel measureregularityextensions of measurescompletion regular compact spaceBorel regular compact space.

MSC classification

Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.) 28A60: Measures on Boolean rings, measure algebras 54G10: $P$-spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 3 , December 1982 , pp. 374 - 385
Copyright
Copyright © Australian Mathematical Society 1982

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