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The necessity of sigma-finiteness in the radon–nikodym theorem

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Wayne C. Bell  and
John W. Hagood
Wayne C. Bell
Affiliation:
Department of Mathematics, Murray State UniversityMurray, Kentucky 42071, U.S.A..
John W. Hagood
Affiliation:
Department of MathematicsMurray State University, Murray, Kentucky 42071, U.S.A..
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Abstract

This note contains characterizations of those sigma-fields for which sigma-finiteness is a necessary condition in the Radon-Nikodym Theorem.

Our purpose is to consider those σ-fields for which σ-finiteness is a necessary condition in the Radon–Nikodym Theorem. We first prove a measure theoretic equivalence in the general case, and then use this to obtain an algebraic characterization in the case when the σ-field is the Borel field of a locally compact separable metric space. For undefined terminology we refer the reader to [1] for measure theoretic and [2] for algebraic properties.

By a measure, we mean a countably additive function from σ-field of sets or a Boolean σ-algebra into the non-negative extended real numbers. We will say that a measure μ on a σ-field of sets Σ is RN provided each μ-continuous finite measure on Σ has a Radon–Nikodym derivative in L1(μ).

MSC classification

Secondary: 28A15: Abstract differentiation theory, differentiation of set functions
Type
Research Article
Information
Mathematika , Volume 28 , Issue 1 , June 1981 , pp. 99 - 101
Copyright
Copyright © University College London 1981

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References

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