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The necessity of sigma‐finiteness in the radon–nikodym theorem
Author(s) -
Bell Wayne C.,
Hagood John W.
Publication year - 1981
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015412
Subject(s) - mathematics , measure (data warehouse) , radon measure , separable space , pure mathematics , field (mathematics) , polish space , discrete mathematics , algebraic number , space (punctuation) , equivalence (formal languages) , locally compact space , algebra over a field , mathematical analysis , linguistics , philosophy , database , computer science
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 L 1 (μ).