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

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).