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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A reformulation of the Radon-Nikodým theorem

Authors: Jonathan Lewin and Mirit Lewin
Journal: Proc. Amer. Math. Soc. 47 (1975), 393-400
MSC: Primary 28A15
MathSciNet review: 0376999
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Abstract: The Radon-Nikodym theorems of Segal and Zaanen are principally concerned with the classification of those measures $ \mu $ for which any $ \lambda \ll \mu $ is given in the form

$\displaystyle ({\text{i}})\quad \lambda (A) = \int_A {gd\mu } $

for all sets $ A$ of finite $ \mu $ measure.

This paper is concerned with the characterization of those pairs $ \lambda ,\mu $ for which the equality (i) holds for every measurable set $ A$, and introduces a notion of compatibility that essentially solves this problem. In addition, some applications are made to Radon-Nikodym theorems for regular Borel measures.

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Keywords: Radon-Nikodym theorem, measure space, Borel measure, absolute continuity, differentiation of measures
Article copyright: © Copyright 1975 American Mathematical Society

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