A reformulation of the Radon-Nikodým theorem

Lewin, Jonathan; Lewin, Mirit

doi:10.1090/S0002-9939-1975-0376999-4

A reformulation of the Radon-Nikodým theorem
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by Jonathan Lewin and Mirit Lewin PDF

Proc. Amer. Math. Soc. 47 (1975), 393-400 Request permission

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 \[ ({\text {i}})\quad \lambda (A) = \int _A {gd\mu } \] 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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