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Proceedings of the American Mathematical Society

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

 

 

Absolute continuity characterization sets


Author: William D. L. Appling
Journal: Proc. Amer. Math. Soc. 55 (1976), 52-56
MSC: Primary 28A25
DOI: https://doi.org/10.1090/S0002-9939-1976-0404566-3
MathSciNet review: 0404566
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Abstract: Suppose $ U$ is a set, $ {\mathbf{F}}$ is a field of subsets of $ U,{\mathfrak{p}_{AB}}$ is the set of all real-valued, bounded finitely additive functions on $ {\mathbf{F}}$, and for each $ \rho $ in $ {\mathfrak{p}_{AB}},{\mathcal{A}_\rho }$ is the set of all elements of $ {\mathfrak{p}_{AB}}$ absolutely continuous with respect to $ \rho ,\mathfrak{p}_A^ + $ is the set of all nonnegative-valued elements of $ {\mathfrak{p}_{AB}}$, and $ {\mathfrak{p}_B}$ is the set of all functions from $ {\mathbf{F}}$ into $ \exp ({\mathbf{R}})$ with bounded range union.

An extension of a previous absolute continuity characterization theorem of the author (Proc. Amer. Math. Soc. 18 (1967), 94-99) is given in the form of a characterization of those subsets $ S$ of $ {\mathfrak{p}_{AB}}$ having the property that if each of $ \xi $ and $ \mu $ is in $ \mathfrak{p}_A^ + $, then $ \xi $ is in $ {\mathcal{A}_\mu }$ iff it is true that if $ \alpha $ is in $ {\mathfrak{p}_B},{\smallint _U}\alpha (I)\mu (I)$ and $ {\smallint _U}\alpha (I)\xi (I)$ exist and the function $ \smallint \alpha \mu $ is in $ S$, then $ \smallint \alpha \xi $ is in $ S$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0404566-3
Keywords: Set function with bounded range union, bounded finitely additive set function, set function integral (variational integral), absolute continuity characterization set
Article copyright: © Copyright 1976 American Mathematical Society