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Proceedings of the American Mathematical Society

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

 

 

A representation characterization theorem


Author: William D. L. Appling
Journal: Proc. Amer. Math. Soc. 50 (1975), 317-321
MSC: Primary 28A10
DOI: https://doi.org/10.1090/S0002-9939-1975-0369643-3
MathSciNet review: 0369643
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Abstract: Given a field $ {\mathbf{F}}$ of subsets of a set $ U$ and a real-valued function $ T$ defined on a set $ S$ of functions from $ {\mathbf{F}}$ into $ \exp ({\mathbf{R}})$ with bounded range unions, a necessary and sufficient condition is given in order that there be a bounded finitely additive function $ \theta $ from $ {\mathbf{F}}$ into $ {\mathbf{R}}$ such that if $ \alpha $ is in $ S$, then the integral $ \int_U {\alpha (I)\theta (I)} $, as a variational integral, i.e., a refinement-wise limit of appropriate sums over (finite) subdivisions, exists and is $ T(\alpha )$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0369643-3
Keywords: Set function, variational integral, functional representation
Article copyright: © Copyright 1975 American Mathematical Society