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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integral representation theorems in topological vector spaces

Author: Alan H. Shuchat
Journal: Trans. Amer. Math. Soc. 172 (1972), 373-397
MSC: Primary 46G10; Secondary 28A45, 47B99
MathSciNet review: 0312264
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Abstract: We present a theory of measure and integration in topological vector spaces and generalize the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems to this setting, using strong integrals. As an application, we find the containing Banach space of the space of continuous $ p$-normed space-valued functions. It is known that Bochner integration in $ p$-normed spaces, using Lebesgue measure, is not well behaved and several authors have developed integration theories for restricted classes of functions. We find conditions under which scalar measures do give well-behaved vector integrals and give a method for constructing examples.

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Keywords: Riesz representation, vectot measures, integral representation, containing Banach space, nonlocally convex spaces
Article copyright: © Copyright 1972 American Mathematical Society