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Transactions of the American Mathematical Society

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

 
 

 

Absolutely continuous functions on idempotent semigroups in the locally convex setting


Author: A. Katsaras
Journal: Trans. Amer. Math. Soc. 206 (1975), 329-337
MSC: Primary 46E40
DOI: https://doi.org/10.1090/S0002-9947-1975-0374904-2
MathSciNet review: 0374904
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Abstract: Let $ E$ be a locally convex space and let $ T$ be a semigroup of semicharacters on an idempotent semigroup. It is shown that there exists an isomorphism between the space of $ E$-valued functions on $ T$ and the space of all $ E$-valued finitely additive measures on a certain algebra of sets. The space of all $ E$-valued functions on $ T$ which are absolutely continuous with respect to a positive definite function $ F$ is identified with the space of all $ E$-valued measures which are absolutely continuous with respect to the measure $ {m_F}$ corresponding to $ F$. Finally a representation is given for the operators on the set of all $ E$-valued finitely additive measures on an algebra of sets which are absolutely continuous with respect to a positive measure.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0374904-2
Keywords: Bounded variation, absolutely continuous, $ \upsilon $-integral, locally convex space, continuous seminorm, semicharacter, semigroup, positive-definite, polygonal function, simple function, finitely additive measure
Article copyright: © Copyright 1975 American Mathematical Society

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