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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Vector valued absolutely continuous functions on idempotent semigroups


Authors: Richard A. Alò, André de Korvin and Richard J. Easton
Journal: Trans. Amer. Math. Soc. 172 (1972), 491-500
MSC: Primary 28A45; Secondary 43A15, 46G10
MathSciNet review: 0310181
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Abstract: In this paper the concept of vector valued, absolutely continuous functions on an idempotent semigroup is studied. For $ F$ a function of bounded variation on the semigroup $ S$ of semicharacters with values of $ F$ in the Banach space $ X$, let $ A = {\text{AC}}(S,X,F)$ be all those functions of bounded variation which are absolutely continuous with respect to $ F$. A representation theorem is obtained for linear transformations from the space $ A$ to a Banach space which are continuous in the BV-norm. A characterization is also obtained fot the collection of functions of $ A$ which are Lipschitz with respect to $ F$. With regards to the new integral being utilized it is shown that all absolutely continuous functions are integrable.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0310181-3
PII: S 0002-9947(1972)0310181-3
Keywords: Bounded variation, absolutely continuous, $ v$-integral, Banach space, linear transformation, dual space, Lipschitz functions, convex set functions, semicharacter, semigroup, set-functions, Möbius function, positive definite, polygonal function, characteristic function, simple function
Article copyright: © Copyright 1972 American Mathematical Society