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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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Keywords: Bounded variation, absolutely continuous, <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$v$">-integral, Banach space, linear transformation, dual space, Lipschitz functions, convex set functions, semicharacter, semigroup, set-functions, M&#246;bius function, positive definite, polygonal function, characteristic function, simple function
Article copyright: © Copyright 1972 American Mathematical Society