Vector valued absolutely continuous functions on idempotent semigroups

Alò, Richard A.; de Korvin, André; Easton, Richard J.

doi:10.1090/S0002-9947-1972-0310181-3

Vector valued absolutely continuous functions on idempotent semigroups
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by Richard A. Alò, André de Korvin and Richard J. Easton PDF

Trans. Amer. Math. Soc. 172 (1972), 491-500 Request permission

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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