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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A characterisation of absolutely continuous measures on topological semigroups


Author: Heneri A. M. Dzinotyiweyi
Journal: Proc. Amer. Math. Soc. 121 (1994), 1103-1109
MSC: Primary 22A10; Secondary 43A05
MathSciNet review: 1189541
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Abstract: Let S be a completely regular topological semigroup and $ \mu $ a bounded regular Borel measure on S. For a very large class of noncompact semigroups S, we show that the map $ x \to {\mu ^ \ast }\bar x$ of S into the space of bounded regular Borel measures on S is norm-continuous if and only if $ {\mu _0}f$ is a left uniformly continuous function on S, for all bounded continuous functions f on S. Here the function $ {\mu _0}f$ is given by

$\displaystyle {\mu _0}f(x): = \int {f(yx)d\mu (y)\quad {\text{on}}\;S.} $


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1189541-7
PII: S 0002-9939(1994)1189541-7
Article copyright: © Copyright 1994 American Mathematical Society