A Characterisation of Absolutely Continuous Measures on Topological Semigroups

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