A characterisation of absolutely continuous measures on topological semigroups

Abstract

Let S be a completely regular topological semigroup and μ \mu a bounded regular Borel measure on S. For a very large class of noncompact semigroups S, we show that the map x → μ ∗ x ¯ 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 μ 0 f {\mu _0}f is a left uniformly continuous function on S, for all bounded continuous functions f on S. Here the function μ 0 f {\mu _0}f is given by \[ μ 0 f ( x ) := ∫ f ( y x ) d μ ( y ) on S . {\mu _0}f(x): = \int {f(yx)d\mu (y)\quad {\text {on}}\;S.} \]