Absolutely continuous and locally quasi-invariant measures on locally compact semigroups

Abstract

Let S be a locally compact semigroup and T a locally compact subsemigroup of S. It is shown that, for a large class of semigroups S, if T is ’reasonably large’, the absolutely continuous measures on T extend to measures whose translates by some points of S are absolutely continuous on S. As a consequence we prove the absolute continuity of translates of some measures related to a locally quasi-invariant measure.