Approximate Identities in Spaces of all Absolutely Continuous Measures on Locally Compact Semigroups

Let G be a locally compact group. Then Ma (G), the space of all absolutely continuous measures on G, has a bounded approximate identity. Baker and Baker proved that ℒ(S) (the space of all measures μ ∈ M(S) so that maps x ↦ εx *|μ| and x ↦ |μ|*εx are weak continuous from a locally compact semigroup S into M(S)) is closed under absolutely continuity and has an approximate identity. The main purpose of this paper is to show that similar results hold true for a locally compact semigroup S and Ma(S) the space of all absolutely continuous measures on S.