For an extensive class of locally
compact semigroups S, foundation semigroups with identity element, we prove that
two subalgebras of M(S) [the algebra of the bounded Radon measures on S]
coincide. Namely, the algebra L(S), generated by the m ∈ M(S)+ for which the
orbits on the compact subsets of S are weakly compact subsets of M(S),
or, equivalently, for which the translations are weakly continuous, and the
algebra Me(S), generated by the m ∈ M(S)+ for which the restrictions
of the orbit of m on S to the compact subsets of S are weakly compact.
In case S is a group, both these algebras consist of the bounded Radon
measures that are absolutely continuous with respect to a Haar measure on
S.
