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Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let 
(S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.
