Let $S$ be a topological semigroup, and $M(S)$ be bounded, regular complex Borel measures on $S$. How can we identify bounded Borel measurable functions with elements in $M^*(S)$?

Please consult pages 3940 and Propositions 4.13 and 4.14 of
for a description of an embedding of the commutative C*algebra of bounded Borel functions on $K$ into $C(\tilde{K})=C(K)^{**}$ which might be of interest to you. 

