# How identify bounded Borel measurable functions

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 $M^*(S)$?

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Do you mean $\langle f,\mu\rangle := \int_S f(x)\mu(\mathrm dx)$ for any $\mu \in M(S)$ and bounded Borel $f$? – Ilya Mar 14 '13 at 16:03
Yes. Thanks for your attention. – Ali Mar 14 '13 at 16:15

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.