# 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

Please consult pages 39-40 and Propositions 4.13 and 4.14 of

H.G. Dales, A.T.-M. Lau and D. Strauss, Second duals of measure algebras, Dissertationes Mathematicae (Rozprawy Matematyczne), 481, 2012, p. 1-121. (available also here)

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.

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