Reference for proof that $C_b^* = rba$ - MathOverflow

Reference for proof that $C_b^* = rba$

2011-07-18T11:34:51Z

The following theorem seems to have folk status:

The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, regular, finitely additive Borel set functions.

This fact is often mentioned (for instance in the answer to http://mathoverflow.net/questions/44183/dual-of-bounded-uniformly-continuous-functions) but I'm having great difficulty actually finding a reference. Often Dunford & Schwartz is mentioned as a reference; D&S defines $rba$, but doesn't prove the connection to the dual of $C_b$. Hildebrandt 1934 proves a characterization in terms of limits of Stieltjes integrals, but that is still some steps away from the characterization above. I haven't been able to find anything coming closer than this.

Does anyone know of a real proof of this statement? Am I maybe overlooking a very simple proof?

Answer by dan232 for Reference for proof that $C_b^* = rba$

2011-08-10T20:11:16Z

In the answer you mentioned, the space $X$ is metrizable, hence normal, so the proof from Dunford & Schwartz that appeared in the comments is aplicable in that case.

Answer by Ljubomir Cukic for Reference for proof that $C_b^* = rba$

2011-08-11T07:53:38Z

The topological dual of the space of bounded continuous functions on a topological space X is isomorphic to the space of finite, zero set regular, finitely additive Baire set functions; see: R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 2 (1983), 97–190 (a proof is on pp. 115-117).