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 [Dual of bounded uniformly continuous functions](https://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](http://www.jstor.org/pss/1989829) 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?



Reference for proof that $C_b^* = rba$
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).


