# Reference for proof that $C_b^* = rba$

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) 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?

• What's wrong with the discussion in IV.6, p.261ff in DS? They prove the duality at least for $X$ normal. – Theo Buehler Jul 18 '11 at 11:51
• What hypotheses (if any) are you putting on your space $X$? – Yemon Choi Jul 18 '11 at 12:56
• Perhaps confusion: Dunford & Schwartz notation $C(X)$ is the space of bounded continuous functions on $X$. The dual (for $X$ normal, but probably completely regular will also work) is Theorem IV.6.2, as Theo says. – Gerald Edgar Jul 18 '11 at 13:47
• Thanks @Theo and @Gerard - I obviously overlooked Theorem IV.6.2, possibly because of the different notation. Thanks for pointing that out! – Mark Peletier Aug 14 '11 at 18:14

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.