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?