The following paradox has got me stumped. I'm hoping someone can point out the error.

Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-valued functions on $X$, bounded for $C_b$ and converging to zero at infinity for $C_0$, and equip the spaces with the supremum norm.

By Dunford & Schwartz Th. IV.6.2, the topological dual $C_b'$ is isometrically isomorphic to the space $rba(X)$ of regular bounded *finitely* additive Borel measures on $X$; by Rudin (Real and Complex Analysis, Th. 6.19) $C_0'$ is isometrically isomorphic to the space $rca(X)$ of regular *countably* additive Borel measures.
In both cases the identification has the form
$\langle \xi,f\rangle = \int_X f\, d\mu_\xi$.

Since $C_0\subset C_b$, and since $C_0$ and $C_b$ share the same norm, a simple calculation gives that $C_b'\subset C_0'$; since the two identification structures above are the same, this implies that $rba\subset rca$.

However, the definitions of $rba$ and $rca$ immediately imply that $rca\subset rba$.

What is going wrong here?

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