Let $X$ be a compact Hausdorff space and let $$C(X) = \{\text{ Continuous complex functions on } X\}$$ $$B(X) = \{ \text{Bounded complex Borel-measurable functions on X}\}$$ both equipped with the sup-norm, and therefore Banach spaces.
We have that $C(X)^*=M(X)$ is the set of complex Radon measures, while $B(X)^* = \operatorname{ba}(X)$ are the bounded, finitely additive measures.
Each element $T$ of $\operatorname{ba}(X)$ is continuous when restricted to $C(X)\subset B(X)$. Thus, by the Riesz representation theorem, there is a countably additive Radon measure on $C(X)$ corresponding to $T$. However, all the elements of $M(X)$ are countably additive, while those in $B(X)$ need not be!
I am wondering where I am going wrong as I get this contradictory conclusion.
Thank you.