I'm working through the proof of the above representation. Let $C_{b}(X)$ be the space of bounded continuous functions on a normal Hausdorf space. I've proven that for any positive linear functional $\phi \in (C_{b}(X))^{*}$, there exists a finitely additive positive measure $\mu$ such that $$ \forall f \in C_{b}(X), \,\,\, \phi(f)=\int_{X} f d\mu. $$
Now I want to extend this to all linear functions $\phi \in (C_{b}(X))^{*}$, not necessarily just positive ones, and prove that there exists a finitely additive signed measure $\mu=\mu^{+}-\mu^{-}$ such that $$ \forall f \in C_{b}(X), \,\,\, \phi(f)=\int_{X} f d\mu^{+}-\int_{X} f d\mu^{-}. $$ The problem is, simply applying the first result for positive linear functionals still seems to leave some serious work to do: Any linear functional $\phi$ can be written as $\phi=\phi^{+}-\phi^{-}$, and hence applying the top result to the positive functionals $\phi^{+}$ and $\phi^{-}$ guarantees me some measures $\mu_{1}$ and $\mu_{2}$ such that $ \forall f \in C_{b}(X)$, $$\phi(f)=\int_{X} f d\mu_{1}-\int_{X} f d\mu_{2}. $$ However, this does not tell me that $\mu_{1}=\mu^{+}$ and $\mu_{2}=\mu^{-}$ for some signed measure $\mu$. Jordan's decomposition theorem tells us that it need not be the case that simply defining $\mu=\mu_{1}-\mu_{2}$ does the job.
So, I suppose my question then is, given any two positive measures $\mu_{1}$ and $\mu_{2}$, does there exist a signed measure $\mu$ such that $\mu_{1}=\mu^{+}$ and $\mu_{2}=\mu^{-}$?