Continuing my work through Folland, trying to prove the following (Chapter 7 #22):
Added: *Let $X$ be a locally compact Hausdorff space.*
Let $\{f_\alpha\}_{\alpha\in A}$ be a subset of $C_0(X)$ and $\{c_\alpha\}_{\alpha\in A}$ be a family of complex numbers. If for each finite set $B\subset A$ there exists $\mu_B\in M(X)$ such that $\|\mu_B\|\le 1$ and $\int f_\alpha\ d\mu_B=c_\alpha$ for $\alpha\in B$, then there exists $\mu\in M(X)$ such that $\|\mu\|\le 1$ and $\int f_\alpha\ d\mu=c_\alpha$ for all $\alpha\in A$.
Work so far: I think the idea is to use the result of Proposition 7.19, which says that if we have a sequence $\mu,\mu_1,\mu_2,\dots \in M(\mathbb{R})$, such that $\sup_n \|\mu_n\|<\infty$ with $F_n(x):=\mu_n((-\infty,x])\to F(x):=\mu((-\infty,x])$ at all $x$ where $F$ is continuously, then $\mu_n\to \mu$ vaguely.
My rough idea is that it suffices to assume that $A$ is countable, since we can use density arguments, and try to construct such a sequence satisfying 7.19, but I don't know where to go from there, let alone how to get there.
Any help would be greatly appreciated.