How to show that there's a continuous function separating convex sets of Radon measures? First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed (weak-* sense) set of probability measures. $m$ is a probability measure out of $K$.
In a paper I read, the author uses the fact that there's a continuous function $g$ s.t. $\int gdm>sup_{\mu \in K}\{\int gd\mu\}$
So far I was able to show that $\exists \psi\in C(X)^{**}$ s.t. $\psi(m)>sup_{\mu \in K}\{\psi(\mu)\}$
And also that the unit ball of $C(X)$ is dense in the weak-* topology in the unit ball of $C(X)^{**}$, meaning that $\exists \{g_n\}\subset B(C(X))$ s.t. $g_n\overset{w-*}{\longrightarrow} \psi$, where the canonical embedding is $g_n(\mu)=\int g_n d\mu$.
Making the final step of showing some continuous function actually fulfills the desired inequality, has been unsuccessful for me so far. Neither could I find any book that show a similar claim.
Any help would be appreciated, especially a simple quick solution that I missed.
 A: I do not think that this result is true, you need that $K$ is closed w.r.t. the weak-* topology.
The following is a counterexample: Take $X = [0,1]$ and $m$ the Lebesgue measure. $K$ is the closed convex hull of the Dirac measures. Then, $m \not\in K$ (see below) but you cannot separate $m$ from $K$ with a continuous function.
It remains to show $m \not\in K$.
It is easy to see that
\begin{equation*}
 L = \Big\{ \sum_{i=1}^n \alpha_i \, \delta_{x_i} \text{ for some } n, x_i, 0 \le \alpha_i, \sum_{i=1}^n \alpha_i = 1\Big\}.
\end{equation*}
is the convex hull of the Dirac measures.
Then, $K$ is the closure of $L$.
Moreover, it is easy to see that the Lebesgue measure $\lambda$ has distance $2$ from all measures in $L$, hence it does not belong to $K$.
Note, however, that $\lambda$ belongs to the weak-* closure of $K$ and this is the reason why you cannot separate it from $K$.
In response to the edit: If $K$ is weak-* closed,
this separation is a standard result. The mentioned paper already includes a reference to Dunford/Schwartz. It can also be found in other books, see also Separation in dual space.
