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 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 $g_n(\mu)=\int gd\mu$.

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.