This is a question about Proposition 9.1 in an article by Cormac Walsh: https://arxiv.org/pdf/1610.07508.pdf.
Let $K$ be a compact Haussdorf space, let $(C(K),\|\cdot\|_{\infty})$ be the space of continuous functions on $K$ with positive cone $C(K)_{+}$ consisting of all positive continuous functions on $K$. It is known that the dual cone of $C(K)_{+}$ is car$_{+}(K)$, the regular Borel measures on $K$ and the set of probability measures $$S(C(K)):=\{\mu\in\text{car}_{+}(K):\mu(K)=1\}$$ is the state space with respect to $b\equiv 1$, the constant $1$ function. The pure states, (extreme points of the state space), are the Dirac Masses $\delta_{x}$, given by $$\int_{K}f\ d\delta_{x}=f(x)\qquad(x\in K,f\in C(K)).$$
The statement I am trying to solve is the following:
Let $g:K\rightarrow\mathbb{R}$ be an upper semicontinuous function on $K$, then $\overline{g}:S(C(K))\rightarrow\mathbb{R}$ given by $$\overline{g}(\mu)=\int_{K}g\ d\mu$$ is a weak*-upper semicontinuous affine function on $S(C(K))$.
The fact that $\overline{g}$ is affine is evident, but I have had trouble to prove weak*-upper semicontinuity. I have tried using the fact that $\{\delta\in\text{car}_{+}(K):|\mu(f_{i})-\delta(f_{i})|<\varepsilon\text{ for all }i\in\{1,...,n\}\}$, for some $f_{1},...,f_{n}\in C(K)$ and $\varepsilon>0$ are open sets in the weak*-topology, but I do not really see how to move forward with that approach.
I have also explored using the fact that $\overline{g}$ restricted to the pure states is weak*-upper semicontinuous and then using Choquet theory, but Choquet theory is definitely not my expertise and I could not find results to utilise.
Any help would be much appreciated.