Let $(A,\mathcal A)$ be a measure space, let $\mathcal P(A)=\{ q : (A, \mathcal A, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(A,\mathcal A)$. We equip $\mathcal P(A)$ with the weakest topology such that for all bounded measurable $f:A\to\mathbb R$, the function $p\to p(f)\triangleq\int_{\mathcal X}f ~dp$ is continuous. When talking about measurability, we equip $\mathcal P(A)$ with the Borel $\sigma$-algebra generated by the previously defined topology. We will work with elements in $\mathcal P(\mathcal P(A))$ that we equip with the same topology. For $p\in\mathcal P(A)$ and $\mu\in\mathcal P(\mathcal P(A))$ we say that $\mu\sim p$ if for all continuous linear form $f:\mathcal P(A)\to\mathbb R$, $\mu(f)\triangleq \int f ~d\mu=f(p)$.
Let us be given a bounded continuous function $g:\mathcal P(A)\to\mathbb R$ and define \begin{align*} \bar g:\mathcal P(A)&\to\mathbb R\\ p&\to \sup\{ \mu(g) : \mu\in\mathcal P(\mathcal P(A)), \mu\sim p \} \end{align*}
Can we show that $\bar g$ is upper semi continuous ? Or can we show that it is measurable ?
The function $\bar g$ is a concave function and I know from Phelps' "Lectures on Choquet theorem" that in the case where $\mathcal P(A)$ is compact, this is exactly the upper concave envelope $p\to\inf\{ h(p):h\text{ is affine with } g \leq h \}$. However in my case it might not be compact and I do not know if the two match.
Here is the beggining of my best attempt. We know that $\bar g$ is upper semi continuous iff for all $y\in \mathbb R$, $\{ p : \bar g(p)<y \}$ is open. By definition of $\bar g$, we can rewrite \begin{align*} \{ p : \bar g(p) < y\} &= \{ p:\exists \varepsilon>0,~\forall \mu\sim p,~\mu(g)<y-\varepsilon \}\\ &=\bigcup_{\varepsilon >0}\{ p:\forall \mu\sim p,~\mu(g)<y-\varepsilon \} \end{align*}
Therefore if we can show that $A_y=\{ p:\exists \mu\sim p,~\mu(g)\geq y \}$ is closed for all $y\in\mathbb R$ we would be done. Let $\{ p_\alpha\}$ be a net in $A_y$ converging to $p$ and let $\{\mu_\alpha\}$ be such that $\mu_\alpha\sim p_\alpha$ and $\mu_\alpha(g)\geq y$ then we have to build from $\mu_\alpha$ a $\mu\sim p$ such that $\mu(g) \geq y$, I am not sure how to do that. Typically (and for isntance in Phelp's Letcure on choquet's theorem, section 3) here we would appeal to weak* compactness to say that $\{ \mu_\alpha\}$ converges to some $\mu$ and we are essentially done, but without weak compactness we cannot do that, however it may still be doable to build such a $\mu$.