Topological structure & compactness of the space of probability measures Let $(X,\tau)$ be a topological space and let $\mathcal P$ denote the set of all probability measures on $\mathscr B$, the Borel $\sigma$-algebra generated by the open subsets of $X$. In what follows, let $\mathrm{BC}(X)$ denote the set of bounded, continuous real-valued functions on $X$.

Define a topology $\upsilon$ on $\mathcal P$ as the one generated by the topological subbasis consisting of sets of the form $$U_{f,y,\varepsilon}\equiv\left\{\mathbb P\in\mathcal P:\left|\int_{x\in X}f(x)\,\mathrm d\mathbb P(x)-y\right|<\varepsilon\right\},\quad\text{where } f\in\mathrm{BC}(X),\,y\in\mathbb R,\,\varepsilon>0.$$ The topology $\upsilon$ is often referred to as the weak-star topology, given the functional-analytic structure with which $\mathcal P$ can be endowed (see also below). It is not difficult to check that a net $(\mathbb P_{\alpha})_{\alpha\in A}$ in $\mathcal P$ (where $A$ is a non-empty directed index set) converges to $\mathbb P\in\mathcal P$ with respect to this topology if and only if the corresponding net $$\left(\int_{x\in X}f(x)\,\mathrm d\mathbb P_{\alpha}(x)\right)_{\alpha\in A}$$ of real numbers converges to $\int_{x\in X}f(x)\,\mathrm d\mathbb P(x)$ with respect to the Euclidean topology on $\mathbb R$ for every fixed $f\in\mathrm{BC}(X)$.

It is quite well-known from functional-analytic probability theory that if $(X,\tau)$ is metrizable and compact, then $(\mathcal P,\upsilon)$ is compact. [And I think, although I am not 100% sure, that the Riesz representation theorem ensures the compactness of $(\mathcal P,\upsilon)$ if $(X,\tau)$ is compact and merely Hausdorff, and not necessarily metrizable.]

What I am trying to do is establish the compactness of $(\mathcal P,\upsilon)$ using only the compactness of the underlying space $(X,\tau)$, using no further topological separation or countability axioms:

Conjecture: If $(X,\tau)$ is compact, then so is $(\mathcal P,\upsilon)$.

Naturally, the applicability of the usual functional-analytic toolkit (such as the Riesz representation theorem) is quite limited without further assumptions. I was thinking of using Alexander’s subbasis theorem as a starting point, but I am actually not even sure that the conjecture is true in its general form.

Any hints, references for proofs (if true), or outlines of counterexamples (if false) would be greatly appreciated.
 A: The answer is yes. First, replace the compact space with a compact Hausdorff space with the "same" continuous functions, using the following theorem:

Theorem: Let $X$ be any topological space. Then there exists a completely regular Hausdorff space $Y$ and a continuous surjection
$\tau:X\to Y$ such that the function $g\mapsto g\circ\tau$ is an
isomorphism from $C_B(Y)$ onto $C_B(X)$.

This is Theorem 3.9 of "Rings of continuous functions" (1960) by Gillman and Jerison. For compact Hausdorff spaces, the space of Radon probability measures is compact by Theorem 4.5.3 of "Weak Convergence of Measures" (2018) by Bogachev.
Now, one can identify Baire measures on the original compact space and the compact Hausdorff space. Baire measures are defined on the $\sigma$-algebra generated by continuous functions. For compact Hausdorff spaces, Baire measures and Radon measures correspond to each other, so it follows that we get weak compactness on the original compact space when we look at Baire probability measures.
But each Baire probability measure on compact space extends to some (not necessarily Radon) probability measure, as explained here. Since the weak topology cannot separate Borel probability measures with the same Baire restriction, this is enough to prove the compactness of the space of Borel probability measures.
