Topological structure & compactness of the space of probability measures

Solution 1:

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.