compactness of probability measures in the weak topology for a non-metrizable compact space

Question

I saw and understood some proofs which showed, that for a compact metric space $X$ the space of probability measures $\mathcal{P}(X)$ with the weak topology ($\mu\mapsto\int f\,\mathrm{d}\mu$ is continuous for every $f\in C_b(X,\mathbb{R})$) is compact. Now I read that this is also true for $X$ not being a compact metric space but just a compact Hausdorff-space, but I couldn't find a good proof for this. The proofs which I have found where all like in one line "that is clear because of duality and Riesz-representation stuff" but unfortunately I'm not so familiar with functional analysis and duality. Do you have a good reference, where this is shown in detail?