Let $X$ be a Polish space and $Prob(X)$ be the set of Borel probability measures on $X$, and let $Prob(X)$ be equipped with the weak-* topology (So that a sequence $\mu_m$ converges to $\mu$ in $Prob(X)$ if and only if $\int_X f d\mu_m \to \int_X f d\mu$ for all $f\in C_b(X)$ ).

In many stochastic process or probability textbooks, it is proven that $Prob(X)$ is a Polish space. However, I do not get any motivation nor application of this theorem. What is the point of proving that $Prob(X)$ is a Polish space? Do we really need this result when dealing with stochastic process?


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