# What is the usage of the fact that $Prob(X)$ is Polish if $X$ is Polish?

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?