Why on a complete metric spaces that are Polish spaces we can define probability measures?

Question

I have a slight background on probability measures and Polish spaces.

I know that a Polish space is a separable completely metrizable topological space. In the lecture notes I am following the author says:

"Since the metric space $(G,d)$ is a Polish space, we may consider the set of probability measures on $G$ denoted by $\mathcal{P}(G)$"

Answer 1

You can always consider the set of (Borel) probability measures on any space $X$. But for Polish spaces it's a nice space with a lot of structure, see Kechris' book (Classical Descriptive Set Theory), p. 109 and onwards for more details. It is Polish and if $X$ is compact then so is $\mathcal{P}(X)$ etc. There is a nice description of convergence of measures as well.