Let $(X, d)$ be a metric space. Then @Nate Eldredge said that
The space $\mathcal{P}(X)$ of probability measures on a Polish space $X$, endowed with the weak topology (induced by bounded continuous functions), is first countable - indeed Polish. This is a standard result that you can find in, say, Billingsley's Convergence of Probability Measures. Note here that the "weak topology" is really a weak$^*$ topology. You can think about the fact that, given a separable Banach space $W$, the weak* topology on $W^*$ is not first countable, but the topology it induces on the ball of $W^*$ is first countable (indeed, Polish!).
- In Onno van Gaans's note Probability measures on metric spaces, I got that
- Theorem 4.2 If $X$ is separable, then the weak topology of $\mathcal{P}(X)$ is separable.
- Proposition 5.3 If $X$ is compact, then the weak topology of $\mathcal{P}(X)$ is compact.
Could you elaborate on which theorem in Billingsley's Convergence of Probability Measures mentiones that if $X$ is Polish then the weak topology of $\mathcal{P}(X)$ is Polish?
- From Riesz–Markov–Kakutani theorem, it seems to me weak$^*$ topology is induced by the space $C_0 (X)$ of continuous functions on $X$ vanishing at infinity, but I'm not sure. Could you elaborate on how the weak$^*$ topology of $\mathcal{P}(X)$ is defined?