I do have various questions regarding the topic of probability measures on polish spaces in general, thus I am trying to divide them in “small” subquestions. Hence, this is my first question on this issue.
Notation:
- $\Omega$ is a Polish space;
- $\mathcal{B}(\Omega)$ is the Borel $\sigma$-algebra on $\Omega$;
- $\Delta (\Omega)$ is the set of probability measures on $\mathcal{B}(\Omega)$.
Quite often I read that $\mathcal{B}(\Omega)$ is endowed with the topology of weak convergence, where the topology of weak convergence states that for every real function $f$ bounded, continuous on $\Omega$, $$\lim_{n \to \infty} \int f d\mu_n = \int f d\mu.$$
Here there are my basic questions:
- How does the topology of weak convergence creates open sets in $\Delta (\Omega)$?
- How do those open sets look like?
As always, any feedback regarding anything will be more than welcome.
Thank you for your time.
PS: Maybe the question looks naive, but I am trying to see why, if $\Omega$ is Polish, then $\Delta (\Omega)$ endowed with the topology of weak convergence is Polish as well (of course, any feedback regarding this issue is welcome as well!). Thus I would like to see how the topology of weak convergence actuallys works.