Let $P$ be the space of all probability measures on a Polish space $X$. Then, $P$ is Polish as well as if its topology is the weak*-topology (a.k.a. the topology of weak convergence). (EDIT: IS THAT STATEMENT RIGHT?? Or is that only right for probability BOREL measures? i.e. our $\sigma$-algebra $\mathcal P$ HAS to be a Borel-$\sigma$-algebra. That would explain the fact that we choose $\mathcal P$ as Borel-$\sigma$-algebra)

Let $\mathcal P$ be the $\sigma$-algebra on $P$. Often, for the definition of random variables which value in $(P,\mathcal P)$ we choose $\mathcal P$ as generated by the open sets of the weak*-topology.

In Schervish's book "Theory of statistics" (2012) on page 27 he states that this implicates that $\mathcal P$ is generated by all sets

$A_{B,t}=\{\mathbb P\in P: \mathbb P(B)\le t\}$ for every $B\in \mathcal B(X),t\in [0,1]$.

Can anyone explain why these choices of $A_{B,t}$ generate $\mathcal P$? And can anyone explain basic and important facts about this weak*-topology that are useful in this context?


migrated from mathoverflow.net Apr 14 at 22:00

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    $\begingroup$ You can find answers to all your questions in the appendices of "Fundamentals of Nonparametric Bayesian Inference" by Ghosal and van der Vaart. $\endgroup$ – Michael Greinecker Apr 3 at 7:23

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