# $\sigma$-algebra on space of measures

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.netApr 14 at 22:00

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