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I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider the topology of weak convergence on the space $E'$ of probability measures on $E$ (thus $E'$ is Polish too).

  • 1) Let $X$ be a random variable taking its values in $E$, and let ${\cal B} \subset {\cal A}$ be a $\sigma$-field. Then the conditional law ${\cal L}(X \mid {\cal B})$ is a $E'$-valued random variable. This should be a consequence of the measurability of the conditional expectations $\Bbb E[f(X) \mid {\cal B}]={\cal L}(X \mid {\cal B})(f)$ but I don't find any standard probability theory book asserting this fact (I'd prefer a "standard" probability theory book rather than a more technical book about random measures).

  • 2) Let $(\mu_n)$ be a sequence of random probability measures on $E$ (in other words the $\mu_n$ are $E'$-valued random variables). Let $\mu_\infty$ be another random probability on $E$ and assume that for each bounded continuous function $f\colon E\to \Bbb R$, the convergence $\mu_n(f) \to \mu_\infty(f)$ holds almost surely. It is tempting to conclude that $\mu_n \to \mu_\infty$ but this is not straightforward since the full set of convergence in $\mu_n(f) \to \mu_\infty(f)$ could depend on $f$. So I'm looking for a reference book showing this almost sure convergence in $E'$. Theorem 7.5.2 in this book by Kuksin's and Shirikyan answers the question but it is stronger than desired because it does not assume the presence of $\mu_\infty$. Moreover I'd prefer a more standard probability/measure theory book.

Actually I would be satisfied to find a reference for the special case when $E$ is compact. Thanks in advance.

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I think a relatively readable book on these issues is "Random Probability Measures on Polish Spaces" by Crauel. –  Michael Greinecker Jun 22 '13 at 10:57
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For (2), in the case that $E$ is compact, you can work on a countable dense subset of $C(E)$. And in the locally compact case, you get weak convergence as soon as you have $\mu_n(f) \to \mu_\infty(f)$ for all $f \in C_0(E)$, or a countable dense subset thereof. –  Nate Eldredge Jun 22 '13 at 11:22
    
Thanks. Here is an interesting reference given in Kuksin & Shirikyan's book tandfonline.com/doi/abs/10.1080/17442500600745359#.UcXQjPnTquk –  Stéphane Laurent Jun 22 '13 at 16:48
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1 Answer

For the first question I read a paper today related to Markov Kernels which state it in a more general case (for Markov Kernels)

It has no demonstration at all, but it is a good reference for this basic facts and definitions.

Check it out in section 2.

http://www.sciencedirect.com/science/article/pii/S0167715212004579 The Author is A.G. Nogales, "The existence of regular conditional Probabilities for Markov Kernels".

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