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 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 as 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.

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.