Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that

$$\int f d \mu_n \to \int f d\mu$$

in probability for each bounded continuous real-valued function. This would be the definition of weak convergence $\mu_n \Rightarrow \mu$, if I dropped "in probability".

Is there any standard way to extract a weakly convergent subsequence from $(\mu_n)$ consisting of almost all members of $(\mu_n)$? Possibly with additional assumptions? Where can I learn about such things?

Sorry if this is a trivial question.

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that

$$\int f d \mu_n \to \int f d\mu$$

in probability for each bounded continuous real-valued function. This would be the definition of weak convergence $\mu_n \Rightarrow \mu$, if I dropped "in probability" and the measures $\mu_n$ were deterministic.

Is there any standard way to extract a weakly convergent subsequence from $(\mu_n)$ consisting of almost all members of $(\mu_n)$? Possibly with additional assumptions? Where can I learn about such things?

Sorry if this is a trivial question.

Let $\mu_1, \mu_2, \dots$ be a sequence of random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that

$$\int f d \mu_n \to \int f d\mu$$

in probability for each bounded continuous real-valued function. If there was usual convergence here, then this would be the definition of weak convergence $\mu_n \Rightarrow \mu$.

Is there any standard way to extract a weakly convergent subsequence from $(\mu_n)$ consisting of almost all members of $(\mu_n)$? Possibly with additional assumptions? Where can I learn about such things?

Sorry if this is a trivial question.

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that

$$\int f d \mu_n \to \int f d\mu$$

in probability for each bounded continuous real-valued function. This would be the definition of weak convergence $\mu_n \Rightarrow \mu$, if I dropped "in probability".

Is there any standard way to extract a weakly convergent subsequence from $(\mu_n)$ consisting of almost all members of $(\mu_n)$? Possibly with additional assumptions? Where can I learn about such things?

Sorry if this is a trivial question.