Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following types of convergence, where $\mu$ is some non-random probability measure :
(A) $ \forall f\in C_b, \quad \mathbb P \left(\int f(x)d\mu_n(x)\rightarrow_{n\rightarrow\infty}\int f(x)d\mu(x)\right)=1$ and
(B) $ \mathbb P \left(\forall f\in C_b,\quad \int f(x)d\mu_n(x)\rightarrow_{n\rightarrow\infty}\int f(x)d\mu(x)\right)=1.$ Whereas (B) clearly implies (A), what about the other direction ?
1) Do you have a counter-example where (A) does not implies (B) ?
2) Do we have a sufficient criterion so that (A) implies (B) ? (I have in mind the use of the Borel-Cantelli lemma to improve the probability convergence to the almost sure one)