Given a Polish space $(E,d)$ with a sigma-algebra $\mathcal{E},$ we say that a family $\mathcal{A}\subset\mathcal{E}$ is a separating class if if two probability measures that agree on $\mathcal{A}$ necessarily agree also on the whole of $\mathcal{E}.$
My question is:
If two RANDOM measures $\mu$, $\nu$ on $(E,d)$ coincide in distribution on $\mathcal{A}$ then are they equal in distribution? Namely,
$$\mu(A)\stackrel{(d)}{=}\nu(A)\text{ for all }A\in\mathcal{A}\quad\stackrel{?}{\Rightarrow}\quad \mu\stackrel{(d)}{=}\nu.$$
Equivalently, is $\mathcal{A}$ a separating class also for the space of random probability measure? If not, what is a separating class for the space of random probability measure?