When I was reading Billingsley's book "Convergence of probability measures", it is claimed that "A convergence-determining class is obviously a separating class". But I don't understand why it's obvious and try to write this claim in a more clear way.
The definitions of separating class and convergence determining class are given in another related question: If $S$ is compact then the separating classes are same as convergence determining classes (Many thanks!)
Let's formulate the problem a bit more formally. In some probability space, we have two probability measures $P$ and $Q$. The sequence of probability measures $\{P_n\}$ weakly converges to $P$. Suppose we have a convergence determining class $\tilde{A}$, and $P$ and $Q$ coincide on $\tilde{A}$. Now we want to show that $P=Q$.
I was thinking to show that for any $A\in\tilde{A}$ with $Q(\partial A)=0$, we have $P_n A\to Q A$. But I don't know how to connect $Q(\partial A)$ and $P(\partial A)$. Thanks ahead for any advice!
