Hello all.

I have the following (perhaps basic) question: Let $X$ be a separable metric space. Does there necessarily exist a countable set $\mathcal{C}$ of Borel sets in $X$ such that any two probability measures which agree on $\mathcal{C}$ must agree on the whole of $\mathcal{B}(X)$?

(And slightly more generally: Let $(X,\Sigma)$ be a countably generated measurable space. Then does there necessarily exist a countable set $\mathcal{C}$ of $\Sigma$-measurable sets such that any two probability measures which agree on $\mathcal{C}$ must agree on the whole of $\Sigma$?)

Thanks, Julian.