Let $\mu$ be a probability measure on the $\sigma$-field $\mathcal{A}$. I have shown that the relation '$A\sim B$', if $\mu(A\mathop\Delta B)=0$, is an equivalence relation on $\mathcal{A}$. Denoting $[A]$ to be the equivalence class containing the set $A$, let $S=\{[A]:A\in\mathcal{A}\}$. I have shown that the function $\rho([A],[B])=\mu(A\mathop\Delta B)$ is an well-defined metric on $S$. I only need to show that if $\mathcal{A}$ is countably generated, then $S$ is separable (under the metric $\rho$), that is there is countable dense set.

I think the approach will be like this. Take the counatble generator of $\mathcal{A}$. The field generated by the generator may be countable. Now for every set $A\in\mathcal{A}$ and $\epsilon>0$, there will exist set $F$ in the field such that $\mu(A\mathop\Delta F)<\epsilon$. But I can't do any better.

Any help will be appreciated.


The algebra generated by a countable collection is countable. So $\mathcal D$, the algebra generated by $\cal A$, is countable. Then we indeed conclude by an approximation argument to prove that $\{[D],D\in\mathcal D\}$ is dense in $S$.

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