Suppose that a sigma-algebra $\Sigma$ of subsets of $X$ is generated by a countable collection $\mathcal A$ of subsets of $X$. Suppose that $\mu$ is a finite, finitely additive measure on $(X, \Sigma)$ with the property that, for all countable, pairwise disjoint sequences $A_1,A_2,...$ in $\mathcal A$, $$\mu(\cup_n A_n) = \sum_n \mu(A_n).$$
Does it follow that $\mu$ is countably additive on all of $\Sigma$?
It seems plausible to me that this holds. My idea was to try to prove it with a generating class argument, but I'm not sure how to set it up. Let $\mathscr S$ be the collection of all countable, pairwise disjoint sequences for which $\mu$ is additive. Then $\mathscr S$ contains sequences of sets that are all in $\mathcal A$. For a normal generating class argument, one then shows that $\mathscr S$ is closed under complements and countable unions, but that doesn't make sense here because $\mathscr S$ is a collection of sequences. I'm not sure how else to proceed.