Let $X$ be a set, $\mathcal{S}\subseteq \mathcal{P}(X)$ be a semiring on $X$, and $\mu: \mathcal{S}\rightarrow [0,\infty]$ with $\mu(\emptyset)=0$. Assume $\mu$ is finitely additive and countably subadditive. Does it imply that $\mu$ is also countably additive? (I know that the converse is true)
The question arises because such a set function is required in some versions of the Extension theorem for measures (see, e.g., the textbook by Billingsley, or by Klenke). I'm wondering to what extent it is more general than a countably additive function (i.e., a premeasure)?
If being finitely additive and countably subadditive and being countably additive are not equivalent in the setup above, could someone provide a counter example on the semiring $\mathcal{S}:=\{(a, b]\mid a,b\in \mathbb{R},a\leq b \}$?
