is a finitely additive measure?
What induction already gave you is that "two-set additivity" implies finite additivity, it can't give you more (e.g. sigma-additivity).
For counterexample, take an ultrafilter containing Fréchet filter on (we don't want to be trivial) and for every set
It is straightforward to verify that is a finitely additive measure but is not sigma additive.
I don't quite understand the remark about continuity though.