is a finitely additive measure?

What induction already gave you is that "two-set additivity" impliesfinite 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.