A lecturer made a passing reference to this... he said that finite additivity plus continuity is equivalent to countable additivity... could someone offer a proof of this or else direct me to one?

I don't understand why the continuity criterion is necessary

If $\displaystyle F_i\cap F_j = \emptyset \ \forall i\neq j \Rightarrow \mu(\bigcup_{i=1}^NF_i) = \sum_{i=1}^N\mu(F_i)$

Then don't we get countable additivity straight away just by induction?

Can someone provide a counterexample? Thanks