I want to find some condition to construct a continuous finitely additive measure on the natural numbers, i.e. $f:P(\mathbb{N})\rightarrow [0,1]$ such that $f(\{n\})=0$, and $f$ is an additive measure.

I know in ZFC we can use an ultrafilter $U$ and define $f$ by $f(A)=1\Leftrightarrow A\in U$, but this is too trival.

How about ZF? or some other condition? like large cardinal.