I want to find some condition to construct a continuous finite additive measure on nature. i.e $f:P(N)\rightarrow [0,1]$s.t f({n})=0,and f is additive measure  .

I know in ZFC can use an untraflter U constract by$f(A)=1\Leftrightarrow A\in U$,but this too trival.

How about ZF? or some others conditon?like large cardinal.

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.