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.