TITLE: How to construct a continuous finite additive measure on the natural numbers
QUESTION [13 upvotes]: 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.

REPLY [8 votes]: Eric van Douwen constructed a whole slew of finitely additive measures on the set of natural numbers, with various shifting and scaling properties in this paper. He discussed how much `choice' was needed for the constructions; often some version of the Hahn-Banach theorem would suffice.