(1) Do I have this correct?

From a non-trivial non-principal ultrafilter UF, I can define a 0-1 measure p by basically saying that p(a,b) = 1 if there exists a member S of UF so that a and b are both members of S, and 0 otherwise.

(2) Once the details of the previous construction are straightened out, how would I go from such an ultrafilter (assuming the existence of a measurable cardinal) to define a real-valued measure? I know that the existence of one is equivalent to the existence of the other, but roughly what would be the details of construction of the real-valued measure?

Thank you in advance.