SCHWEITZER 1970 Problem 7

Let us use the word $N$ -measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $0$ on finite sets, and equal to $1$ on the whole set. We say that the system $\Upsilon$ of sets determines the $N$ -measure $\mu$ if any $N$ -measure coinciding with $\mu$ on all elements of $\Upsilon$ is necessarily identical with $\mu$ . Prove the existence of an $N$ -measure $\mu$ that cannot be determined by a system of cardinality less than continuum.

I. Juhasz