This is a report on a paper of the same title with Erik Ellentuck that appeared in Fund. Math. 65 (1969), 33 - 42. The measures that are considered, though finitely additive, are defined for all sets of natural numbers. Various natural sets of measures are introduced and a set of natural numbers is considered “measurable&rdquo with respect to the set of measures if all measures agree on the set. One goal was to give a setting to explain “Benford's law” that about 30% of the numbers in a naturally occurring list have first digit 1.
Related work will be mentioned.
Richard Bumby