Extreme points of the set of density measures

We study finitely additive measures on the set N which extend the asymptotic density (density measures). We show that there is a one-to-one correspondence between density measures and positive functionals in ℓ∞⁎, which extend Cesàro mean. Then we study maximal possible value attained by a density measure for a given set A and the corresponding question for the positive functionals extending Cesàro mean. Using the obtained results, we can find a set of functionals such that their closed convex hull in ℓ∞⁎ with weak⁎ topology is precisely the set of all positive functionals extending Cesàro mean. This also describes a set of density measures, from which all density measures can be obtained as the closed convex hull.