Extreme points of the set of density measures

Abstract

We study finitely additive measures on the set $\mathbb N$ which extend the asymptotic density (density measures). We show that there is a one-to-one correspondence between density measures and positive functionals in $\ell_\infty^*$, which extend Ces\`{a}ro 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 Ces\`{a}ro mean. Using the obtained results, we can find a set of functionals such that their closed convex hull in $\ell_\infty^*$ with weak${}^*$ topology is precisely the set of all positive functionals extending Ces\`{a}ro mean. Since we have a one-to-one correspondence between such functionals and density measures, this also gives a set of density measures, from which all density measures can be obtained as the closed convex hull.