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 ${\ell }_{\infty }^{\ast }$, 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 ${\ell }_{\infty }^{\ast }$ with weak${}^{\ast }$ topology is precisely the set of all positive functionals extending Cesàro 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.

Cited by in corpus (1)

• Functional analytic approach to Cesàro mean