Extreme points of the set of density measures

TL;DR

This paper investigates finitely additive measures extending asymptotic density, characterizes their extremal points via positive functionals extending Cesàro mean, and describes the structure of the set of all such measures.

Contribution

It establishes a one-to-one correspondence between density measures and positive functionals extending Cesàro mean, and characterizes the extremal structure of these measures.

Findings

Identifies a one-to-one correspondence between density measures and positive functionals in ll__^*.

Describes the set of all density measures as the closed convex hull of a specific set of extremal measures.

Provides a characterization of the maximal values of density measures for given sets.

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.