L\'evy group and density measures

TL;DR

This paper explores finitely additive measures extending asymptotic density on integers and characterizes their invariance under the Lévy group of permutations, linking measure extension to group invariance.

Contribution

It provides a new characterization of the Lévy group and establishes that measures extending density are exactly those invariant under this group.

Findings

Finitely additive measures extending density are characterized by invariance under the Lévy group.

A new characterization of the Lévy group of permutations is introduced.

The equivalence between measure extension and group invariance is proven.

Abstract

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the L\'evy group $\mathcal{G}$ of permutations of $\mathbb N$. Using a new characterization of the L\'evy group $\mathcal G$ we will prove that a finitely additive measure extends density if and only if it is $\mathcal{G}$-invariant.