Densitometria I. Discrete groups

TL;DR

This paper investigates the properties and characterizations of subadditive functionals called means on functions over commutative groups, focusing on their extremal forms, uniqueness, and the conditions under which they are well-defined.

Contribution

It introduces the concepts of lowest and uppermost means, providing characterizations and expressions, and explores the conditions for a functional to be a mean and its relation to densities.

Findings

Defined and characterized extremal means, called lowest and uppermost.

Provided expressions for extremal means and conditions for their uniqueness.

Explored the relationship between means and densities, with partial results.

Abstract

An upper mean here is a subadditive functional $\overline M$ defined on bounded functions on a commutative group which has, beside some natural requirements, the property we call restricted additivity: if $g(x)= f(x)+f(x+t)$, then $\overline{M} (g)= 2 \overline{M} (f)$. This tries to grasp that it should not depend on local properties. This naturally induces a lower mean, and when they coincide it is the mean. Restriction to 0--1 valued functions (sets) is a density. We answer the following questions: Given a functional defined on a subset of all functions, when is it a mean? Given a functional, which is a mean, how do we find the upper mean it came from? Is it unique? Given a function $f$, what are the possible values of $\overline M(f)$, for upper means $\overline M$? In particular, we find the extremal means and give several expressions for it. We propose the names ``lowest and uppermost mean'' for them to replace the not really justified names ``lower and upper Banach mean and density''. We also consider analogous questions for densities, with partial answers only.