Range of density measures

TL;DR

This paper studies properties of finitely additive density measures on natural numbers, introduces a new class based on cluster points, and characterizes their possible value ranges, simplifying previous descriptions.

Contribution

It introduces a new class of density measures using cluster points and characterizes their value ranges, simplifying prior complex descriptions.

Findings

Range of density measures for any subset of N is characterized.

Values attained by these measures are explicitly studied.

The description simplifies previous complex characterizations.

Abstract

We investigate some properties of density measures -- finitely additive measures on the set of natural numbers $\N$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence $\big(\frac{A(n)}{n}\big)$ as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of $\N$. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao \cite{brbr} for general finitely additive measures. Also the values which can be attained by the measures defined in the first part of the paper are studied.