Density measures on a certain flow

TL;DR

This paper explores finitely additive density measures on natural numbers, examining their relationships like absolute continuity and singularity, especially those constructed from ultrafilters, and provides necessary and sufficient conditions for these relations.

Contribution

It introduces a class of density measures from ultrafilters and characterizes their absolute continuity and singularity relations with precise conditions.

Findings

Characterization of absolute continuity between density measures

Criteria for singularity among density measures

Analysis of weak absolute continuity and strong singularity

Abstract

We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and investigate absolute continuity and singularity for those density measures. In particular, for any pair of such density measures we prove necessary and sufficient conditions that one is absolutely continuous with respect to the other and that they are singular. Also we prove the same results for weak absolute continuity and strong singularity.