Completeness and additive property for submeasures

TL;DR

This paper characterizes when the pseudometric induced by an extended real-valued submeasure on a set algebra is complete, providing conditions, applications, and counterexamples, and relates these to the Stone space of the Boolean algebra.

Contribution

It offers necessary and sufficient conditions for the completeness of the pseudometric from submeasures, correcting previous gaps and exploring various classes of submeasures.

Findings

Completeness characterized by specific conditions on submeasures.

Lower semicontinuous submeasures yield complete pseudometrics.

Upper densities like the upper Banach density do not produce complete pseudometrics.

Abstract

Given an extended real-valued submeasure $\nu$ defined on a field of subsets $\Sigma$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_\nu$ defined by $d_{\nu}(A,B):=\min\{1,\nu(A\bigtriangleup B)\}$ for all $A,B \in \Sigma$ is complete. As an application, we show that if $\varphi: \mathcal{P}(\omega)\to [0,\infty]$ is a lower semicontinuous submeasure and $\nu(A):=\lim_n \varphi(A\setminus \{0, 1, \ldots, n-1\})$ for all $A\subseteq \omega$, then $d_\nu$ is complete. This includes the case of all weighted upper densities, fixing a gap in a proof by Just and Krawczyk in [Trans.~Amer.~Math.~Soc.~\textbf{285} (1984), 803--816]. In contrast, we prove that if $\nu$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_\nu$ is not complete. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $\Sigma/\nu$.