Completeness and additive property for submeasures

Abstract

Given an extended real-valued submeasure $\hat{I}½$ defined on a field of subsets $\hat{I}\pounds$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_{\hat{I}}½$ defined by $d_{\hat{I}}½(A,B):=\min \{1,\hat{I}½(A△B)\}$ for all $A,B\in \hat{I}\pounds$ is complete. As an application, we show that if $\ddot{I}†:\mathcal{P}(\ddot{I}‰)\to [0,\infty ]$ is a lower semicontinuous submeasure and $\hat{I}½(A):={\lim }_n\ddot{I}†(A\setminus \{0,1,\dots ,n-1\})$ for all $A\subseteq \ddot{I}‰$, then $d_{\hat{I}}½$ 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 $\hat{I}½$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_{\hat{I}}½$ is not complete. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $\hat{I}\pounds /\hat{I}½$.