Completeness and additive property for submeasures

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 $\Sigma/\nu$.