Completeness and additive property for submeasures
Jonathan M. Keith, Paolo Leonetti
TL;DR
The paper characterizes when the pseudometric space built from a submeasure, (Σ,d_ν) with d_ν(A,B)=min{1,ν(AΔB)}, is complete, proving an AP0-like criterion for submeasures: completeness holds iff increasing sequences with finite total ν- increment admit a limit set A with ν(A_n∖A)=0 and ν(A∖A_n)→0. It then shows a broad positive result: every lower semicontinuous submeasure φ on P(ω) yields a complete space (P(ω), d_{||·||_φ}) and extends to weighted upper densities, addressing gaps in earlier claims by Just–Krawczyk and Farah. Conversely, it establishes negative results for upper densities, proving that any μ^* dominating bd^* (in particular, bd^* itself and Buck density) yields an incomplete pseudometric space, highlighting a clear contrast between additively-defined densities and non-additive submeasures. The proofs blend Stone space techniques with careful Cauchy-sequence constructions and explicit counterexamples, advancing understanding of when submeasures induce complete metric structures and clarifying the behavior of upper densities in this context.
Abstract
Given an extended real-valued submeasure $ν$ defined on a field of subsets $Σ$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_ν$ defined by $d_ν(A,B):=\min\{1,ν(A\bigtriangleup B)\}$ for all $A,B \in Σ$ is complete. As an application, we show that if $\varphi: \mathcal{P}(ω)\to [0,\infty]$ is a lower semicontinuous submeasure and $ν(A):=\lim_n \varphi(A\setminus \{0, 1, \ldots, n-1\})$ for all $A\subseteq ω$, then $d_ν$ 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 $ν$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_ν$ is not complete.