Completeness and additive property for submeasures

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 B)\ for all A,B ∈ Σ is complete. As an application, we show that if φ: P(ω)→ [0,∞] is a lower semicontinuous submeasure and ν(A):=limₙ φ(A \0, 1, …, n-1\) for all A⊆ 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.~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. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $Σ/ν$.