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 $d_ν(A,B):=\min{1,ν(A\bigtriangleup B)}$ for all $A,B \in Σ$ is complete. As an application, we show that if $φ: \mathcal{P}(ω)\to [0,\infty]$ is a lower semicontinuous submeasure and $ν(A):=\lim_n φ(A\setminus {0, 1, \ldots, n-1})$ 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. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $Σ/ν$.

💡 Research Summary

The paper investigates the completeness of the pseudometric space induced by an extended real‑valued submeasure ν defined on a field Σ of subsets of a set X. The pseudometric is given by dν(A,B)=min{1,ν(A△B)}. While completeness criteria are well‑understood for finitely additive (measure‑like) maps, the authors extend these results to the non‑additive setting of submeasures.

The central result (Theorem 2.1) provides three equivalent conditions for completeness: (i) (Σ,dν) is complete; (ii) for every dν‑Cauchy sequence {An} the difference ν⁺(An)−ν⁻(An) admits an exact Hahn decomposition; (iii) for every increasing sequence {An} with Σ ν(An+1\An)<∞ there exists A∈Σ such that ν(An\A)=0 for all n and ν(A\An)→0. Condition (iii) is a natural analogue of the classical AP(null) property for measures, replacing “finite symmetric difference” with “ν‑measure zero”.

Using this theorem, the authors recover the known completeness characterization for finitely additive bounded measures (Corollary 2.2) and show that any σ‑subadditive submeasure on a σ‑field yields a complete space (Corollary 2.3).

A major positive application concerns lower‑semicontinuous submeasures (lscsm). For an lscsm φ on ℘(ω) they define a derived submeasure ‖·‖φ via the limit of φ on tails of a set. Theorem 2.5 proves that (℘(ω),d‖·‖φ) is complete. This unifies and extends earlier results: Solecki’s theorem that every analytic P‑ideal is the exhaustion of some lscsm, and the authors’ previous work on the completeness of the pseudometric induced by the upper asymptotic density d⋆. Consequently, all weighted upper densities (defined via Erdős–Ulam functions) are shown to generate complete pseudometrics, filling a gap in the proof of Just and Krawczyk (1984).

On the negative side, the paper demonstrates that the upper Banach density bd⋆ (and any upper density dominating it) does not give a complete pseudometric. Proposition 2.6 constructs a specific Cauchy sequence whose limit fails to exist, and Theorem 2.7 extends this non‑completeness to a broad class of upper densities, including the upper analytic, upper Polya, and upper Buck densities.

Finally, the authors relate completeness to the Stone space of the Boolean algebra Σ/ν. They define a submeasure ν̂ on the power set of the Stone space and prove that (Σ,dν) is complete iff (℘(Stone(Σ/ν)),dν̂) is complete (Theorem 2.9). This provides a topological characterization and connects the problem to well‑studied properties of Boolean algebras.

Overall, the paper delivers a comprehensive framework for understanding when the Fréchet–Nikodým metric associated with a submeasure is complete, bridges gaps in the literature concerning weighted densities, and offers new perspectives via Hahn decompositions and Stone duality.