Completeness and additive property for submeasures
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910063859335168 |
|---|---|
| author | Keith, Jonathan M. Leonetti, Paolo |
| author_facet | Keith, Jonathan M. Leonetti, Paolo |
| contents | 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 $Σ/ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completeness and additive property for submeasures Keith, Jonathan M. Leonetti, Paolo Functional Analysis Classical Analysis and ODEs General Topology 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 $Σ/ν$. |
| title | Completeness and additive property for submeasures |
| topic | Functional Analysis Classical Analysis and ODEs General Topology |
| url | https://arxiv.org/abs/2501.13615 |