Vector lattices with a Hausdorff uo-Lebesgue topology

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:505

Article

Deng, Yang¹  de Jeu, Marcel^2,3 

[1] Southwestern Univ Finance & Econ, Sch Econ Math, Chengdu 611130, Sichuan, Peoples R China

[2] Leiden Univ, Math Inst, POB 9512, NL-2300 RA Leiden, Netherlands

[3] Univ Pretoria, Dept Math & Appl Math, Corner Lynnwood Rd & Roper St, ZA-0083 Hatfield, South Africa

关键词: Vector lattice;    Banach lattice;    Unbounded order convergence;    Lebesgue topology;    uo-Lebesgue topology;   

DOI  :  10.1016/j.jmaa.2021.125455

来源: Elsevier

【 摘 要 】

We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of L-0(X, Sigma, mu) for a semi-finite measure mu falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context. (c) 2021 Published by Elsevier Inc.

【 授权许可】

Free