Vector lattices with a Hausdorff uo-Lebesgue topology

TL;DR

This paper explores the construction and properties of a Hausdorff uo-Lebesgue topology on vector lattices, linking it to measure convergence and providing insights into uo-convergence behavior.

Contribution

It introduces a method to construct Hausdorff uo-Lebesgue topologies from ideals and analyzes their properties, especially in relation to measure convergence and sequence convergence.

Findings

Hausdorff uo-Lebesgue topology can be constructed from a dense ideal with a separating dual.

In measure spaces, convergence in this topology aligns with convergence in measure on finite subsets.

Sequences can be extracted from nets converging in the topology, ensuring uo-convergence under certain conditions.

Abstract

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 $\mathrm{L}_0(X,\Sigma,\mu)$ 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.