Vector lattices with a Hausdorff uo-Lebesgue topology

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.

Automated summary

The study investigates the construction of a Hausdorff uo-Lebesgue topology from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and examines the properties of the obtained topologies. When a vector lattice has an order dense ideal with a separating order continuous dual, it can always be equipped with such a topology, and the restriction of this topology to a regular sublattice is also a Hausdorff uo-Lebesgue topology. If a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology contains a sequence that is uo-convergent to the same limit.