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- W3031599872 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." @default.
- W3031599872 created "2020-06-05" @default.
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- W3031599872 date "2022-01-01" @default.
- W3031599872 modified "2023-10-18" @default.
- W3031599872 title "Vector lattices with a Hausdorff uo-Lebesgue topology" @default.
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- W3031599872 doi "https://doi.org/10.1016/j.jmaa.2021.125455" @default.
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