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 ${\mathrm{L}}_0(X,\mathrm{\Sigma },\mu )$ for a semi-finite measure μ 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.

我们研究了在一个向量点阵上的 Hausdorff uo-Lebesgue 拓扑结构，该拓扑结构是在一个有序稠密理想上的 Hausdorff (o)-Lebesgue 拓扑结构，以及由此获得的拓扑结构的性质是什么。当向量格有一个具有分离阶连续对偶的阶密理想时，总是可以以这种方式为其提供这样的拓扑，并且这种拓扑对规则子格的限制也是豪斯多夫 uo-Lebesgue 拓扑. 的正则向量子格${\mathrm{升}}_0(X,\mathrm{\Sigma },\mu )$对于半有限测度μ属于这一类，并且其 Hausdorff uo-Lebesgue 拓扑中的网络收敛是有限测度子集的测度收敛。当向量格不仅具有具有分离阶连续对偶的阶密理想，而且具有可数 sup 性质时，我们证明了在其 Hausdorff uo-Lebesgue 拓扑中收敛的正则向量亚格中的每个网络总是包含一个序列u 收敛到相同的极限。这使我们能够在这种情况下对有关 uo 收敛的各种拓扑问题给出令人满意的答案。