论文标题
带有Hausdorff Uo-Lebesgue拓扑的矢量晶格
Vector lattices with a Hausdorff uo-Lebesgue topology
论文作者
论文摘要
我们研究了Hausdorff Uo-Lebesgue拓扑的构建，该拓扑是由Hausdorff（O） - Lebesgue拓扑的矢量晶格，以密集的理想顺序以及所获得的拓扑的性质。当矢量晶格具有订单密集的理想和分离顺序连续双重的订单密集的理想时，总是有可能以这种方式为其提供这样的拓扑，并且将这种拓扑结构限制为常规的Sublattice，也是Hausdorff Uo-Lebesgue拓扑。 $ \ mathrm {l} _0（x，σ，μ）的常规矢量sublatice，用于半限制度量$μ$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $，然后在其Hausdorff uo-lebesgue拓扑中的网络收敛是在有限程度的子集中的融合中的。当矢量晶格不仅具有连续分离的订单连续双重的订单密度理想，而且具有可数的SUP特性，我们表明，常规矢量sublattice中的每个网中的每个网中的每个网中的每个网都会在其Hausdorff uo-lebesgue拓扑中收敛，始终包含一个序列，该序列始终是uo convergent to Sames convergent。这使我们能够在这种情况下对有关UO-Convergence的各种拓扑问题给出令人满意的答案。
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 $\mathrm{L}_0(X,Σ,μ)$ 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.
