In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that both first‐ and second‐order stochastic dominance induce Dedekind super complete lattices, that is, lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably bounded set of probability measures is directed (for example, a lattice), then the supremum and infimum with respect to first‐order or second‐order stochastic dominance can be approximated by sequences in the weak topology or in the Wasserstein‐1 topology, respectively. As a consequence, we are able to prove that a sublattice of probability measures is complete with respect to first‐order stochastic dominance or second‐order stochastic dominance and increasing convex order if and only if it is compact in the weak topology or in the Wasserstein‐1 topology, respectively. This complements a set of characterizations of tightness and uniform integrability, which are discussed in a preliminary section.

中文翻译：

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关于随机优势，均匀可整合性和晶格性质的注释

在这项工作中，我们讨论一阶和二阶随机优势的晶格阶的完备性。主要结果表明，一阶和二阶随机优势都诱导了Dedekind超完备晶格，也就是说，其中每个有界非空子集都有一个具有相同的最小上界和最大下界的可数子集。此外，我们表明，如果定向了一组适当有界的概率测度（例如，格子），则一阶或二阶随机优势度的最高和最低点可以通过弱拓扑中的序列来近似或分别在Wasserstein-1拓扑中。作为结果，我们能够证明，当且仅当在弱拓扑或Wasserstein-1拓扑中是紧凑的时，关于一阶随机优势或二阶随机优势和递增凸序的概率测度的子格才是完整的，分别。这补充了一组紧密度和均匀可集成性的特征，这些特征在初步部分中进行了讨论。