随机集和Choquet型表示
Random sets and Choquet-type representations
摘要：
作为具有不可数项的凸组合的适当推广，我们引入了所谓的Choquet组合、Choquet分解和Choquet凸分解，以及它们相应的作用在Lebesgue-Bochner空间幂集上的壳算子.我们表明，Choquet壳符合凸包在有限维设置，但Choquet壳往往是更大的无限维。我们还提供了Choquet船体的定量表征。进一步证明了一个集合的Choquet可分解壳与它的（强）闭可分解壳重合，一个集合的Choquet凸可分解壳与它的凸包的Choquet可分解壳重合.证明了闭值函数的所有可测选择的集合是Choquet可分解的，而闭凸值函数的所有可测选择的集合是Choquet凸可分解的。最后，我们研究了Choquet可分解和Choquet凸可分解壳算子连续应用时的算子型特征。
英文摘要：
As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decompositions and Choquet convex decompositions, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull. Furthermore, we show that Choquet decomposable hull of a set coincides with its (strongly) closed decomposable hull and the Choquet convex decomposable hull of a set coincides with its Choquet decomposable hull of the convex hull. It turns out that the collection of all measurable selections of a closed-valued multifunction is Choquet decomposable and those of a closed convex-valued multifunction is Choquet convex decomposable. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.
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