逼近理论在非compactness的Wasserstein度量的相对Hausdorff测度中的应用
An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric
摘要：
在简要回顾了R. Lowen在1989年的文章中，我们证明了这一理论与完备可分度量空间上具有有限一阶矩的概率测度空间上著名的Wasserstein度量密切相关.更确切地说，我们引入了一个规范的方法结构，称为压缩方法结构，并证明它是度量的Wasserstein度量。证明这个结果的关键要素是Dini定理，Ascoli定理，以及度量空间上的实值压缩类具有一些很好的稳定性。然后，我们结合所得到的结果与Prokhorov定理建立不等式之间的相对Hausdorff措施的非紧性的Wasserstein度量和规范措施的非一致可积性。
英文摘要：
After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.
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