An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

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• 作者：Ben Berckmoes ; ¹ ; ^{ben.berckmoes@uantwerpen.be} ; Tim Hellemans ; Mark Sioen ; Jan Van Casteren
• 关键词：Approach theory ; Ascoli's Theorem ; Dini's Theorem ; (Relative) Hausdorff measure of non-compactness ; Prokhorov's Theorem ; Wasserstein metric
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：449
• 期：2
• 页码：1770-1789
• 全文大小：420 K
• 卷排序：449

文摘

After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related to 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.