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

TL;DR

This paper explores the relationship between approach theory and the Wasserstein metric on probability measures, establishing a new approach structure and linking it to measures of non-compactness and non-uniform integrability.

Contribution

It introduces the contractive approach structure, showing it is metrized by the Wasserstein metric, and connects this to measures of non-compactness and integrability.

Findings

The contractive approach structure is metrized by the Wasserstein metric.

Established inequalities between Hausdorff measure of non-compactness and non-uniform integrability.

Linked approach theory with classical probability measure compactness results.

Abstract

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.