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After shortly reviewing the fundamentals of approach theory as introduced byR. Lowen in 1989, we show that this theory is intimately related with thewell-known Wasserstein metric on the space of probability measures with afinite first moment on a complete and separable metric space. More precisely,we introduce a canonical approach structure, called the contractive approachstructure, and prove that it is metrized by the Wasserstein metric. The keyingredients 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 hassome nice stability properties. We then combine the obtained result withProkhorov's Theorem to establish inequalities between the relative Hausdorffmeasure of non-compactness for the Wasserstein metric and a canonical measureof non-uniform integrability.

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