Compact probabilistic metrics on bounded closed intervals of distribution functions

A Menger space (a special type of probabilistic metric spaces) is said to be compact if its strong uniformity is compact. We construct, in a natural way, Menger T-metrics on the set of distribution functions, one for each copula T (a special type of continuous t-norms). We show that, on each bounded closed interval of distribution functions, the strong uniformities of all our spaces are induced by the modified Lévy metric. Hence, they are compact. We establish an alternative description and a number of good properties of our Menger spaces, which may render them well-behaved examples for workers in the field.