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 modi/ed L1 evy 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 /eld. c © 2004 Elsevier B.V. All rights reserved.