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We study a new class of distances between Radon measures similar to thosestudied in a recent paper of Dolbeault-Nazaret-Savar\'e [DNS]. These distances(more correctly pseudo-distances because can assume the value $+\infty$) aredefined generalizing the dynamical formulation of the Wasserstein distance bymeans of a concave mobility function. We are mainly interested in the physicalinteresting case (not considered in [DNS]) of a concave mobility functiondefined in a bounded interval. We state the basic properties of the space ofmeasures endowed with this pseudo-distance. Finally, we study in detail twocases: the set of measures defined in $R^d$ with finite moments and the set ofmeasures defined in a bounded convex set. In the two cases we give sufficientconditions for the convergence of sequences with respect to the distance and weprove a property of boundedness.

On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals