Riemannian manifolds , spaces of measures and the Gromov - Hausdorff distance ∗

We equip the space M(X) of all Borel probability measures an a compact Riemannian manifold X with a canonical distance function which induces the weak-∗ topology on M(X) and has the following property: the map X 7→ M(X) is Lipschitz continous with respect to the Gromov-Hausdorff distance on the space of all the (isometry classes of) compact metric spaces. Introduction Last century brought several attempts of metrization of spaces of compact subsets of metric spaces. The most known is that due to Hausdorff [Ha]: given two compact subsets A and B of a metric space X , the Hausdorff distance dH(A,B) is defined as the lower bound of non-negative such that A is contained in the -neighbourhood of B and B is contained in the -neighbourhood of A. This distance doesn’t recognize the topology of A and B: for instance, the Hausdorff distance of I = [0, 1] and Ik = {0, 1/k, 2/k, . . . , 1} converges to 0 as k → ∞ but topologically I and Ik are pretty different. This is probably why Borsuk [Bo1] (see also [Bo2]) defined other distances (called by him continuity and homotopy metrics) which reflect topological similarity of the sets. A bit unfortunate feature of Borsuk metrics is that the continuity metric dB is simple and intuitive but not complete while his homotopy distance dB has a clear sense only for ANR’s (i.e. absolute neighbourhood retracts). Given a compact metric space X of finite dimension, the class of all ANR’s contained in X is complete with respect to dB. Therefore, d h B seems to be adequate to measure topological difference of ANR’s. AMS Subject Classification: 53 C 23, 60 B 05.