RANDOM METRIC SPACES AND THE UNIVERSAL URYSOHN SPACE

ANATOLY VERSHIK

Abstract. We introduce a model of the set of all Polish (=separable complete metric) spaces which is the cone R of distance matrices, and consider the geometrical and probabilistic problems connected with this object. We prove that the generic Polish space in the sense of this model is the so called universal Urysohn space which was defined by P.S.Urysohn in the 1920-th. Then we consider the metric spaces with measures (metric triples) and define a complete invariant of its - matrix distribution. We give an intrinsic characterization of matrix distribution and using the ergodic theorem give a new proof of Gromov’s reconstruction theorem. A natural construction of a wide class of measures on the cone R is given and for these we show that with probability one the random Polish space is again the Urysohn space. There is a tight link of these questions with metric classification of measurable functions of several arguments and classification of the actions of infinite symmetric group. Applications to the statistical theory of metric space will follow.