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Given a metric space with a Borel probability measure, for each integer  N , we obtain a probability distribution on  N  ×  N  distance matrices by considering the distances between pairs of points in a sample consisting of  N  points chosen independently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.

Lipschitz Correspondence between Metric Measure Spaces and Random Distance Matrices