  Given a metric space with a Borel probability measure, for each integer $N$
we obtain a probability distribution on $N\times N$ distance matrices by
considering the distances between pairs of points in a sample consisting of $N$
points chosen indepenedently 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.
