A combinatorial idea of Gromov is to assign to each metric space X the collection of all distance matrices corresponding to all possible n-tuples of points in X. Given a filtration the functor F on finite metric spaces we consider the set of all possible F-persistence diagrams generated by metric subsets of X of cardinality n. For a class of filtration functors which we call compatible, the answer is positive, and these admit stability results in the Gromov-Hausdorff sense. In order to capture frequency or statistics, it is more useful to consider that, in addition to a metric structure, a probability measure has been specified. Then, given n, to an mm-space X one assigns a probability measure Un induced by pushing forward the reference probability measure on X into the space of all F-persistence diagrams on n-point samples of X. The stability of these constructions can now be expressed in Gromov-Wasserstein sense. As a consequence of this, one can now establish the concentration of the measure Un as n goes to infinity.