A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces

Abstract

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.the root and endowed with a finite measure. Theorem 2.3 ensures that (K, d c GHP ) is a Polish metric space. We extend those results by considering the Gromov-Hausdorff-Prokhorov metric, d GHP , on the set L of (isometry classes of) rooted locally compact, complete length spaces, endowed with a locally finite measure. Theorem 2.7 ensures that (L, d GHP ) is also a Polish metric space. The proof of the completeness of L relies on a pre-compactness criterion given in Theorem 2.9. The methods used are similar to the methods used in [4] to derive properties about the Gromov-Hausdorff topology of the set of locally compact complete length spaces. This work extends some of the results from [9], which doesn't take into account the geometrical structure of the spaces, as well as the results from [11], which consider only the compact case and probability measures. This comes at the price of having to restrict ourselves to the context of length spaces. In [12] the Gromov-Hausdorff-Prokhorov topology is considered for general Polish spaces (instead of length spaces) but endowed with locally finite measures satisfying the doubling condition. We also mention the different approach of [2], using the ideas of correspondences between metric spaces and couplings of measures.This work was developed for applications in the setting of weighted real trees (which are elements of L), see Abraham, Delmas and Hoscheit [1]. We give an hint of those applications by stating that the construction of a weighted tree coded in a continuous function with compact support is measurable with respect to the topology induced by d c GHP on K or by d GHP on L. This construction allows us to define random variables on K using continuous random processes on R, in particular the Lévy trees of [6] that describe the genealogy of the so-called critical or sub-critical continuous state branching processes that become a.s. extinct. The measure m is then a "uniform" measure on the leaves of the tree which has finite mass. The construction can be generalized to super-critical continuous state branching processes which can live forever; in that case the corresponding genealogical tree is infinite and the measure m on the leaves is also infinite. This paper gives an appropriate framework to handle such tree-valued random variables and also tree-valued Markov processes as in [1].The structure of the paper is as follow. Section 2 collects the main results of the paper. The applicat...

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