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 L\'evy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.
Highlights
In the present work, we aim to give a topological framework to certain classes of measured metric spaces
Evans [7] and Evans, Pitman and Winter [8] considered the space of real trees, which is Polish when endowed with the Gromov-Hausdorff metric
In the monograph by Evans [7], the author describes a topology on the space of compact real trees, equipped with a probability measure, using the Prokhorov metric to compare the measures, defining the so-called weighted Gromov-Hausdorff metric
Summary
We aim to give a topological framework to certain classes of measured metric spaces. Evans [7] and Evans, Pitman and Winter [8] considered the space of real trees, which is Polish when endowed with the Gromov-Hausdorff metric This has given a framework to the theory of continuum random trees, which originated with Aldous [3]. In the monograph by Evans [7], the author describes a topology on the space of compact real trees, equipped with a probability measure, using the Prokhorov metric to compare the measures, defining the so-called weighted Gromov-Hausdorff metric. Theorem 2.3 ensures that (K, dcGHP) is a Polish metric space We extend those results by considering the Gromov-Hausdorff-Prokhorov metric, dGHP, on the set L of (isometry classes of) rooted locally compact, complete length spaces, endowed with a locally finite measure.
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