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

TL;DR

This paper extends the Gromov-Hausdorff-Prokhorov metric to non-compact, locally compact length spaces with measures, establishing it as a Polish space, facilitating the study of Lévy trees with locally finite measures.

Contribution

It introduces a new metric for non-compact measure spaces and proves its properties, enabling analysis of Lévy trees in a rigorous metric framework.

Findings

Extended the Gromov-Hausdorff-Prokhorov metric to non-compact spaces.

Proved the space with this metric is a Polish space.

Facilitated the rigorous study of Lévy trees with locally finite measures.

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.