# Metrization of the Gromov-Hausdorff (-Prokhorov) Topology for Boundedly-Compact Metric Spaces

## TL;DR

This paper introduces new metrics on boundedly-compact metric and measured metric spaces that generate the Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov topologies, ensuring completeness and separability, and extends classical theorems for these spaces.

## Contribution

It defines and analyzes metrics for boundedly-compact (measured) metric spaces that generate relevant topologies, extending previous work and providing foundational tools for studying random such spaces.

## Key findings

- Metrics generate the Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov topologies.
- Metrics are complete and separable.
- Generalizes Strassen's theorem for approximate couplings.

## Abstract

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. Completeness and separability are also proved for these metrics. Hence, they provide the measure theoretic requirements to study random (measured) boundedly-compact pointed metric spaces, which is the main motivation of this work. In addition, we present a generalization of the classical theorem of Strassen which is of independent interest. This generalization proves an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in term of approximate coupling. A Strassen-type result is also proved for the Gromov-Hausdorff-Prokhorov metric for compact spaces.