Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

Abstract

In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov–Hausdorff topology, is a Polish space. The same is done for the Gromov–Hausdorff–Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement to study random boundedly-compact pointed (measured) metric spaces, which is the main motivation of this work. In particular, this provides a unified framework for studying random graphs, random discrete spaces and random length spaces. The proofs use a generalization of the classical theorem of Strassen, presented here, which is of independent interest. This generalization provides an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in terms of approximate couplings. A Strassen-type result is also provided for the Gromov–Hausdorff–Prokhorov metric for compact spaces.