A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
Metric Geometry
2013-01-28 v1 Probability
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.
Keywords
Cite
@article{arxiv.1202.5464,
title = {A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces},
author = {Romain Abraham and Jean-Francois Delmas and Patrick Hoscheit},
journal= {arXiv preprint arXiv:1202.5464},
year = {2013}
}