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

Romain Abraham (1) , Jean-François Delmas (2) , Patrick Hoscheit (2, 1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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 Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.

Mots clés: Gromov-Hausdorff; Prokhorov metric; length space; Levy tree; boundedly finite measure

Domaines: Géométrie métrique [math.MG]; Probabilités [math.PR]

Romain Abraham, Jean-François Delmas, Patrick Hoscheit. A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces. Electronic Journal of Probability, 2013, 18, pp.14. ⟨10.1214/EJP.v18-2116⟩. ⟨hal-00673921⟩