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

Romain Abraham (MAPMO, Université d'Orléans)
Jean-François Delmas (CERMICS, École des Ponts Paristech)
Patrick Hoscheit (CERMICS, École des Ponts Paristech)


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 boundedly 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 boundedly finite measure.

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Pages: 1-21

Publication Date: January 24, 2013

DOI: 10.1214/EJP.v18-2116


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