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

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 Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.
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https://hal.archives-ouvertes.fr/hal-00673921
Contributor : Patrick Hoscheit <>
Submitted on : Friday, February 24, 2012 - 3:14:21 PM
Last modification on : Saturday, May 30, 2020 - 3:39:47 AM
Document(s) archivé(s) le : Friday, May 25, 2012 - 2:26:03 AM

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

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