## Electronic Journal of Probability

### A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric 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 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.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 14, 21 pp.

Dates
Accepted: 24 January 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064239

Digital Object Identifier
doi:10.1214/EJP.v18-2116

Mathematical Reviews number (MathSciNet)
MR3035742

Zentralblatt MATH identifier
1285.60004

Subjects
Primary: 60B05: Probability measures on topological spaces

Rights

#### Citation

Abraham, Romain; Delmas, Jean-François; Hoscheit, Patrick. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 (2013), paper no. 14, 21 pp. doi:10.1214/EJP.v18-2116. https://projecteuclid.org/euclid.ejp/1465064239

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