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A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. (English) Zbl 1285.60004
Summary: 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.

MSC:
60B05 Probability measures on topological spaces
54E50 Complete metric spaces
05C80 Random graphs (graph-theoretic aspects)
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