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

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

2012-02-24

arxiv.org/abs/1202.5464v1

Mathematics

Metric Geometry

Scientific paper

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 L\'evy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.

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