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
