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.
"A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces." Electron. J. Probab. 18 1 - 21, 2013. https://doi.org/10.1214/EJP.v18-2116