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In this paper, an approach for generalizing the Gromov-Hausdorff metric ispresented, which applies to metric spaces equipped with some additionalstructure. A special case is the Gromov-Hausdorff-Prokhorov metric betweenmeasured metric spaces. This abstract framework unifies several existingGromov-Hausdorff-type metrics for metric spaces equipped with a measure, apoint, a closed subset, a curve, a tuple of such structures, etc. Along withreviewing these special cases in the literature, several new examples are alsopresented. Two frameworks are provided, one for compact metric spaces and theother for boundedly-compact pointed metric spaces. In both cases, aGromov-Hausdorff-type metric is defined and its topological properties arestudied. In particular, completeness and separability is proved under someconditions. This enables one to study random metric spaces equipped withadditional structures, which is the main motivation of this work.

A Unified Framework for Generalizing the Gromov-Hausdorff Metric