A note for Gromovs distance functions on the space of mm-spaces

This is just a note for \cite[Chapter$3{1/2}_+$]{gromov}. Maybe this note is obvious for a reader who knows metric geometry. I wish that someone study further in this direction.

💡 Research Summary

The paper is a concise note that revisits the distance functions introduced by Mikhail Gromov on the space of metric‑measure (mm) spaces. Although the original material appears in Chapter 3½ of Gromov’s monograph, many readers find the definitions and basic properties scattered across the text. This note gathers those definitions, clarifies subtle points, and records a few elementary but useful facts that are often taken for granted.

First, the author recalls the notion of an mm‑space ((X,d,\mu)), where ((X,d)) is a complete separable metric space and (\mu) is a Borel probability measure. Three principal distances are considered: the Gromov‑Hausdorff distance (d_{GH}), which compares only the metric structures; the Gromov‑Prokhorov distance (d_{GP}), which compares the measures via the Prokhorov metric after an isometric embedding; and the combined Gromov‑Hausdorff‑Prokhorov distance (d_{GHP}), defined as

\