Mass and radius of balls in Gromov-Hausdorff-Prokhorov convergent sequences

Abstract

We survey some properties of Gromov--Hausdorff--Prokhorov convergent sequences $(\mathsf{X}_n, d_{\mathsf{X}_n}, \nu_{\mathsf{X}_n})_{n \ge 1}$ of random compact metric spaces equipped with Borel probability measures. We formalize that if the limit is almost surely non-atomic, then for large $n$ each open ball in $\mathsf{X}_n$ with small radius must have small mass. Conversely, if the limit is almost surely fully supported, then each closed ball in $\mathsf{X}_n$ with small mass must have small radius. We do not claim any new results, but justifications are provided for properties for which we could not find explicit references.