Gromov-Hausdorff-Prokhorov收敛序列中球的质量和半径
Mass and radius of balls in Gromov-Hausdorff-Prokhorov convergent sequences
摘要：
本文研究了随机紧度量空间中具有Borel概率测度的Gromov-Hausdorff-Prokhorov收敛序列$(\mathsf{X}_n, d_{\mathsf{X}_n}, ν_{\mathsf{X}_n})_{n \ge 1}$的一些性质。我们形式化了，如果极限几乎必然是非原子的，那么对于大的$n$，$\mathsf{X}_n$中每个小半径的开球必须有小质量。相反，如果极限几乎必然完全支持，则$\mathsf{X}_n$中每个小质量的闭球必须具有小半径。我们不要求任何新的结果，但提供的理由，我们无法找到明确的参考属性。
英文摘要：
We survey some properties of Gromov--Hausdorff--Prokhorov convergent sequences $(\mathsf{X}_n, d_{\mathsf{X}_n}, ν_{\mathsf{X}_n})_{n \ge 1}$ 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.
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