I am trying to understand [this survey article](https://arxiv.org/abs/1810.02664) by Le Gall on Brownian geometry, especially the statement of Theorem 1.


The basic statement of the theorem is
$$(m\_n,d\_n) \to (m\_{\infty}, d\_{\infty})$$
"in the Gromov–Hausdorff sense" as $n \to \infty$, where the convergence is in distribution.


Here $(m\_n,d\_n)$ and $(m\_{\infty},d\_{\infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.


For every compact metric space $(X,d)$ and $R > 0$, we have
$$(\*) \, \, \, \mathbb{P} \left[ d\_{GH}[ (m\_n,d\_n), (X,d) ] < R \right] \to \mathbb{P} \left[ d\_{GH}[ (m\_{\infty},d\_{\infty}), (X,d) ] < R \right]$$
as $n \to \infty$.


But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (\*) is too strong?



What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
Following the notation of the paper, let $\mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $\mathrm{d\_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : \mathbb{K} \to \mathbb{R}$, we have $\mathbb{E}[F((m\_n, d\_n))] \to \mathbb{E}[F((m\_\infty, d\_\infty))]$. The [portmanteau theorem](https://en.wikipedia.org/wiki/Convergence_of_measures#Weak_convergence_of_measures) gives you several other equivalent statements.


In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (\mathbb{K}, \mathrm{d\_{GH}})$, the metric space of all compact metric spaces.


In particular, if $(X,d)$ is a fixed compact metric space, the function $\mathrm{d\_{GH}}(\cdot, (X,d)) : \mathbb{K} \to \mathbb{R}$ is a continuous function. So if we let $Y\_n = \mathrm{d\_{GH}}((m\_n, d\_n), (X, d))$, then the scalar-valued random variables $Y\_n$ converge in distribution to $Y$. So your formula (\*) holds, but as you say, only for values of $R$ at which the function $R \mapsto \mathbb{P}[\mathrm{d\_{GH}}((m\_\infty, d\_\infty), (X,d)) < R]$ is continuous.


