# What does convergence in distribution "in the Gromov–Hausdorff" sense mean?

I am trying to understand this survey article 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?

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 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.