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?