Convergence of metric measure spaces via embeddings in the Urysohn universal space

Abstract

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb{U}$. Due to the universality of $\mathbb{U}$, the collection ${\mathbb{X}}_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) ${\mathcal{P}}_{\sim }(\mathbb{U})=\mathcal{P}(\mathbb{U})/\sim$ of the space $\mathcal{P}(\mathbb{U})$ of Borel probability measures on $\mathbb{U}$, where $\mu \sim \nu$ if $\nu$ is the pushforward of $\mu$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb{U}$, we show that, under the above identification, Gromov's box topology on ${\mathbb{X}}_1$ coincides with the quotient topology induced by the weak topology of $\mathcal{P}(\mathbb{U})$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathcal{P}(\mathbb{U})$ induces a complete and separable distance ${\mathsf{d}}_{\mathrm{mG}}$ on ${\mathbb{X}}_1\cong {\mathcal{P}}_{\sim }(\mathbb{U})$, which metrises the quotient topology of ${\mathcal{P}}_{\sim }(\mathbb{U})$ and is Hölder equivalent to the box distance $\square$.

12 pages