{
  "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) $\\mathscr P_\\sim(\\mathbb U)=\\mathscr P(\\mathbb U)/\\sim$ of the space $\\mathscr P(\\mathbb U)$ of Borel probability measures on $\\mathbb U$, where $μ\\simν$ if $ν$ is the pushforward of $μ$ 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 $\\mathscr P(\\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\\mathscr P(\\mathbb U)$ induces a complete and separable distance ${\\sf d}_{\\rm mG}$ on $\\mathbb X_1\\cong\\mathscr P_\\sim(\\mathbb U)$, which metrises the quotient topology of $\\mathscr P_\\sim(\\mathbb U)$ and is Hölder equivalent to the box distance $\\square$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24207",
  "author_total": 3,
  "authors": [
    {
      "name": "Milica Caković"
    },
    {
      "name": "Enrico Pasqualetto"
    },
    {
      "name": "Timo Schultz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fx127fhjyj22z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24207",
  "fulltext_url": null,
  "id": "fx127fhjyj22z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24207",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214039068",
  "pdf_url": "https://arxiv.org/pdf/2609.24207",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fx127fhjyj22z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24207",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24207"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Convergence of metric measure spaces via embeddings in the Urysohn universal space",
  "title_text": "Convergence of metric measure spaces via embeddings in the Urysohn universal space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fx127fhjyj22z",
  "venue": "arXiv",
  "versions": []
}