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

  • Milica CakoviΔ‡
  • Enrico Pasqualetto
  • Timo Schultz
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Abstract

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space π•Œ. Due to the universality of π•Œ, the collection 𝕏1 of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) π’«βˆΌ(π•Œ)=𝒫(π•Œ)/∼ of the space 𝒫(π•Œ) of Borel probability measures on π•Œ, where μ∼ν if Ξ½ is the pushforward of ΞΌ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of π•Œ, we show that, under the above identification, Gromov's box topology on 𝕏1 coincides with the quotient topology induced by the weak topology of 𝒫(π•Œ). More quantitatively, the truncated $1$-Wasserstein distance on 𝒫(π•Œ) induces a complete and separable distance sf dmG on 𝕏1β‰…π’«βˆΌ(π•Œ), which metrises the quotient topology of π’«βˆΌ(π•Œ) and is HΓΆlder equivalent to the box distance β–‘.

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Cite

BibTeXRIS
BibTeX
@misc{cakovic2026convergence,
  title = {{Convergence of metric measure spaces via embeddings in the Urysohn universal space}},
  author = {Milica Caković and Enrico Pasqualetto and Timo Schultz},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24207},
  eprint = {2609.24207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24207},
}
RIS
TY  - UNPB
TI  - Convergence of metric measure spaces via embeddings in the Urysohn universal space
AU  - Milica Caković
AU  - Enrico Pasqualetto
AU  - Timo Schultz
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24207
UR  - https://arxiv.org/abs/2609.24207
AB  - We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space π•Œ. Due to the universality of π•Œ, the collection 𝕏_1 of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) 𝒫_∼(π•Œ)=𝒫(π•Œ)/∼ of the space 𝒫(π•Œ) of Borel probability measures on π•Œ, where μ∼ν if Ξ½ is the pushforward of ΞΌ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of π•Œ, we show that, under the above identification, Gromov's box topology on 𝕏_1 coincides with the quotient topology induced by the weak topology of 𝒫(π•Œ). More quantitatively, the truncated $1$-Wasserstein distance on 𝒫(π•Œ) induces a complete and separable distance sf d_mG on 𝕏_1≅𝒫_∼(π•Œ), which metrises the quotient topology of 𝒫_∼(π•Œ) and is HΓΆlder equivalent to the box distance β–‘.
ER  -