{"id":"https://openalex.org/W7214039068","doi":"https://doi.org/10.48550/arxiv.2609.24207","title":"Convergence of metric measure spaces via embeddings in the Urysohn universal space","display_name":"Convergence of metric measure spaces via embeddings in the Urysohn universal space","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214039068","doi":"https://doi.org/10.48550/arxiv.2609.24207"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24207","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24207","pdf_url":"https://arxiv.org/pdf/2609.24207","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.24207","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5145617815","display_name":"Milica Cakovi\u0107","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cakovi\u0107, Milica","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009413352","display_name":"Enrico Pasqualetto","orcid":"https://orcid.org/0000-0002-7532-5577","observed_orcids":["https://orcid.org/0000-0002-7532-5577"]},"institutions":[{"id":"https://openalex.org/I170726198","display_name":"University of Novi Sad","ror":"https://ror.org/00xa57a59","country_code":"RS","type":"education","lineage":["https://openalex.org/I170726198"]},{"id":"https://openalex.org/I94722563","display_name":"University of Jyv\u00e4skyl\u00e4","ror":"https://ror.org/05n3dz165","country_code":"FI","type":"education","lineage":["https://openalex.org/I94722563"]}],"countries":["FI","RS"],"is_corresponding":false,"raw_author_name":"Pasqualetto, Enrico","raw_affiliation_strings":["Depart- ment of Mathematics and Informatics , Faculty of Sciences , University of Novi Sad , Trg Dositeja Obradovi\u0107a 3 , 21000 Novi Sad , Serbia","Department of Mathematics and Statistics , P.O. Box 35 , FI-40014 University of Jyvaskyla ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Depart- ment of Mathematics and Informatics , Faculty of Sciences , University of Novi Sad , Trg Dositeja Obradovi\u0107a 3 , 21000 Novi Sad , Serbia","institution_ids":["https://openalex.org/I170726198"]},{"raw_affiliation_string":"Department of Mathematics and Statistics , P.O. Box 35 , FI-40014 University of Jyvaskyla ;","institution_ids":["https://openalex.org/I94722563"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152484520","display_name":"Timo Schultz","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I94722563","display_name":"University of Jyv\u00e4skyl\u00e4","ror":"https://ror.org/05n3dz165","country_code":"FI","type":"education","lineage":["https://openalex.org/I94722563"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Schultz, Timo","raw_affiliation_strings":["Department of Mathematics and Statistics , P.O. Box 35 , FI-40014 University of Jyvaskyla ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , P.O. Box 35 , FI-40014 University of Jyvaskyla ;","institution_ids":["https://openalex.org/I94722563"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/metric-measure-spaces","display_name":"metric measure spaces","score":0.9010000228881836},{"id":"https://openalex.org/keywords/isometric-embedding","display_name":"isometric embedding","score":0.9449999928474426},{"id":"https://openalex.org/keywords/borel-probability-measures","display_name":"Borel probability measures","score":0.796999990940094},{"id":"https://openalex.org/keywords/quotient-topology","display_name":"quotient topology","score":0.718999981880188},{"id":"https://openalex.org/keywords/weak-topology","display_name":"weak topology","score":0.8399999737739563},{"id":"https://openalex.org/keywords/wasserstein-1-distance","display_name":"1-Wasserstein distance","score":0.9139999747276306}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8216000199317932},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.6747999787330627},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.5756000280380249},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.5468999743461609},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5033000111579895},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.49399998784065247},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.47099998593330383},{"id":"https://openalex.org/C129089157","wikidata":"https://www.wikidata.org/wiki/Q621550","display_name":"General topology","level":3,"score":0.4575999975204468},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.4422000050544739},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4171000123023987},{"id":"https://openalex.org/C47776270","wikidata":"https://www.wikidata.org/wiki/Q1455249","display_name":"Topological vector space","level":3,"score":0.4142000079154968},{"id":"https://openalex.org/C105433845","wikidata":"https://www.wikidata.org/wiki/Q1194053","display_name":"Metrization theorem","level":3,"score":0.4088999927043915},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4023999869823456},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.4004000127315521},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C191226916","wikidata":"https://www.wikidata.org/wiki/Q5384710","display_name":"Equivalence of metrics","level":4,"score":0.3817000091075897},{"id":"https://openalex.org/C145084290","wikidata":"https://www.wikidata.org/wiki/Q2713824","display_name":"Metric map","level":4,"score":0.36579999327659607},{"id":"https://openalex.org/C84496893","wikidata":"https://www.wikidata.org/wiki/Q1135782","display_name":"Urysohn and completely Hausdorff spaces","level":4,"score":0.35670000314712524},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35530000925064087},{"id":"https://openalex.org/C93361540","wikidata":"https://www.wikidata.org/wiki/Q848569","display_name":"Complete metric space","level":3,"score":0.3546000123023987},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C41261874","wikidata":"https://www.wikidata.org/wiki/Q3855712","display_name":"Intrinsic metric","level":4,"score":0.350600004196167},{"id":"https://openalex.org/C129866940","wikidata":"https://www.wikidata.org/wiki/Q505760","display_name":"Weak topology (polar topology)","level":5,"score":0.34700000286102295},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.34450000524520874},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C129301229","wikidata":"https://www.wikidata.org/wiki/Q741865","display_name":"Uniform continuity","level":3,"score":0.32600000500679016},{"id":"https://openalex.org/C130805567","wikidata":"https://www.wikidata.org/wiki/Q1162676","display_name":"Haar measure","level":2,"score":0.32499998807907104},{"id":"https://openalex.org/C154497576","wikidata":"https://www.wikidata.org/wiki/Q7247777","display_name":"Product metric","level":3,"score":0.32409998774528503},{"id":"https://openalex.org/C166359235","wikidata":"https://www.wikidata.org/wiki/Q1150180","display_name":"Baire space","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C5278979","wikidata":"https://www.wikidata.org/wiki/Q4944905","display_name":"Borel equivalence relation","level":4,"score":0.3068999946117401},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.296999990940094},{"id":"https://openalex.org/C6419321","wikidata":"https://www.wikidata.org/wiki/Q1669863","display_name":"Compact-open topology","level":5,"score":0.2957000136375427},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.2791999876499176},{"id":"https://openalex.org/C31316033","wikidata":"https://www.wikidata.org/wiki/Q1207972","display_name":"Polish space","level":3,"score":0.27880001068115234},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.27790001034736633},{"id":"https://openalex.org/C162329640","wikidata":"https://www.wikidata.org/wiki/Q332648","display_name":"Covering space","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C165263651","wikidata":"https://www.wikidata.org/wiki/Q2443460","display_name":"Topological conjugacy","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.25609999895095825},{"id":"https://openalex.org/C55606962","wikidata":"https://www.wikidata.org/wiki/Q1572094","display_name":"Product measure","level":3,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24207","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24207","pdf_url":"https://arxiv.org/pdf/2609.24207","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.24207","doi":"https://doi.org/10.48550/arxiv.2609.24207","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24207","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.24207","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24207","pdf_url":"https://arxiv.org/pdf/2609.24207","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214039068.pdf","grobid_xml":"https://content.openalex.org/works/W7214039068.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W3129359075"],"related_works":["https://openalex.org/W4394722982","https://openalex.org/W3043009603","https://openalex.org/W2047098573","https://openalex.org/W2516124909","https://openalex.org/W7101081117","https://openalex.org/W4323706380","https://openalex.org/W2414373602","https://openalex.org/W1021519951","https://openalex.org/W2053844940","https://openalex.org/W1996508495"],"abstract_inverted_index":{"We":[0],"study":[1],"different":[2],"notions":[3],"of":[4,6,12,27,34,37,55,61,75,88,91,117,150],"convergence":[5],"metric":[7,19,39],"measure":[8,40],"spaces":[9,41],"by":[10,113],"means":[11],"isometric":[13],"embeddings":[14],"into":[15],"the":[16,25,30,47,56,73,89,98,109,114,123,147,159],"Urysohn":[17],"universal":[18],"space":[20,57],"$\\mathbb":[21,28,32,66,92,105,141],"U$.":[22],"Due":[23],"to":[24,158],"universality":[26],"U$,":[29,67,93],"collection":[31],"X_1$":[33,106],"isomorphism":[35],"classes":[36],"normalised":[38],"can":[42],"be":[43],"canonically":[44],"identified":[45],"with":[46,108],"quotient":[48,110,148],"(set)":[49],"$\\mathscr":[50,58,118,128,151],"P_\\sim(\\mathbb":[51,143,152],"U)=\\mathscr":[52],"P(\\mathbb":[53,59,119,129],"U)/\\sim$":[54],"U)$":[60,130,153],"Borel":[62],"probability":[63],"measures":[64],"on":[65,104,127,140],"where":[68],"$\u03bc\\sim\u03bd$":[69],"if":[70],"$\u03bd$":[71],"is":[72,155],"pushforward":[74],"$\u03bc$":[76],"under":[77,97],"an":[78],"isometry":[79],"between":[80],"their":[81],"respective":[82],"supports.":[83],"By":[84],"making":[85],"crucial":[86],"use":[87],"ultrahomogeneity":[90],"we":[94],"show":[95],"that,":[96],"above":[99],"identification,":[100],"Gromov's":[101],"box":[102,160],"topology":[103,111,116,149],"coincides":[107],"induced":[112],"weak":[115],"U)$.":[120],"More":[121],"quantitatively,":[122],"truncated":[124],"$1$-Wasserstein":[125],"distance":[126,136,161],"induces":[131],"a":[132],"complete":[133],"and":[134,154],"separable":[135],"${\\sf":[137],"d}_{\\rm":[138],"mG}$":[139],"X_1\\cong\\mathscr":[142],"U)$,":[144],"which":[145],"metrises":[146],"H\u00f6lder":[156],"equivalent":[157],"$\\square$.":[162]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}