{"id":"https://openalex.org/W7204829891","doi":"https://doi.org/10.48550/arxiv.2608.28330","title":"Realising atomless laws as distance distributions on metric measure spaces","display_name":"Realising atomless laws as distance distributions on metric measure spaces","publication_year":2026,"publication_date":"2026-08-28","ids":{"openalex":"https://openalex.org/W7204829891","doi":"https://doi.org/10.48550/arxiv.2608.28330"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.28330","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28330","pdf_url":"https://arxiv.org/pdf/2608.28330","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.28330","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045045693","display_name":"Christoph Th\u00e4le","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Th\u00e4le, Christoph","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5115616809","display_name":"Philipp Tuchel","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tuchel, Philipp","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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":0.6963000297546387,"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":0.6963000297546387,"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.1656000018119812,"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"}},{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.10029999911785126,"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/distance-distribution","display_name":"distance distribution","score":0.8059999942779541},{"id":"https://openalex.org/keywords/metric-measure-spaces","display_name":"metric measure spaces","score":0.9390000104904175},{"id":"https://openalex.org/keywords/complete-separable-metric-spaces","display_name":"complete separable metric spaces","score":0.796999990940094},{"id":"https://openalex.org/keywords/ultrametric-spaces","display_name":"ultrametric spaces","score":0.7250000238418579},{"id":"https://openalex.org/keywords/bi-lipschitz-equivalence","display_name":"bi-Lipschitz equivalence","score":0.9419999718666077},{"id":"https://openalex.org/keywords/singular-measures","display_name":"singular measures","score":0.4399999976158142}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8842999935150146},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.7263000011444092},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.6934999823570251},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.6313999891281128},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.589900016784668},{"id":"https://openalex.org/C118733216","wikidata":"https://www.wikidata.org/wiki/Q332504","display_name":"Absolute continuity","level":2,"score":0.5666999816894531},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5626999735832214},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5590000152587891},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4749999940395355},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.4625999927520752},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4562999904155731},{"id":"https://openalex.org/C166758865","wikidata":"https://www.wikidata.org/wiki/Q892585","display_name":"Borel measure","level":3,"score":0.42730000615119934},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.4253999888896942},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4025999903678894},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.3935999870300293},{"id":"https://openalex.org/C39952206","wikidata":"https://www.wikidata.org/wiki/Q21086747","display_name":"Borel set","level":2,"score":0.3749000132083893},{"id":"https://openalex.org/C5278979","wikidata":"https://www.wikidata.org/wiki/Q4944905","display_name":"Borel equivalence relation","level":4,"score":0.3490000069141388},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.3450999855995178},{"id":"https://openalex.org/C31316033","wikidata":"https://www.wikidata.org/wiki/Q1207972","display_name":"Polish space","level":3,"score":0.3384000062942505},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.3328000009059906},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.3255000114440918},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.3019999861717224},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.298799991607666},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C25942563","wikidata":"https://www.wikidata.org/wiki/Q893487","display_name":"Borel hierarchy","level":4,"score":0.2955999970436096},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.2883000075817108},{"id":"https://openalex.org/C41261874","wikidata":"https://www.wikidata.org/wiki/Q3855712","display_name":"Intrinsic metric","level":4,"score":0.27570000290870667},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.26579999923706055},{"id":"https://openalex.org/C93361540","wikidata":"https://www.wikidata.org/wiki/Q848569","display_name":"Complete metric space","level":3,"score":0.2639999985694885},{"id":"https://openalex.org/C143924541","wikidata":"https://www.wikidata.org/wiki/Q258374","display_name":"Outer measure","level":5,"score":0.2581999897956848}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.28330","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28330","pdf_url":"https://arxiv.org/pdf/2608.28330","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.28330","doi":"https://doi.org/10.48550/arxiv.2608.28330","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.28330","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.28330","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28330","pdf_url":"https://arxiv.org/pdf/2608.28330","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3617462173","display_name":null,"funder_award_id":"SPP 2265","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8141651824","display_name":"Grenzwerts\u00e4tze f\u00fcr das Volumen zuf\u00e4lliger Projektionen von l_pB\u00e4llen","funder_award_id":"516672205","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204829891.pdf","grobid_xml":"https://content.openalex.org/works/W7204829891.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4415736337","https://openalex.org/W2031176639","https://openalex.org/W2067616646","https://openalex.org/W7114925592","https://openalex.org/W4416686522","https://openalex.org/W1614609926","https://openalex.org/W2779223392","https://openalex.org/W2026858861","https://openalex.org/W1602822304","https://openalex.org/W1607357020"],"abstract_inverted_index":{"We":[0,32],"study":[1],"which":[2],"probability":[3,30,38],"measures":[4,72],"on":[5],"$[0,\\infty)$":[6],"can":[7,40],"occur":[8],"as":[9],"the":[10,13,53,57,88,102,114,119],"distribution":[11],"of":[12,59,113],"distance":[14],"between":[15],"two":[16],"independent":[17],"points":[18],"sampled":[19],"from":[20,104],"a":[21,28],"complete":[22],"separable":[23],"metric":[24,90],"space":[25,103],"equipped":[26],"with":[27,76],"Borel":[29,37],"measure.":[31,80],"prove":[33],"that":[34,73,87],"an":[35,97],"atomless":[36,71],"measure":[39],"be":[41,92],"realised":[42],"in":[43],"this":[44],"way":[45],"if":[46,49],"and":[47,62,68],"only":[48],"its":[50],"support":[51],"contains":[52],"origin.":[54],"This":[55],"settles":[56],"conjecture":[58],"Aldous,":[60],"Blanc,":[61],"Curien":[63],"for":[64,82],"absolutely":[65],"continuous":[66],"laws":[67],"also":[69],"covers":[70],"are":[74,121],"singular":[75],"respect":[77],"to":[78,96,123],"Lebesgue":[79],"Moreover,":[81],"every":[83],"$L>1$,":[84],"we":[85],"show":[86],"realising":[89],"may":[91],"chosen":[93],"$L$-bi-Lipschitz":[94],"equivalent":[95],"ultrametric.":[98],"The":[99],"proof":[100],"constructs":[101],"compact":[105],"components":[106,120],"whose":[107],"mutual":[108],"distances":[109,117],"produce":[110],"prescribed":[111],"parts":[112],"measure,":[115],"while":[116],"within":[118],"assigned":[122],"smaller":[124],"scales.":[125]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-09-01T00:00:00"}
