{"id":"https://openalex.org/W1974561446","doi":"https://doi.org/10.2307/2275608","title":"Lebesque measure zero subsets of the real line and an infinite game","display_name":"Lebesque measure zero subsets of the real line and an infinite game","publication_year":1996,"publication_date":"1996-03-01","ids":{"openalex":"https://openalex.org/W1974561446","doi":"https://doi.org/10.2307/2275608","mag":"1974561446"},"language":"en","primary_location":{"id":"doi:10.2307/2275608","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275608","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060683305","display_name":"Marion Scheepers","orcid":"https://orcid.org/0000-0001-8037-8158"},"institutions":[{"id":"https://openalex.org/I120156002","display_name":"Boise State University","ror":"https://ror.org/02e3zdp86","country_code":"US","type":"education","lineage":["https://openalex.org/I120156002"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Marion Scheepers","raw_affiliation_strings":["Department of Mathematics, Boise State University, Boise, Idaho 83725, E-mail:","Boise State University,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Boise State University, Boise, Idaho 83725, E-mail:","institution_ids":["https://openalex.org/I120156002"]},{"raw_affiliation_string":"Boise State University,","institution_ids":["https://openalex.org/I120156002"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5060683305"],"corresponding_institution_ids":["https://openalex.org/I120156002"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.4652,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.55752212,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"61","issue":"1","first_page":"246","last_page":"249"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9908999800682068,"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"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9908999800682068,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9200999736785889,"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/mathematics","display_name":"Mathematics","score":0.7657632827758789},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.7466458678245544},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.686933696269989},{"id":"https://openalex.org/keywords/lebesgue-measure","display_name":"Lebesgue measure","score":0.6820091605186462},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.6654927134513855},{"id":"https://openalex.org/keywords/assertion","display_name":"Assertion","score":0.6036838293075562},{"id":"https://openalex.org/keywords/real-line","display_name":"Real line","score":0.5798830389976501},{"id":"https://openalex.org/keywords/ideal","display_name":"Ideal (ethics)","score":0.5759087800979614},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.5458508133888245},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5347297191619873},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.505964457988739},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5040713548660278},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4607810080051422},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.4522545337677002},{"id":"https://openalex.org/keywords/real-number","display_name":"Real number","score":0.44484689831733704},{"id":"https://openalex.org/keywords/lebesgue-integration","display_name":"Lebesgue integration","score":0.3119678497314453},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07722070813179016}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7657632827758789},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.7466458678245544},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.686933696269989},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.6820091605186462},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.6654927134513855},{"id":"https://openalex.org/C40422974","wikidata":"https://www.wikidata.org/wiki/Q741248","display_name":"Assertion","level":2,"score":0.6036838293075562},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.5798830389976501},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.5759087800979614},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.5458508133888245},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5347297191619873},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.505964457988739},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5040713548660278},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4607810080051422},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4522545337677002},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.44484689831733704},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.3119678497314453},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07722070813179016},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2275608","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275608","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183744937","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183744937","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W2032234063","https://openalex.org/W4255866735"],"related_works":["https://openalex.org/W2230063236","https://openalex.org/W2917590914","https://openalex.org/W2074721578","https://openalex.org/W2374237741","https://openalex.org/W2035556505","https://openalex.org/W2038835257","https://openalex.org/W2318453636","https://openalex.org/W2387614002","https://openalex.org/W1974561446","https://openalex.org/W2057097927"],"abstract_inverted_index":{"Let":[0],"denote":[1,121],"the":[2,10,17,47,54,77,122,128,137,171,174],"ideal":[3],"of":[4,9,20,23,30,40,61,80,115,130,154,173],"Lebesgue":[5],"measure":[6],"zero":[7],"subsets":[8,79],"real":[11],"line.":[12],"Then":[13,170],"add(":[14,41,44],")":[15,45],"denotes":[16],"minimal":[18],"cardinality":[19,138],"a":[21,71],"subset":[22],"whose":[24],"union":[25],"is":[26,46,70,111,142,177],"not":[27,112,143],"an":[28,36,113],"element":[29,114],".":[31],"In":[32,127],"[1]":[33],"Bartoszynski":[34,133],"gave":[35],"elegant":[37],"combinatorial":[38],"characterization":[39],"),":[42],"namely:":[43],"least":[48],"cardinal":[49,215],"number":[50,216],"\u03ba":[51],"for":[52,98,158],"which":[53],"following":[55],"assertion":[56,123,175],"fails:":[57],"For":[58,85,187],"every":[59],"family":[60],"at":[62,90,192],"most\u03ba":[63],"functions":[64],"from":[65,74],"\u03c9":[66,68,75,81],"to":[67,76],"there":[69,102],"function":[72],"F":[73],"finite":[78],"such":[82,108,156,165],"that:":[83],"1.":[84],"each":[86,99,159,188],"m,":[87,189],"F(m)":[88,190],"has":[89,191],"most":[91,193],"m":[92,107,148,160],"+":[93],"1":[94,141,181],"elements,":[95],"and":[96,164,212],"2.":[97],"f":[100],"in":[101,140,180],"are":[103],"only":[104],"finitely":[105],"many":[106],"that":[109,136,157,166,185],"f(m)":[110],"F(m).":[116],"The":[117],"symbol":[118],"A(\u03ba)":[119,176],"will":[120],"above":[124],"about":[125,214],"\u03ba.":[126,217],"course":[129],"his":[131],"proof,":[132],"also":[134],"shows":[135],"restriction":[139],"sharp.":[144],"Indeed,":[145],"let":[146],"(Rm:":[147],"<":[149],"\u03c9)":[150],"be":[151],"any":[152],"sequence":[153],"integers":[155],"Rm,":[161],"\u2264":[162],"Rm+1,":[163],"limm\u2192\u221eRm":[167],"=":[168],"\u221e.":[169],"truth":[172],"preserved":[178],"if":[179],"we":[182],"say":[183],"instead":[184],"1\u2032.":[186],"Rm":[194],"elements.":[195],"We":[196,203],"shall":[197],"use":[198],"this":[199],"observation":[200],"later":[201],"on.":[202],"now":[204],"define":[205],"three":[206],"more":[207],"statements,":[208],"denoted":[209],"B(\u03ba),":[210],"C(\u03ba)":[211],"D(\u03ba),":[213]},"counts_by_year":[{"year":2017,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}