{"id":"https://openalex.org/W2026065145","doi":"https://doi.org/10.2478/forma-2013-0003","title":"Random Variables and Product of Probability Spaces","display_name":"Random Variables and Product of Probability Spaces","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2026065145","doi":"https://doi.org/10.2478/forma-2013-0003","mag":"2026065145"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2013-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2013-0003","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2013-0003","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://sciendo.com/pdf/10.2478/forma-2013-0003","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048322538","display_name":"Hiroyuki Okazaki","orcid":"https://orcid.org/0009-0000-1022-361X"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroyuki Okazaki","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I137975476"],"apc_list":null,"apc_paid":null,"fwci":1.4899,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.82428196,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"21","issue":"1","first_page":"33","last_page":"39"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9865999817848206,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9865999817848206,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.9178000092506409,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.90829998254776,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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":0.792456865310669},{"id":"https://openalex.org/keywords/algebra-of-random-variables","display_name":"Algebra of random variables","score":0.7302427291870117},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.6607365608215332},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.6302742958068848},{"id":"https://openalex.org/keywords/random-element","display_name":"Random element","score":0.555336058139801},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5051708817481995},{"id":"https://openalex.org/keywords/sum-of-normally-distributed-random-variables","display_name":"Sum of normally distributed random variables","score":0.49800777435302734},{"id":"https://openalex.org/keywords/random-compact-set","display_name":"Random compact set","score":0.4952212870121002},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.49395662546157837},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4847671389579773},{"id":"https://openalex.org/keywords/convergence-of-random-variables","display_name":"Convergence of random variables","score":0.40057554841041565},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.13961687684059143}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.792456865310669},{"id":"https://openalex.org/C89231458","wikidata":"https://www.wikidata.org/wiki/Q4723976","display_name":"Algebra of random variables","level":4,"score":0.7302427291870117},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.6607365608215332},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.6302742958068848},{"id":"https://openalex.org/C52386014","wikidata":"https://www.wikidata.org/wiki/Q166211","display_name":"Random element","level":3,"score":0.555336058139801},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5051708817481995},{"id":"https://openalex.org/C182514268","wikidata":"https://www.wikidata.org/wiki/Q7636889","display_name":"Sum of normally distributed random variables","level":4,"score":0.49800777435302734},{"id":"https://openalex.org/C52627884","wikidata":"https://www.wikidata.org/wiki/Q4424101","display_name":"Random compact set","level":4,"score":0.4952212870121002},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.49395662546157837},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4847671389579773},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.40057554841041565},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.13961687684059143},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.2478/forma-2013-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2013-0003","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2013-0003","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:86b3afda-d592-4c28-8e64-0afed7479f02/oai:repozytorium.uwb.edu.pl:11320/3670","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401248","display_name":"Jagiellonian University Repository (Jagiellonian University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I126596746","host_organization_name":"Jagiellonian University","host_organization_lineage":["https://openalex.org/I126596746"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:doaj.org/article:c8b658a9ddb34299a4116071f26ef286","is_oa":false,"landing_page_url":"https://doaj.org/article/c8b658a9ddb34299a4116071f26ef286","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 21, Iss 1, Pp 33-39 (2013)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/3670","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/3670","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2013-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2013-0003","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2013-0003","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2026065145.pdf","grobid_xml":"https://content.openalex.org/works/W2026065145.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W1979928600","https://openalex.org/W2105559527","https://openalex.org/W2119506873","https://openalex.org/W2148014389","https://openalex.org/W2148468670","https://openalex.org/W2149602715","https://openalex.org/W2150348213","https://openalex.org/W2163641194","https://openalex.org/W2296232473","https://openalex.org/W2304533832","https://openalex.org/W2552817710","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W121312646","https://openalex.org/W2250503047","https://openalex.org/W2183541783","https://openalex.org/W2259932122","https://openalex.org/W2037640110","https://openalex.org/W4246232962","https://openalex.org/W3195391652","https://openalex.org/W2389966714","https://openalex.org/W2383059660","https://openalex.org/W2548885433"],"abstract_inverted_index":{"Summary":[0],"We":[1],"have":[2],"been":[3],"working":[4],"on":[5,62,71],"the":[6,9,12,23,28,39,49,57,76],"formalization":[7,41],"of":[8,30,42,52,78],"probability":[10,32,81],"and":[11,16,27,44,55,66],"randomness.":[13],"In":[14,34],"[15]":[15,43],"[16],":[17],"we":[18,37,47,74],"formalized":[19],"some":[20],"theorems":[21],"concerning":[22],"real-valued":[24,68],"random":[25,50,60,69],"variables":[26,51],"product":[29,77],"two":[31],"spaces.":[33,82],"this":[35],"article,":[36],"present":[38],"generalized":[40],"[16].":[45],"First,":[46],"formalize":[48,75],"arbitrary":[53],"set":[54],"prove":[56],"equivalence":[58],"between":[59],"variable":[61,70],"\u03a3,":[63],"Borel":[64],"sets":[65],"a":[67],"\u03a3.":[72],"Next,":[73],"countably":[79],"infinite":[80]},"counts_by_year":[{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2025-10-10T00:00:00"}