{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23263"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23263","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Boolean-valued probabilistic metric spaces","abstract":"In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($\\Omega,{\\cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $V\\sp{\\cal B}$, correspond to random variables with domain $\\Omega$, and the sets which act as binary operations on $\\Re$ in the model correspond to binary operations on the class of random variables with domain $\\Omega$.","abstract_html":"In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($\\Omega,{\\cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $V\\sp{\\cal B}$, correspond to random variables with domain $\\Omega$, and the sets which act as binary operations on $\\Re$ in the model correspond to binary operations on the class of random variables with domain $\\Omega$.","abstract_has_math":true,"creators":["Holsey, Mickey Charles"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Takeuti, Gaisi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:07:55Z","date_published":"2011-05-07T14:07:55Z","updated_at":"2026-07-22T22:25:21Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Holsey, Mickey Charles"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9010890","(UMI)AAI9010890"],"render_values":[{"text":"AAI9010890","href":null,"code":true},{"text":"(UMI)AAI9010890","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23263","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Takeuti, Gaisi"]},{"key":"dc:creator","label":"Author","values":["Holsey, Mickey Charles"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:07:55Z","10000-01-01","1989"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1989 Holsey, Mickey Charles"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9010890","(UMI)AAI9010890","http://hdl.handle.net/2142/23263"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($\\Omega,{\\cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $V\\sp{\\cal B}$, correspond to random variables with domain $\\Omega$, and the sets which act as binary operations on $\\Re$ in the model correspond to binary operations on the class of random variables with domain $\\Omega$.","The next step is to generalize the theory of metric spaces by requiring the triangle inequality be satisfied for some generalized addition $\\oplus$ (i.e. some commutative, associative, nondecreasing binary operation on $\\Re\\sp+$ which has 0 as identity). If the generalized addition is continuous then the generalized metric space has a natural topology on it which satisfies the same general properties that metric topologies satisfy.","A Boolean valued probabilistic metric space is a triple ($\\tilde S,{\\cal D},{\\cal G}$) which corresponds to some triple ($S,d,\\oplus$) in $V\\sp{\\cal B}$ which acts as a generalized metric space in the nonstandard model. Thus $\\tilde S$ is a subset of $V\\sp{\\cal B},{\\cal D}$ is a function which maps ordered pairs of elements of $\\tilde S$ to a.e. nonnegative random variables, and ${\\cal G}$ is a binary operation on the class of a.e. nonnegative random variables. The topology on the generalized metric space in $V\\sp{\\cal B}$ corresponds to a topology on $\\tilde S$ which is uniform.","There are a large number of probabilistic metric spaces which can be isomorphically injected into some Boolean valued probabilistic metric space. This mapping is generally not a surjection due to the richer structure of the Boolean valued probabilistic metric space. There are also probabilistic metric spaces which cannot be isomorphically injected into any Boolean valued probabilistic metric space.","This paper leads to several interesting lines of inquiry. One of these is whether all uniform topological spaces may be obtained from some generalized metric space by the Boolean valued method and another is what systems in the sciences can Boolean valued probabilistic metric spaces be used to model.","Made available in DSpace on 2011-05-07T14:07:55Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010890.pdf: 2924585 bytes, checksum: ea98b546309833650445d59ec0ecf053 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:03:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:30:10-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Boolean-valued probabilistic metric spaces"]}]}],"canonical_facts":{"dc:contributor":["Takeuti, Gaisi"],"dc:creator":["Holsey, Mickey Charles"],"dc:date":["2011-05-07T14:07:55Z","10000-01-01","1989"],"dc:description":["In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($\\Omega,{\\cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $V\\sp{\\cal B}$, correspond to random variables with domain $\\Omega$, and the sets which act as binary operations on $\\Re$ in the model correspond to binary operations on the class of random variables with domain $\\Omega$.","The next step is to generalize the theory of metric spaces by requiring the triangle inequality be satisfied for some generalized addition $\\oplus$ (i.e. some commutative, associative, nondecreasing binary operation on $\\Re\\sp+$ which has 0 as identity). If the generalized addition is continuous then the generalized metric space has a natural topology on it which satisfies the same general properties that metric topologies satisfy.","A Boolean valued probabilistic metric space is a triple ($\\tilde S,{\\cal D},{\\cal G}$) which corresponds to some triple ($S,d,\\oplus$) in $V\\sp{\\cal B}$ which acts as a generalized metric space in the nonstandard model. Thus $\\tilde S$ is a subset of $V\\sp{\\cal B},{\\cal D}$ is a function which maps ordered pairs of elements of $\\tilde S$ to a.e. nonnegative random variables, and ${\\cal G}$ is a binary operation on the class of a.e. nonnegative random variables. The topology on the generalized metric space in $V\\sp{\\cal B}$ corresponds to a topology on $\\tilde S$ which is uniform.","There are a large number of probabilistic metric spaces which can be isomorphically injected into some Boolean valued probabilistic metric space. This mapping is generally not a surjection due to the richer structure of the Boolean valued probabilistic metric space. There are also probabilistic metric spaces which cannot be isomorphically injected into any Boolean valued probabilistic metric space.","This paper leads to several interesting lines of inquiry. One of these is whether all uniform topological spaces may be obtained from some generalized metric space by the Boolean valued method and another is what systems in the sciences can Boolean valued probabilistic metric spaces be used to model.","Made available in DSpace on 2011-05-07T14:07:55Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010890.pdf: 2924585 bytes, checksum: ea98b546309833650445d59ec0ecf053 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:03:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:30:10-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9010890","(UMI)AAI9010890","http://hdl.handle.net/2142/23263"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Holsey, Mickey Charles"],"dc:subject":["Mathematics"],"dc:title":["Boolean-valued probabilistic metric spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:21Z"}