{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68171"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68171","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Pettis Integral","abstract":"The Pettis integral of a weakly measurable vector-valued function is the most natural integral for use in Banach spaces. Although first defined over forty years ago, the integral has stubbornly defied analysis and has long been considered unmanageable. My thesis presents the first successful analysis of the Pettis integral. I show that a slight restriction on the measure spaces under consideration leads to a theory of Pettis integration very analogous to the theory of the better known, but more restrictive, Bochner integral. The resulting characterization of the Pettis integrable functions is much simpler than was previously believed possible.","abstract_html":"The Pettis integral of a weakly measurable vector-valued function is the most natural integral for use in Banach spaces. Although first defined over forty years ago, the integral has stubbornly defied analysis and has long been considered unmanageable. My thesis presents the first successful analysis of the Pettis integral. I show that a slight restriction on the measure spaces under consideration leads to a theory of Pettis integration very analogous to the theory of the better known, but more restrictive, Bochner integral. The resulting characterization of the Pettis integrable functions is much simpler than was previously believed possible.","abstract_has_math":false,"creators":["Geitz, Robert Frederick"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-14T13:09:46Z","date_published":"2014-12-14T13:09:46Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8026497"],"render_values":[{"text":"(UMI)AAI8026497","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68171","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Geitz, Robert Frederick"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:46Z","10000-01-01","1980"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68171","(UMI)AAI8026497"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The Pettis integral of a weakly measurable vector-valued function is the most natural integral for use in Banach spaces. Although first defined over forty years ago, the integral has stubbornly defied analysis and has long been considered unmanageable. My thesis presents the first successful analysis of the Pettis integral. I show that a slight restriction on the measure spaces under consideration leads to a theory of Pettis integration very analogous to the theory of the better known, but more restrictive, Bochner integral. The resulting characterization of the Pettis integrable functions is much simpler than was previously believed possible.","The thesis falls naturally into three parts. I first consider a vector-valued function f : (OMEGA) (--->) X in terms of the associated family {x*f : (VBAR)(VBAR) x* (VBAR)(VBAR) (LESSTHEQ) 1} of scalar-valued functions. This gives new insight into the various types of measurability for vector-valued functions. I next make an extensive study of the properties of a function that are determined by the geometry of its range. Here I characterize the functions that are equivalent to strongly measurable functions and give the first necessary and sufficient conditions for a function to be Pettis integrable. The deep connection between perfect measure spaces and the Pettis integral also becomes apparent here. The final chapter of the thesis contains its most important results. Here I prove a dominated convergence theorem for the Pettis integral and characterize the Pettis integrable functions as limits, in a certain sense, of sequences of simple funtions.","Made available in DSpace on 2014-12-14T13:09:46Z (GMT). No. of bitstreams: 1 8026497.pdf: 2335674 bytes, checksum: 8a5658dd111649a1e931ce2d2691f051 (MD5) Previous issue date: 1980","Embargo set by: Seth Robbins for item 68349 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","85 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980."]},{"key":"dc:title","label":"Title","values":["The Pettis Integral"]}]}],"canonical_facts":{"dc:creator":["Geitz, Robert Frederick"],"dc:date":["2014-12-14T13:09:46Z","10000-01-01","1980"],"dc:description":["The Pettis integral of a weakly measurable vector-valued function is the most natural integral for use in Banach spaces. Although first defined over forty years ago, the integral has stubbornly defied analysis and has long been considered unmanageable. My thesis presents the first successful analysis of the Pettis integral. I show that a slight restriction on the measure spaces under consideration leads to a theory of Pettis integration very analogous to the theory of the better known, but more restrictive, Bochner integral. The resulting characterization of the Pettis integrable functions is much simpler than was previously believed possible.","The thesis falls naturally into three parts. I first consider a vector-valued function f : (OMEGA) (--->) X in terms of the associated family {x*f : (VBAR)(VBAR) x* (VBAR)(VBAR) (LESSTHEQ) 1} of scalar-valued functions. This gives new insight into the various types of measurability for vector-valued functions. I next make an extensive study of the properties of a function that are determined by the geometry of its range. Here I characterize the functions that are equivalent to strongly measurable functions and give the first necessary and sufficient conditions for a function to be Pettis integrable. The deep connection between perfect measure spaces and the Pettis integral also becomes apparent here. The final chapter of the thesis contains its most important results. Here I prove a dominated convergence theorem for the Pettis integral and characterize the Pettis integrable functions as limits, in a certain sense, of sequences of simple funtions.","Made available in DSpace on 2014-12-14T13:09:46Z (GMT). No. of bitstreams: 1 8026497.pdf: 2335674 bytes, checksum: 8a5658dd111649a1e931ce2d2691f051 (MD5) Previous issue date: 1980","Embargo set by: Seth Robbins for item 68349 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","85 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980."],"dc:identifier":["http://hdl.handle.net/2142/68171","(UMI)AAI8026497"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["The Pettis Integral"],"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":"2026-07-22T22:25:58Z"}