{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/38643"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/38643","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Representation theory, Borel cross-sections, and minimal measures","abstract":"Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \\ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C<sub>b</sub>(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f ∈ C<sub>b</sub>(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f in C<sub>b</sub>(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.","abstract_html":"Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \\ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C&lt;sub&gt;b&lt;/sub&gt;(E) → L&lt;sup&gt;∞&lt;/sup&gt;(μ) with Γ(f∘Π) = f for every f ∈ C&lt;sub&gt;b&lt;/sub&gt;(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L&lt;sup&gt;∞&lt;/sup&gt;(μ) with Γ(f∘Π) = f for every f in C&lt;sub&gt;b&lt;/sub&gt;(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.","abstract_has_math":false,"creators":["Miller, Janice E."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Olin, Robert F."],"committee_members":["Ball, Joseph A.","McCoy, Robert A.","Aull, Charles E.","Johnson, Lee W."],"year":1993,"date_issued":"1993","date_published":"1993","updated_at":"2026-07-22T22:19:26Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-06192006-125737"],"render_values":[{"text":"etd-06192006-125737","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/38643","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Olin, Robert F."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Ball, Joseph A.","McCoy, Robert A.","Aull, Charles E.","Johnson, Lee W."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Miller, Janice E."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:15:08Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:15:08Z","2006-06-19"]},{"key":"dc:date.issued","label":"Date","values":["1993"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-06192006-125737"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/38643"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \\ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C<sub>b</sub>(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f ∈ C<sub>b</sub>(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f in C<sub>b</sub>(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Representation theory, Borel cross-sections, and minimal measures"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Olin, Robert F."],"dc:contributor.committeemember":["Ball, Joseph A.","McCoy, Robert A.","Aull, Charles E.","Johnson, Lee W."],"dc:contributor.department":["Mathematics"],"dc:creator":["Miller, Janice E."],"dc:date.accessioned":["2014-03-14T21:15:08Z"],"dc:date.available":["2014-03-14T21:15:08Z","2006-06-19"],"dc:date.issued":["1993"],"dc:description.abstract":["Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \\ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C<sub>b</sub>(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f ∈ C<sub>b</sub>(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f in C<sub>b</sub>(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E."],"dc:description.degree":["Ph. D."],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-06192006-125737"],"dc:identifier.uri":["http://hdl.handle.net/10919/38643"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Representation theory, Borel cross-sections, and minimal measures"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:26Z"}