{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T05:01:45Z","timestamp":1772427705648,"version":"3.50.1"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2841,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2006,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>T<\/jats:italic> be a positive <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub>-<jats:italic>L<\/jats:italic><jats:sub>\u221e<\/jats:sub> contraction. We prove that the following statements are equivalent in constructive mathematics.<\/jats:p><jats:p>(1) The projection in <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub>, on the space of invariant functions exists:<\/jats:p><jats:p>(2) The sequence (<jats:italic>T<jats:sup>n<\/jats:sup><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic>\u2208N<\/jats:sub> Ces\u00e1ro-converges in the <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub> norm:<\/jats:p><jats:p>(3) The sequence (<jats:italic>T<jats:sup>n<\/jats:sup><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic>\u2208N<\/jats:sub> Ces\u00e1ro-converges almost everywhere.<\/jats:p><jats:p>Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.<\/jats:p><jats:p>As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations.<\/jats:p><jats:p>This answers a question posed by Bishop.<\/jats:p>","DOI":"10.2178\/jsl\/1146620162","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:22:51Z","timestamp":1198081371000},"page":"611-623","source":"Crossref","is-referenced-by-count":2,"title":["A constructive view on ergodic theorems"],"prefix":"10.1017","volume":"71","author":[{"given":"Bas","family":"Spitters","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006241_ref011","unstructured":"Spitters Bas , Constructive and intuitionistic integration theory and functional analysis, Ph.D. thesis. University of Nijmegen, 2002."},{"key":"S0022481200006241_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511608728"},{"key":"S0022481200006241_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1972-0291411-3"},{"key":"S0022481200006241_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70740-7"},{"key":"S0022481200006241_ref004","volume-title":"Foundations of constructive analysis","author":"Bishop","year":"1967"},{"key":"S0022481200006241_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61667-9"},{"key":"S0022481200006241_ref001","volume-title":"Annals of Pure and Applied Logic","author":"Avigad"},{"key":"S0022481200006241_ref007","doi-asserted-by":"publisher","DOI":"10.1515\/9783110844641"},{"key":"S0022481200006241_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5775-2"},{"key":"S0022481200006241_ref009","first-page":"393","article-title":"Erratum to \u2018A constructive ergodic theorem","volume":"216","author":"Nuber","year":"1976","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200006241_ref005","volume-title":"Linear operators. Part I: General theory","author":"Dunford","year":"1958"},{"key":"S0022481200006241_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-03-07067-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200006241","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T20:16:21Z","timestamp":1556828181000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200006241\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2006,6]]}},"alternative-id":["S0022481200006241"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1146620162","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,6]]}}}