{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T20:14:28Z","timestamp":1773260068800,"version":"3.50.1"},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6676,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,12]]},"abstract":"<jats:p>Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.<\/jats:p><jats:p>The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an <jats:italic>F<\/jats:italic><jats:sub><jats:italic>\u03c3<\/jats:italic><\/jats:sub> equivalence relation on a Polish space <jats:italic>X<\/jats:italic> is induced by the continuous action of a Polish group <jats:italic>G<\/jats:italic> on <jats:italic>X<\/jats:italic>, then exactly one of the following alternatives holds:<\/jats:p><jats:p>(I) Elements of <jats:italic>X<\/jats:italic> can be classified up to <jats:italic>E<\/jats:italic>-equivalence by \u201cconcrete invariants\u201d computable in a reasonably definable way, i.e., there is a Borel function <jats:italic>f<\/jats:italic>: <jats:italic>X<\/jats:italic> \u2192 <jats:italic>Y<\/jats:italic>, <jats:italic>Y<\/jats:italic> a Polish space, such that <jats:italic>xEy<\/jats:italic> \u21d4 <jats:italic>f(x)<\/jats:italic> = <jats:italic>f(y)<\/jats:italic>, or else<\/jats:p><jats:p>(II) <jats:italic>E<\/jats:italic> contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation <jats:italic>xE<\/jats:italic><jats:sub>0<\/jats:sub><jats:italic>y<\/jats:italic> \u21d4 \u2203<jats:italic>n<\/jats:italic>\u2200<jats:italic>m<\/jats:italic> \u2265 <jats:italic>n<\/jats:italic>(<jats:italic>x<\/jats:italic>(<jats:italic>n<\/jats:italic>) = <jats:italic>y<\/jats:italic>(<jats:italic>n<\/jats:italic>)) on the Cantor space 2<jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> (<jats:italic>\u03c9<\/jats:italic> = {0,1,2, \u2026}), i.e., there is a continuous embedding <jats:italic>g<\/jats:italic>: 2<jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> \u2192 <jats:italic>X<\/jats:italic> such that <jats:italic>xE<\/jats:italic><jats:sub>0<\/jats:sub><jats:italic>y<\/jats:italic> \u21d4 <jats:italic>g<\/jats:italic>(<jats:italic>x<\/jats:italic>)<jats:italic>Eg<\/jats:italic>(<jats:italic>y<\/jats:italic>).<\/jats:p><jats:p>Moreover, alternative (II) is equivalent to:<\/jats:p><jats:p>(II)\u2032 There exists an <jats:italic>E<\/jats:italic>-ergodic, nonatomic probability Borel measure on <jats:italic>X<\/jats:italic>, where <jats:italic>E<\/jats:italic>-ergodic means that every <jats:italic>E<\/jats:italic>-invariant Borel set has measure 0 or 1 and <jats:italic>E<\/jats:italic>-nonatomic means that every <jats:italic>E<\/jats:italic>-equivalence class has measure 0.<\/jats:p>","DOI":"10.2307\/2275888","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:56:55Z","timestamp":1146956215000},"page":"1273-1300","source":"Crossref","is-referenced-by-count":22,"title":["Analytic equivalence relations and Ulm-type classifications"],"prefix":"10.1017","volume":"60","author":[{"given":"Greg","family":"Hjorth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander 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