1130:
360:; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp.
1208:
1225:
533:
392:
216:
if it is Borel-isomorphic to a Borel subset of a Polish space. Kuratowski's theorem then states that two standard Borel spaces
1048:
879:
365:
327:
419:
1040:
1291:
826:
1220:
337:
Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations".
1177:
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977:
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650:
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1215:
1162:
1056:
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1025:
949:
669:
385:
1203:
982:
944:
896:
1108:
1076:
1066:
987:
954:
585:
1125:
1030:
806:
734:
1115:
1198:
644:
575:
511:
967:
725:
685:
378:
1250:
1150:
972:
694:
540:
811:
764:
759:
754:
596:
479:
437:
1120:
1086:
994:
704:
659:
501:
424:
311:
259:
212:
32:
8:
1103:
1093:
939:
903:
729:
458:
415:
781:
1255:
1015:
1000:
699:
580:
558:
316:
299:
1172:
908:
869:
864:
771:
689:
474:
447:
361:
350:
323:
1189:
1098:
874:
859:
849:
834:
801:
796:
786:
664:
639:
454:
357:
346:
289:
52:
1265:
1245:
1020:
918:
913:
891:
749:
714:
634:
528:
250:
1155:
1010:
1005:
816:
791:
744:
674:
654:
614:
604:
401:
99:
1280:
1260:
923:
844:
839:
739:
709:
679:
629:
624:
619:
609:
523:
442:
204:
40:
854:
776:
516:
25:
553:
719:
192:
17:
275:
563:
303:
545:
489:
484:
294:
277:
570:
429:
370:
195:
except for a definability restriction on the witnessing mapping.
262: â Topological space with a notion of uniform properties
278:"A GlimmâEffros Dichotomy for Borel equivalence relations"
276:
Harrington, L. A.; A. S. Kechris; A. Louveau (Oct 1990).
225:
264:
Pages displaying short descriptions of redirect targets
310:
255:
Pages displaying wikidata descriptions as a fallback
315:
1278:
336:
183:has a lesser or equal "Borel cardinality" than
253: â levels of complexity for sets of reals
386:
1131:RieszâMarkovâKakutani representation theorem
282:Journal of the American Mathematical Society
1226:Vitale's random BrunnâMinkowski inequality
393:
379:
293:
198:
1279:
374:
1239:Applications & related
191:, where "Borel cardinality" is like
13:
400:
58:Given Borel equivalence relations
14:
1303:
1168:Lebesgue differentiation theorem
1049:Carathéodory's extension theorem
318:Classical Descriptive Set Theory
246:Hyperfinite equivalence relation
171:is "not more complicated" than
1:
269:
351:10.1016/0003-4843(80)90002-9
339:Annals of Mathematical Logic
98:, if and only if there is a
74:respectively, one says that
7:
1221:PrĂ©kopaâLeindler inequality
239:
10:
1308:
1163:Lebesgue's density theorem
22:Borel equivalence relation
1292:Equivalence (mathematics)
1238:
1216:MinkowskiâSteiner formula
1186:
1146:
1139:
1039:
1031:Projection-valued measure
932:
825:
594:
467:
408:
175:, and the quotient space
1199:Isoperimetric inequality
1178:VitaliâHahnâSaks theorem
507:Carathéodory's criterion
1204:BrunnâMinkowski theorem
1073:Decomposition theorems
1287:Descriptive set theory
1251:Descriptive set theory
1151:Disintegration theorem
586:Universally measurable
163:is Borel reducible to
1053:Convergence theorems
512:Cylindrical Ï-algebra
312:Kechris, Alexander S.
224:are Borel-isomorphic
1121:Minkowski inequality
995:Cylinder set measure
880:Infinite-dimensional
495:equivalence relation
425:Lebesgue integration
260:Entourage (topology)
213:standard Borel space
199:Kuratowski's theorem
33:equivalence relation
1116:Hölder's inequality
978:of random variables
940:Measurable function
827:Particular measures
416:Absolute continuity
322:. Springer-Verlag.
1256:Probability theory
581:Transverse measure
559:Non-measurable set
541:Locally measurable
116:such that for all
1274:
1273:
1234:
1233:
963:almost everywhere
909:Spherical measure
807:Strictly positive
735:Projection-valued
475:Almost everywhere
448:Probability space
366:978-0-8218-4453-3
358:Kanovei, Vladimir
329:978-0-387-94374-9
159:Conceptually, if
66:on Polish spaces
1299:
1209:Milman's reverse
1192:
1190:Lebesgue measure
1144:
1143:
548:
534:infimum/supremum
455:Measurable space
395:
388:
381:
372:
371:
354:
333:
321:
307:
297:
265:
256:
53:product topology
1307:
1306:
1302:
1301:
1300:
1298:
1297:
1296:
1277:
1276:
1275:
1270:
1266:Spectral theory
1246:Convex analysis
1230:
1187:
1182:
1135:
1035:
983:in distribution
928:
821:
651:Logarithmically
590:
546:
529:Essential range
463:
404:
399:
330:
295:10.2307/1990906
272:
263:
254:
251:Wadge hierarchy
242:
201:
93:
80:Borel reducible
12:
11:
5:
1305:
1295:
1294:
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1272:
1271:
1269:
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1184:
1183:
1181:
1180:
1175:
1173:Sard's theorem
1170:
1165:
1160:
1159:
1158:
1156:Lifting theory
1147:
1141:
1137:
1136:
1134:
1133:
1128:
1123:
1118:
1113:
1112:
1111:
1109:FubiniâTonelli
1101:
1096:
1091:
1090:
1089:
1084:
1079:
1071:
1070:
1069:
1064:
1059:
1051:
1045:
1043:
1037:
1036:
1034:
1033:
1028:
1023:
1018:
1013:
1008:
1003:
997:
992:
991:
990:
988:in probability
985:
975:
970:
965:
959:
958:
957:
952:
947:
936:
934:
930:
929:
927:
926:
921:
916:
911:
906:
901:
900:
899:
889:
884:
883:
882:
872:
867:
862:
857:
852:
847:
842:
837:
831:
829:
823:
822:
820:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
768:
767:
762:
757:
747:
742:
737:
732:
722:
717:
712:
707:
702:
697:
695:Locally finite
692:
682:
677:
672:
667:
662:
657:
647:
642:
637:
632:
627:
622:
617:
612:
607:
601:
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592:
591:
589:
588:
583:
578:
573:
568:
567:
566:
556:
551:
543:
538:
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536:
526:
521:
520:
519:
509:
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499:
498:
497:
487:
482:
477:
471:
469:
465:
464:
462:
461:
452:
451:
450:
440:
435:
427:
422:
412:
410:
409:Basic concepts
406:
405:
402:Measure theory
398:
397:
390:
383:
375:
369:
368:
355:
334:
328:
308:
288:(2): 903â928.
271:
268:
267:
266:
257:
248:
241:
238:
200:
197:
157:
156:
114:
113:
100:Borel function
91:
9:
6:
4:
3:
2:
1304:
1293:
1290:
1288:
1285:
1284:
1282:
1267:
1264:
1262:
1261:Real analysis
1259:
1257:
1254:
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1247:
1244:
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1191:
1185:
1179:
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1174:
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1166:
1164:
1161:
1157:
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1152:
1149:
1148:
1145:
1142:
1140:Other results
1138:
1132:
1129:
1127:
1126:RadonâNikodym
1124:
1122:
1119:
1117:
1114:
1110:
1107:
1106:
1105:
1102:
1100:
1099:Fatou's lemma
1097:
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1092:
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1085:
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1080:
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1068:
1065:
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1029:
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1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
998:
996:
993:
989:
986:
984:
981:
980:
979:
976:
974:
971:
969:
966:
964:
961:Convergence:
960:
956:
953:
951:
948:
946:
943:
942:
941:
938:
937:
935:
931:
925:
922:
920:
917:
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910:
907:
905:
902:
898:
895:
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868:
866:
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856:
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851:
848:
846:
843:
841:
838:
836:
833:
832:
830:
828:
824:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
766:
765:Outer regular
763:
761:
760:Inner regular
758:
756:
755:Borel regular
753:
752:
751:
748:
746:
743:
741:
738:
736:
733:
731:
727:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
687:
683:
681:
678:
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673:
671:
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666:
663:
661:
658:
656:
652:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
600:
598:
593:
587:
584:
582:
579:
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572:
569:
565:
562:
561:
560:
557:
555:
552:
550:
544:
542:
539:
535:
532:
531:
530:
527:
525:
522:
518:
515:
514:
513:
510:
508:
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500:
496:
493:
492:
491:
488:
486:
483:
481:
478:
476:
473:
472:
470:
466:
460:
456:
453:
449:
446:
445:
444:
443:Measure space
441:
439:
436:
434:
432:
428:
426:
423:
421:
417:
414:
413:
411:
407:
403:
396:
391:
389:
384:
382:
377:
376:
373:
367:
363:
359:
356:
352:
348:
344:
340:
335:
331:
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320:
319:
313:
309:
305:
301:
296:
291:
287:
283:
279:
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273:
261:
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252:
249:
247:
244:
243:
237:
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231:
227:
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219:
215:
214:
209:
206:
205:measure space
196:
194:
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186:
182:
178:
174:
170:
166:
162:
154:
150:
146:
142:
138:
134:
131:
130:
129:
127:
123:
119:
112:
108:
104:
103:
102:
101:
97:
89:
86:, in symbols
85:
81:
77:
73:
69:
65:
61:
56:
54:
50:
47: Ă
46:
42:
38:
34:
30:
27:
23:
19:
1041:Main results
777:Set function
705:Metric outer
660:Decomposable
517:Cylinder set
494:
430:
342:
338:
317:
285:
281:
233:
229:
221:
217:
211:
210:is called a
207:
202:
188:
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180:
176:
172:
168:
164:
160:
158:
152:
148:
144:
140:
136:
132:
125:
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117:
115:
110:
106:
95:
87:
83:
79:
75:
71:
67:
63:
59:
57:
48:
44:
36:
28:
26:Polish space
21:
15:
1001:compact set
968:of measures
904:Pushforward
897:Projections
887:Logarithmic
730:Probability
720:Pre-measure
502:Borel space
420:of measures
345:(1): 1â28.
193:cardinality
18:mathematics
1281:Categories
973:in measure
700:Maximising
670:Equivalent
564:Vitali set
270:References
128:, one has
43:subset of
39:that is a
1087:Maharam's
1057:Dominated
870:Intensity
865:Hausdorff
772:Saturated
690:Invariant
595:Types of
554:Ï-algebra
524:đ-system
490:Borel set
485:Baire set
105:Î :
1104:Fubini's
1094:Egorov's
1062:Monotone
1021:variable
999:Random:
950:Strongly
875:Lebesgue
860:Harmonic
850:Gaussian
835:Counting
802:Spectral
797:Singular
787:s-finite
782:Ï-finite
665:Discrete
640:Complete
597:Measures
571:Null set
459:function
314:(1994).
240:See also
51:(in the
1016:process
1011:measure
1006:element
945:Bochner
919:Trivial
914:Tangent
892:Product
750:Regular
728:)
715:Perfect
688:)
653:)
645:Content
635:Complex
576:Support
549:-system
438:Measure
304:1990906
167:, then
90: â€
1082:Jordan
1067:Vitali
1026:vector
955:Weakly
817:Vector
792:Signed
745:Random
686:Quasi-
675:Finite
655:Convex
615:Banach
605:Atomic
433:spaces
418:
364:
326:
302:
143:' â Î(
139:
135:
94:
31:is an
924:Young
845:Euler
840:Dirac
812:Tight
740:Radon
710:Outer
680:Inner
630:Brown
625:Borel
620:Besov
610:Baire
300:JSTOR
232:| = |
41:Borel
24:on a
1188:For
1077:Hahn
933:Maps
855:Haar
726:Sub-
480:Atom
468:Sets
362:ISBN
324:ISBN
220:and
124:' â
70:and
62:and
20:, a
347:doi
290:doi
236:|.
226:iff
155:').
82:to
78:is
55:).
35:on
16:In
1283::
343:18
341:.
298:.
284:.
280:.
203:A
151:Î(
147:)
109:â
724:(
684:(
649:(
547:Ï
457:/
431:L
394:e
387:t
380:v
353:.
349::
332:.
306:.
292::
286:3
234:Y
230:X
228:|
222:Y
218:X
208:X
189:F
187:/
185:Y
181:E
179:/
177:X
173:F
169:E
165:F
161:E
153:x
149:F
145:x
141:x
137:E
133:x
126:X
122:x
120:,
118:x
111:Y
107:X
96:F
92:B
88:E
84:F
76:E
72:Y
68:X
64:F
60:E
49:X
45:X
37:X
29:X
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