940:-algebra of all subsets of the real line), the above shows us that unlike equivalence relations which admit transversals, many examples of group actions which appear naturally in ergodic theory give rise to hyperfinite orbit equivalence relations (in particular, whenever the underlying space is a standard Borel space and the group is countable and amenable).
263:). Therefore, it is natural to ask whether certain equivalence relations, which are not necessarily finite, can be approximated by finite equivalence relations. This turns out to be a notion which is both rich enough to encapsulate many natural equivalence relations appearing in mathematics, yet restrictive enough to allow deep theorems to develop.
286:
in general, and therefore it is not clear that this process generates Borel approximations. Indeed, there are countable Borel equivalence relations that are not hyperfinite, and so in particular the process described above will fail to generate Borel equivalence relations approximating the larger
818:
Under certain conditions, it is known that a countable increasing union of hyperfinite equivalence relations is hyperfinite. For example, if the union of the equivalence relations has a property known as "Borel-boundedness" (which roughly means that any Borel assignment of functions
510:
is equipped with the σ-algebra of all its subsets). Moreover, it turns out that any hyperfinite equivalence relation is equal to the orbit equivalence relation generated by some Borel action of the integers, making this an equivalent definition to hyperfiniteness that is often more
684:
being countable; the equivalence relation on the real line identifying every two points is Borel and can be reduced to any other Borel equivalence relation, and in particular to any hyperfinite Borel equivalence relation, but it has an uncountable class, so it cannot be
919:
that admits a Borel transversal is a finite equivalence relation on a subset of full measure (this is essentially
Feldman-Moore, together with Vitali's argument in his classical proof of the non-existence of a nontrivial invariant measure on the
855:
to points on the space can be "eventually bounded" by such a Borel assignment which is constant on equivalence classes), then it is hyperfinite. However, it is unknown whether every such union satisfies this property.
383:
706:
on which it is hyperfinite. Explicitly, this means that one can remove a collection of equivalence classes which is meagre, and get that the equivalence relation is hyperfinite on the remaining space.
853:
203:
678:
879:, the theory is much better understood. For instance, if the equivalence relation is generated by a Borel action of a countable amenable group, the resulting orbit equivalence relation is "
504:
567:
433:
815:
Another open problem in the area is whether a countable increasing union of hyperfinite equivalence relations is hyperfinite. This is often referred to as the union problem.
775:
938:
741:
230:
915:
599:
457:
Any Borel action of the integers on a standard Borel space generates a hyperfinite orbit equivalence relation (recall that a Borel action of a countable group
259:, and in particular those which are countable. Among these, finite equivalence relations are considered to be the simplest (for instance, they admit Borel
807:
on a standard Borel space induces a hyperfinite orbit equivalence relation. While this is still an open problem, some partial results are known.
317:
1261:
883:-hyperfinite", meaning that it is hyperfinite on a subset of the space of full measure (it is worthwhile to note that the action need not be
450:
Any countable Borel equivalence relation that admits a Borel transversal is hyperfinite; this can be shown by a simple application of the
884:
1323:
1094:
Connes, Alain; Feldman, Joel; Weiss, Benjamin (1995), "An amenable equivalence relation is generated by a single transformation",
1125:
1318:
1166:
Dougherty, Randall; Jackson, Steve; Kechris, Alexander S. (1994), "The structure of hyperfinite Borel equivalence relations",
822:
278:
of every class into classes of size two, then joining two classes in the new equivalence relation which are within the same
282:-class to form a partition with classes of size four, and so forth. The key observation is that this process requires the
157:
604:
260:
962:
957:
693:
386:
256:
40:
778:
689:
713:
admits a Borel action on a standard Borel space which induces an equivalence relation that is not hyperfinite.
471:
888:
451:
124:
is an uncountable standard Borel space, the equivalence relation will be uncountable when considered as a
17:
777:
by the shift maps is not hyperfinite. This fact is often considered to be a combinatorial variant of the
1224:
Jackson, Steve; Kechris, Alexander S.; Louveau, Alain (2002), "Countable Borel equivalence relations",
540:
1270:
Marquis, Timothée; Sabok, Marcin (2020), "Hyperfiniteness of boundary actions of hyperbolic groups",
307:
296:
Any finite Borel equivalence relation is hyperfinite. Indeed, it is a finite approximation of itself.
33:
29:
869:
796:
271:
1196:
Gao, Su; Jackson, Steve (2007), "Countable abelian groups and hyperfinite equivalence relations",
1297:
1279:
1213:
1185:
1155:
1137:
1113:
1081:
407:
275:
944:
Similarly, a countable increasing union of hyperfinite equivalence relations on such a space is
746:
1257:
125:
98:
923:
1289:
1249:
1233:
1205:
1175:
1147:
1103:
1076:
Conley, Clinton; Jackson, Steve; Marks, Andrew; Seward, Brandon; Tucker-Drob, Robin (2020),
59:
719:
208:
900:
699:
The action of a finitely-generated hyperbolic group on its Gromov boundary is hyperfinite.
283:
804:
800:
710:
584:
1180:
1312:
1301:
1117:
515:
114:
1217:
1189:
1159:
466:
63:
70:
239:
Intuitively, this means that there is a sequence finite equivalence relations on
21:
1293:
299:
A subequivalence relation of a hyperfinite equivalence relation is hyperfinite.
1237:
1209:
1108:
703:
110:
397:
into finite Borel equivalence relations, thereby proving its hyperfiniteness.
274:
of finite equivalence relations. This can be done, for instance, by taking a
255:
A major area of research in descriptive set theory is the classification of
74:
1151:
795:
The above examples seem to indicate that Borel actions of "tame" countable
696:
on a standard Borel space induces a hyperfinite orbit equivalence relation.
518:
on a standard Borel space induces a hyperfinite orbit equivalence relation.
378:{\displaystyle F_{0}\subseteq F_{1}\subseteq F_{2}\subseteq ...\subseteq G}
876:
78:
39:
with countable classes, that can, in a certain sense, be approximated by
105:
with itself, when equipped with the product σ-algebra. We will say that
266:
It is also worthwhile to note that any countable equivalence relation
1126:"Cardinal characteristics and countable Borel equivalence relations"
680:). Note that the above does not hold if we remove the assumption of
525:
is a countable Borel equivalence relation on a standard Borel space
142:
be a countable Borel equivalence relation on a standard Borel space
1284:
1253:
1086:
232:
is an increasing sequence of finite Borel equivalence relations on
1142:
385:
into finite subgroups naturally gives rise to a filtration of the
1078:
Borel asymptotic dimension and hyperfinite equivalence relations
702:
Any countable Borel equivalence relation can be restricted to a
533:
is a hyperfinite equivalence relation on a standard borel space
1248:, Lecture Notes in Mathematics, vol. 1852, Springer,
1038:
310:
group acting Borel-measurably on a standard Borel space
1049:
435:
is hyperfinite and of finite index (meaning that every
848:{\displaystyle f:\mathbb {N} \rightarrow \mathbb {N} }
1005:
926:
903:
825:
749:
722:
607:
587:
543:
474:
410:
320:
211:
160:
891:). Since every countable Borel equivalence relation
601:
as above is a Borel reduction whenever it satisfies
1244:Kechris, Alexander S.; Miller, Benjamin D. (2004),
198:{\displaystyle E=\bigcup _{n\in \mathbb {N} }F_{n}}
983:
932:
909:
847:
769:
735:
673:{\displaystyle \forall x,y\in X,f(x)Ef(y)\iff xEy}
672:
593:
561:
498:
427:
377:
243:, each finer then its predecessors, approximating
224:
197:
1168:Transactions of the American Mathematical Society
895:on a standard non-atomic Borel probability space
803:conjectured that any Borel action of a countable
514:More generally, any Borel action of a countable
1060:
864:Under the assumption that the underlying space
120:The above names might be misleading, since if
1027:
109:is finite (respectively, countable) if E has
8:
1016:
979:
977:
58:be a standard Borel space, that is; it is a
994:
875:and that one is willing to remove sets of
799:induce hyperfinite equivalence relations.
660:
656:
1283:
1179:
1141:
1124:Coskey, Samuel; Schneider, Scott (2017),
1107:
1085:
925:
902:
841:
840:
833:
832:
824:
759:
754:
748:
727:
721:
606:
586:
542:
473:
409:
351:
338:
325:
319:
216:
210:
189:
179:
178:
171:
159:
499:{\displaystyle a:G\times X\rightarrow X}
404:is a countable equivalence relation and
973:
1050:Dougherty, Jackson & Kechris 1994
270:can be written down as an increasing
7:
1096:Ergodic Theory and Dynamical Systems
1006:Jackson, Kechris & Louveau 2002
608:
14:
1181:10.1090/S0002-9947-1994-1149121-0
984:Connes, Feldman & Weiss 1995
562:{\displaystyle f:X\rightarrow Y}
26:hyperfinite equivalence relation
743:on two generators on the space
837:
657:
653:
647:
638:
632:
553:
490:
439:-class contains finitely many
85:be an equivalence relation on
1:
1226:Journal of Mathematical Logic
716:The action of the free group
1130:Mathematical Logic Quarterly
581:is hyperfinite (recall that
62:which arises by equipping a
1246:Topics in orbit equivalence
1061:Coskey & Schneider 2017
428:{\displaystyle E'\subset E}
257:Borel equivalence relations
41:Borel equivalence relations
1340:
1294:10.1007/s00208-020-02001-9
963:Borel equivalence relation
958:Hyperfinite type II factor
790:
461:on a standard Borel space
387:orbit equivalence relation
43:that have finite classes.
1324:Equivalence (mathematics)
1238:10.1142/S0219061302000138
1210:10.1007/s00222-015-0603-y
1109:10.1017/S014338570000136X
1028:Kechris & Miller 2004
868:is equipped with a Borel
860:Measure-theoretic results
770:{\displaystyle 2^{F_{2}}}
291:Examples and non-examples
97:is a Borel subset of the
1198:Inventiones Mathematicae
1017:Marquis & Sabok 2020
889:quasi-measure preserving
690:finitely-generated group
569:is a Borel reduction of
933:{\displaystyle \sigma }
709:Any group which is not
1319:Descriptive set theory
1152:10.1002/malq.201400111
995:Gao & Jackson 2007
948:-hyperfinite as well.
934:
911:
849:
771:
737:
688:Any Borel action of a
674:
595:
563:
500:
465:is a Borel-measurable
429:
379:
287:equivalence relation.
226:
199:
128:of ordered pairs from
18:descriptive set theory
1272:Mathematische Annalen
935:
912:
850:
779:Banach–Tarski paradox
772:
738:
736:{\displaystyle F_{2}}
675:
596:
564:
501:
452:Feldman-Moore theorem
430:
380:
227:
225:{\displaystyle F_{n}}
200:
20:and related areas of
924:
910:{\displaystyle \mu }
901:
823:
747:
720:
605:
585:
541:
472:
408:
318:
314:. Then a filtration
209:
158:
77:(and forgetting the
34:equivalence relation
30:standard Borel space
870:probability measure
146:. We will say that
89:. We will say that
1039:Conley et al. 2020
930:
907:
885:measure-preserving
845:
791:Weiss's conjecture
767:
733:
670:
591:
559:
496:
425:
375:
247:arbitrarily well.
222:
195:
184:
1263:978-3-540-22603-1
811:The union problem
694:polynomial growth
594:{\displaystyle f}
389:of the action of
167:
99:cartesian product
1331:
1304:
1287:
1278:(4): 1129–1153,
1266:
1240:
1220:
1192:
1183:
1162:
1145:
1136:(3–4): 211–227,
1120:
1111:
1090:
1089:
1063:
1058:
1052:
1047:
1041:
1036:
1030:
1025:
1019:
1014:
1008:
1003:
997:
992:
986:
981:
939:
937:
936:
931:
916:
914:
913:
908:
854:
852:
851:
846:
844:
836:
776:
774:
773:
768:
766:
765:
764:
763:
742:
740:
739:
734:
732:
731:
679:
677:
676:
671:
600:
598:
597:
592:
568:
566:
565:
560:
505:
503:
502:
497:
443:-classes), then
434:
432:
431:
426:
418:
384:
382:
381:
376:
356:
355:
343:
342:
330:
329:
231:
229:
228:
223:
221:
220:
204:
202:
201:
196:
194:
193:
183:
182:
60:measurable space
1339:
1338:
1334:
1333:
1332:
1330:
1329:
1328:
1309:
1308:
1307:
1269:
1264:
1243:
1223:
1195:
1165:
1123:
1093:
1075:
1071:
1066:
1059:
1055:
1048:
1044:
1037:
1033:
1026:
1022:
1015:
1011:
1004:
1000:
993:
989:
982:
975:
971:
954:
943:
922:
921:
899:
898:
862:
821:
820:
813:
793:
788:
755:
750:
745:
744:
723:
718:
717:
603:
602:
583:
582:
539:
538:
470:
469:
447:is hyperfinite.
411:
406:
405:
347:
334:
321:
316:
315:
293:
284:axiom of choice
253:
212:
207:
206:
185:
156:
155:
113:(respectively,
49:
12:
11:
5:
1337:
1335:
1327:
1326:
1321:
1311:
1310:
1306:
1305:
1267:
1262:
1254:10.1007/b99421
1241:
1221:
1193:
1163:
1121:
1102:(4): 431–450,
1091:
1072:
1070:
1067:
1065:
1064:
1053:
1042:
1031:
1020:
1009:
998:
987:
972:
970:
967:
966:
965:
960:
953:
950:
929:
906:
861:
858:
843:
839:
835:
831:
828:
812:
809:
805:amenable group
792:
789:
787:
784:
783:
782:
762:
758:
753:
730:
726:
714:
707:
700:
697:
686:
669:
666:
663:
659:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
590:
558:
555:
552:
549:
546:
519:
512:
495:
492:
489:
486:
483:
480:
477:
455:
448:
424:
421:
417:
414:
398:
374:
371:
368:
365:
362:
359:
354:
350:
346:
341:
337:
333:
328:
324:
308:locally finite
300:
297:
292:
289:
252:
249:
219:
215:
192:
188:
181:
177:
174:
170:
166:
163:
48:
45:
13:
10:
9:
6:
4:
3:
2:
1336:
1325:
1322:
1320:
1317:
1316:
1314:
1303:
1299:
1295:
1291:
1286:
1281:
1277:
1273:
1268:
1265:
1259:
1255:
1251:
1247:
1242:
1239:
1235:
1231:
1227:
1222:
1219:
1215:
1211:
1207:
1203:
1199:
1194:
1191:
1187:
1182:
1177:
1173:
1169:
1164:
1161:
1157:
1153:
1149:
1144:
1139:
1135:
1131:
1127:
1122:
1119:
1115:
1110:
1105:
1101:
1097:
1092:
1088:
1083:
1079:
1074:
1073:
1068:
1062:
1057:
1054:
1051:
1046:
1043:
1040:
1035:
1032:
1029:
1024:
1021:
1018:
1013:
1010:
1007:
1002:
999:
996:
991:
988:
985:
980:
978:
974:
968:
964:
961:
959:
956:
955:
951:
949:
947:
941:
927:
918:
904:
894:
890:
886:
882:
878:
874:
871:
867:
859:
857:
829:
826:
816:
810:
808:
806:
802:
798:
786:Open problems
785:
780:
760:
756:
751:
728:
724:
715:
712:
708:
705:
701:
698:
695:
691:
687:
683:
667:
664:
661:
650:
644:
641:
635:
629:
626:
623:
620:
617:
614:
611:
588:
580:
576:
572:
556:
550:
547:
544:
536:
532:
528:
524:
520:
517:
516:abelian group
513:
509:
493:
487:
484:
481:
478:
475:
468:
464:
460:
456:
453:
449:
446:
442:
438:
422:
419:
415:
412:
403:
399:
396:
392:
388:
372:
369:
366:
363:
360:
357:
352:
348:
344:
339:
335:
331:
326:
322:
313:
309:
305:
301:
298:
295:
294:
290:
288:
285:
281:
277:
273:
269:
264:
262:
258:
250:
248:
246:
242:
237:
235:
217:
213:
190:
186:
175:
172:
168:
164:
161:
153:
149:
145:
141:
137:
136:Definition 2.
133:
131:
127:
123:
118:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
75:Borel subsets
72:
68:
65:
61:
57:
53:
52:Definition 1.
46:
44:
42:
38:
35:
32:X is a Borel
31:
27:
23:
19:
1275:
1271:
1245:
1232:(1): 1–144,
1229:
1225:
1201:
1197:
1171:
1167:
1133:
1129:
1099:
1095:
1077:
1056:
1045:
1034:
1023:
1012:
1001:
990:
945:
942:
896:
892:
880:
877:measure zero
872:
865:
863:
817:
814:
794:
704:comeagre set
685:hyperfinite.
681:
578:
574:
570:
534:
530:
526:
522:
507:
462:
458:
444:
440:
436:
401:
394:
390:
311:
303:
279:
267:
265:
261:transversals
254:
244:
240:
238:
233:
151:
147:
143:
139:
135:
134:
129:
121:
119:
106:
102:
94:
93:is Borel if
90:
86:
82:
66:
64:Polish space
55:
51:
50:
36:
25:
15:
1174:: 193–225,
511:accessible.
152:hyperfinite
117:) classes.
47:Definitions
22:mathematics
1313:Categories
1285:1907.09928
1087:2009.06721
1069:References
887:, or even
251:Discussion
1302:198179841
1143:1103.2312
1118:119385336
928:σ
905:μ
838:→
658:⟺
621:∈
609:∀
554:→
491:→
485:×
420:⊂
370:⊆
358:⊆
345:⊆
332:⊆
276:partition
176:∈
169:⋃
115:countable
71:σ-algebra
69:with its
1218:24460955
1190:29620563
1160:41429052
952:See also
711:amenable
506:, where
416:′
302:Suppose
205:, where
79:topology
577:, then
81:). Let
1300:
1260:
1216:
1188:
1158:
1116:
797:groups
467:action
111:finite
1298:S2CID
1280:arXiv
1214:S2CID
1204:(1),
1186:S2CID
1156:S2CID
1138:arXiv
1114:S2CID
1082:arXiv
969:Notes
801:Weiss
692:with
306:is a
272:union
28:on a
1258:ISBN
537:and
138:Let
54:Let
24:, a
1290:doi
1276:377
1250:doi
1234:doi
1206:doi
1202:201
1176:doi
1172:341
1148:doi
1104:doi
897:(X,
573:to
521:If
400:If
393:on
154:if
150:is
126:set
101:of
73:of
16:In
1315::
1296:,
1288:,
1274:,
1256:,
1228:,
1212:,
1200:,
1184:,
1170:,
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1134:63
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1098:,
1080:,
976:^
529:,
441:E'
236:.
132:.
1292::
1282::
1252::
1236::
1230:2
1208::
1178::
1150::
1140::
1106::
1100:1
1084::
946:μ
917:)
893:E
881:μ
873:μ
866:X
842:N
834:N
830::
827:f
781:.
761:2
757:F
752:2
729:2
725:F
682:F
668:y
665:E
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651:y
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645:f
642:E
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636:x
633:(
630:f
627:,
624:X
618:y
615:,
612:x
589:f
579:F
575:E
571:F
557:Y
551:X
548::
545:f
535:Y
531:E
527:X
523:F
508:G
494:X
488:X
482:G
479::
476:a
463:X
459:G
454:.
445:E
437:E
423:E
413:E
402:E
395:X
391:G
373:G
367:.
364:.
361:.
353:2
349:F
340:1
336:F
327:0
323:F
312:X
304:G
280:E
268:E
245:E
241:X
234:X
218:n
214:F
191:n
187:F
180:N
173:n
165:=
162:E
148:E
144:X
140:E
130:X
122:X
107:E
103:X
95:E
91:E
87:X
83:E
67:X
56:X
37:E
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