658:
698:). However, they are isomorphic as measurable spaces. Indeed, the closed interval is evidently isomorphic to a shorter closed subinterval of the open interval. Also the open interval is evidently isomorphic to a part of the closed interval (just itself, for instance).
44:. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem.
991:
1028:
841:
809:
755:
1110:
863:
777:
726:
381:
323:
297:
159:
125:
503:
1063:
647:
617:
410:
355:
456:
433:
587:
567:
543:
523:
251:
231:
199:
179:
90:
70:
546:
17:
36:. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of
1100:
657:
202:
865:(as measurable spaces, of course, not as topological spaces) can be made by a well-known trick with interspersed digits; for example,
1105:
993:
is clearly injective. It is easy to check that it is bimeasurable. (However, it is not bijective; for example, the number
953:
996:
814:
782:
731:
37:
326:
254:
846:
760:
709:
691:
360:
302:
263:
129:
95:
476:
1033:
622:
592:
25:
687:
386:
331:
1085:
See
Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
438:
415:
572:
552:
528:
508:
236:
216:
184:
164:
75:
55:
1094:
683:
695:
461:
An isomorphism (between two measurable spaces) is, by definition, a bimeasurable
21:
462:
41:
33:
258:
40:
contains very strong results about isomorphic measurable spaces, see
92:
be measurable spaces. If there exist injective, bimeasurable maps
465:. If it exists, these measurable spaces are called isomorphic.
757:
are isomorphic as measurable spaces. It is immediate to embed
1036:
999:
956:
849:
817:
785:
763:
734:
712:
625:
595:
575:
555:
531:
511:
479:
441:
418:
389:
363:
334:
305:
266:
239:
219:
187:
167:
132:
98:
78:
58:
1057:
1022:
986:{\displaystyle g:\mathbb {R} ^{2}\to \mathbb {R} }
985:
857:
835:
803:
771:
749:
720:
641:
611:
581:
561:
537:
517:
497:
450:
427:
404:
375:
349:
317:
291:
245:
225:
193:
173:
153:
119:
84:
64:
547:proof of the Cantor–Bernstein–Schroeder theorem
8:
1111:Theorems in the foundations of mathematics
1035:
1003:
998:
979:
978:
969:
965:
964:
955:
851:
850:
848:
824:
820:
819:
816:
792:
788:
787:
784:
765:
764:
762:
741:
737:
736:
733:
714:
713:
711:
630:
624:
600:
594:
574:
554:
530:
510:
478:
440:
417:
388:
362:
333:
304:
271:
265:
238:
218:
186:
166:
131:
97:
77:
57:
656:
32:since measurable spaces are also called
569:is measurable, since it coincides with
7:
233:is bimeasurable" means that, first,
1023:{\displaystyle 1/11=0.090909\dots }
357:is measurable for every measurable
299:is measurable for every measurable
473:First, one constructs a bijection
30:Borel Schroeder–Bernstein theorem,
18:Cantor–Bernstein–Schroeder theorem
14:
836:{\displaystyle \mathbb {R} ^{2}.}
804:{\displaystyle \mathbb {R} ^{2}.}
750:{\displaystyle \mathbb {R} ^{2}}
690:are evidently non-isomorphic as
412:must be a measurable subset of
1052:
1040:
975:
619:on its complement. Similarly,
489:
399:
393:
344:
338:
286:
280:
142:
108:
1:
589:on a measurable set and with
858:{\displaystyle \mathbb {R} }
772:{\displaystyle \mathbb {R} }
721:{\displaystyle \mathbb {R} }
811:The converse, embedding of
203:Schröder–Bernstein property
1127:
1101:Theorems in measure theory
435:not necessarily the whole
376:{\displaystyle A\subset X}
318:{\displaystyle B\subset Y}
292:{\displaystyle f^{-1}(B)}
154:{\displaystyle g:Y\to X,}
120:{\displaystyle f:X\to Y,}
498:{\displaystyle h:X\to Y}
28:, sometimes called the
1106:Descriptive set theory
1076:A Course on Borel Sets
1059:
1058:{\displaystyle g(x,y)}
1024:
987:
859:
837:
805:
773:
751:
722:
674:
643:
642:{\displaystyle h^{-1}}
613:
612:{\displaystyle g^{-1}}
583:
563:
539:
519:
499:
452:
429:
406:
377:
351:
319:
293:
247:
227:
195:
175:
155:
121:
86:
66:
24:has a counterpart for
1060:
1025:
988:
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838:
806:
774:
752:
723:
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644:
614:
584:
564:
540:
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453:
430:
407:
378:
352:
320:
294:
248:
228:
196:
176:
156:
122:
87:
67:
38:standard Borel spaces
1034:
997:
954:
847:
815:
783:
761:
732:
710:
686:(0, 1) and the
623:
593:
573:
553:
529:
509:
477:
439:
416:
405:{\displaystyle f(X)}
387:
361:
350:{\displaystyle f(A)}
332:
303:
264:
237:
217:
201:are isomorphic (the
185:
165:
130:
96:
76:
56:
42:Kuratowski's theorem
1030:is not of the form
325:), and second, the
1055:
1020:
983:
855:
833:
801:
769:
747:
718:
692:topological spaces
675:
639:
609:
579:
559:
545:exactly as in the
535:
515:
495:
451:{\displaystyle Y.}
448:
428:{\displaystyle Y,}
425:
402:
373:
347:
315:
289:
243:
223:
191:
171:
151:
117:
82:
62:
1078:, Springer, 1998.
1074:S.M. Srivastava,
582:{\displaystyle f}
562:{\displaystyle h}
538:{\displaystyle g}
518:{\displaystyle f}
246:{\displaystyle f}
226:{\displaystyle f}
194:{\displaystyle Y}
174:{\displaystyle X}
85:{\displaystyle Y}
65:{\displaystyle X}
26:measurable spaces
1118:
1064:
1062:
1061:
1056:
1029:
1027:
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1021:
1007:
992:
990:
989:
984:
982:
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968:
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839:
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823:
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748:
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618:
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411:
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382:
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324:
322:
321:
316:
298:
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279:
278:
252:
250:
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244:
232:
230:
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200:
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126:
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123:
118:
91:
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88:
83:
71:
69:
68:
63:
1126:
1125:
1121:
1120:
1119:
1117:
1116:
1115:
1091:
1090:
1071:
1032:
1031:
995:
994:
963:
952:
951:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
897:
894:
891:
888:
885:
881:
877:
845:
844:
818:
813:
812:
786:
781:
780:
759:
758:
735:
730:
729:
708:
707:
704:
688:closed interval
680:
668:
662:
655:
649:is measurable.
626:
621:
620:
596:
591:
590:
571:
570:
551:
550:
527:
526:
507:
506:
475:
474:
471:
437:
436:
414:
413:
385:
384:
359:
358:
330:
329:
301:
300:
267:
262:
261:
235:
234:
215:
214:
211:
183:
182:
163:
162:
128:
127:
94:
93:
74:
73:
54:
53:
50:
12:
11:
5:
1124:
1122:
1114:
1113:
1108:
1103:
1093:
1092:
1089:
1088:
1087:
1086:
1080:
1079:
1070:
1067:
1054:
1051:
1048:
1045:
1042:
1039:
1019:
1016:
1013:
1010:
1006:
1002:
981:
977:
972:
967:
962:
959:
948:
947:
853:
832:
827:
822:
800:
795:
790:
767:
744:
739:
728:and the plane
716:
706:The real line
703:
700:
694:(that is, not
679:
676:
654:
651:
636:
633:
629:
606:
603:
599:
578:
558:
534:
514:
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491:
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482:
470:
467:
447:
444:
424:
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401:
398:
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372:
369:
366:
346:
343:
340:
337:
314:
311:
308:
288:
285:
282:
277:
274:
270:
257:(that is, the
242:
222:
210:
207:
190:
170:
150:
147:
144:
141:
138:
135:
116:
113:
110:
107:
104:
101:
81:
61:
49:
46:
13:
10:
9:
6:
4:
3:
2:
1123:
1112:
1109:
1107:
1104:
1102:
1099:
1098:
1096:
1084:
1083:
1082:
1081:
1077:
1073:
1072:
1068:
1066:
1049:
1046:
1043:
1037:
1017:
1014:
1011:
1008:
1004:
1000:
970:
960:
957:
882:271.82818 28…
875:
871:
868:
867:
866:
830:
825:
798:
793:
742:
701:
699:
697:
693:
689:
685:
684:open interval
677:
671:
665:
661:Example maps
659:
652:
650:
634:
631:
627:
604:
601:
597:
576:
556:
548:
532:
512:
492:
486:
483:
480:
468:
466:
464:
459:
445:
442:
422:
419:
396:
390:
370:
367:
364:
341:
335:
328:
312:
309:
306:
283:
275:
272:
268:
260:
256:
240:
220:
208:
206:
204:
188:
168:
148:
145:
139:
136:
133:
114:
111:
105:
102:
99:
79:
59:
47:
45:
43:
39:
35:
31:
27:
23:
19:
1075:
949:
878:3.14159 265…
873:
869:
705:
696:homeomorphic
681:
669:
667::(0,1)→ and
663:
472:
460:
213:The phrase "
212:
51:
34:Borel spaces
29:
15:
872:(π,100e) =
48:The theorem
1095:Categories
1069:References
549:. Second,
255:measurable
22:set theory
1018:…
976:→
702:Example 2
678:Example 1
632:−
602:−
490:→
463:bijection
383:. (Thus,
368:⊂
310:⊂
273:−
143:→
109:→
1015:0.090909
950:The map
673::→(0,1).
653:Examples
259:preimage
209:Comments
505:out of
843:into
779:into
469:Proof
327:image
161:then
884:) =
682:The
525:and
181:and
72:and
52:Let
16:The
1065:).
253:is
205:).
20:of
1097::
1009:11
946:….
880:,
458:)
1053:)
1050:y
1047:,
1044:x
1041:(
1038:g
1012:=
1005:/
1001:1
980:R
971:2
966:R
961::
958:g
944:5
941:8
938:6
935:2
932:2
929:8
926:9
923:1
920:5
917:8
914:1
911:2
908:4
905:8
902:1
900:.
898:1
895:3
892:7
889:0
886:2
876:(
874:g
870:g
852:R
831:.
826:2
821:R
799:.
794:2
789:R
766:R
743:2
738:R
715:R
670:g
664:f
635:1
628:h
605:1
598:g
577:f
557:h
533:g
513:f
493:Y
487:X
484::
481:h
446:.
443:Y
423:,
420:Y
400:)
397:X
394:(
391:f
371:X
365:A
345:)
342:A
339:(
336:f
313:Y
307:B
287:)
284:B
281:(
276:1
269:f
241:f
221:f
189:Y
169:X
149:,
146:X
140:Y
137::
134:g
115:,
112:Y
106:X
103::
100:f
80:Y
60:X
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