22:
295:
1178:
188:
1256:
1273:
320:
preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as
94:
66:
581:
440:
43:
73:
1096:
927:
467:
1088:
80:
1334:
874:
1268:
113:
244:
62:
51:
1225:
1215:
1025:
934:
698:
204:
47:
554:
345:
is not invariant under translation (like
Lebesgue measure is), but is quasi-invariant under all translations.
1263:
1210:
1104:
1010:
1129:
1109:
1073:
997:
717:
433:
231:
223:
1251:
1030:
992:
944:
1339:
1156:
1124:
1114:
1035:
1002:
633:
542:
402:
208:
1173:
1078:
854:
782:
87:
1163:
1246:
692:
623:
32:
559:
1015:
773:
426:
36:
1298:
1198:
1020:
742:
588:
364:
405: â Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
328:
859:
812:
807:
802:
644:
527:
485:
235:
149:
1168:
1134:
1042:
752:
707:
549:
472:
327:
In general, the 'freedom' of moving within a measure class by multiplication gives rise to
8:
1151:
1141:
987:
951:
777:
506:
463:
829:
1303:
1063:
1048:
747:
628:
606:
153:
1220:
956:
917:
912:
819:
737:
522:
495:
408:
1237:
1146:
922:
907:
897:
882:
849:
844:
834:
712:
687:
502:
335:
192:
1313:
1293:
1068:
966:
961:
939:
797:
762:
682:
576:
387:
353:
339:
196:
165:
1203:
1058:
1053:
864:
839:
792:
722:
702:
662:
652:
449:
219:
176:
1328:
1308:
971:
892:
887:
787:
757:
727:
677:
672:
667:
657:
571:
490:
367:
142:
902:
824:
564:
356:
305:
601:
767:
127:
611:
593:
537:
532:
21:
618:
477:
418:
374:
that is quasi-invariant under all translations by elements of
230:
should exist everywhere; or that the two measures should be
308:. Considering the whole equivalence class of measures
247:
203:
on Ό is locally expressible as multiplication by the
226:
of the transformed measure ÎŒ′ with respect to
289:
1326:
290:{\displaystyle \mu '=T_{*}(\mu )\approx \mu .}
434:
160:. An important class of examples occurs when
1179:RieszâMarkovâKakutani representation theorem
152:which, roughly speaking, is multiplied by a
50:. Unsourced material may be challenged and
1274:Vitale's random BrunnâMinkowski inequality
441:
427:
114:Learn how and when to remove this message
1327:
218:To express this idea more formally in
422:
331:, when transformations are composed.
1287:Applications & related
48:adding citations to reliable sources
15:
13:
448:
316:, it is also the same to say that
304:preserves the concept of a set of
14:
1351:
300:That means, in other words, that
187:is any measure that locally is a
137:with respect to a transformation
1216:Lebesgue differentiation theorem
1097:Carathéodory's extension theorem
20:
207:determinant of the derivative (
275:
269:
1:
414:
222:terms, the idea is that the
7:
1269:PrĂ©kopaâLeindler inequality
396:
10:
1356:
1211:Lebesgue's density theorem
1335:Measures (measure theory)
1286:
1264:MinkowskiâSteiner formula
1234:
1194:
1187:
1087:
1079:Projection-valued measure
980:
873:
642:
515:
456:
63:"Quasi-invariant measure"
1247:Isoperimetric inequality
1226:VitaliâHahnâSaks theorem
555:Carathéodory's criterion
348:It can be shown that if
224:RadonâNikodym derivative
1252:BrunnâMinkowski theorem
1121:Decomposition theorems
382:) < +â or
322:invariant measure class
132:quasi-invariant measure
1299:Descriptive set theory
1199:Disintegration theorem
634:Universally measurable
403:CameronâMartin theorem
291:
1101:Convergence theorems
560:Cylindrical Ï-algebra
292:
236:absolutely continuous
199:. Then the effect of
1169:Minkowski inequality
1043:Cylinder set measure
928:Infinite-dimensional
543:equivalence relation
473:Lebesgue integration
245:
44:improve this article
1164:Hölder's inequality
1026:of random variables
988:Measurable function
875:Particular measures
464:Absolute continuity
1304:Probability theory
629:Transverse measure
607:Non-measurable set
589:Locally measurable
378:, then either dim(
287:
154:numerical function
1340:Dynamical systems
1322:
1321:
1282:
1281:
1011:almost everywhere
957:Spherical measure
855:Strictly positive
783:Projection-valued
523:Almost everywhere
496:Probability space
409:Invariant measure
189:measure with base
124:
123:
116:
98:
1347:
1257:Milman's reverse
1240:
1238:Lebesgue measure
1192:
1191:
596:
582:infimum/supremum
503:Measurable space
443:
436:
429:
420:
419:
393: ⥠0.
336:Gaussian measure
312:, equivalent to
296:
294:
293:
288:
268:
267:
255:
193:Lebesgue measure
148:to itself, is a
119:
112:
108:
105:
99:
97:
56:
24:
16:
1355:
1354:
1350:
1349:
1348:
1346:
1345:
1344:
1325:
1324:
1323:
1318:
1314:Spectral theory
1294:Convex analysis
1278:
1235:
1230:
1183:
1083:
1031:in distribution
976:
869:
699:Logarithmically
638:
594:
577:Essential range
511:
452:
447:
417:
399:
388:trivial measure
340:Euclidean space
334:As an example,
263:
259:
248:
246:
243:
242:
234:(i.e. mutually
197:Euclidean space
166:smooth manifold
120:
109:
103:
100:
57:
55:
41:
25:
12:
11:
5:
1353:
1343:
1342:
1337:
1320:
1319:
1317:
1316:
1311:
1306:
1301:
1296:
1290:
1288:
1284:
1283:
1280:
1279:
1277:
1276:
1271:
1266:
1261:
1260:
1259:
1249:
1243:
1241:
1232:
1231:
1229:
1228:
1223:
1221:Sard's theorem
1218:
1213:
1208:
1207:
1206:
1204:Lifting theory
1195:
1189:
1185:
1184:
1182:
1181:
1176:
1171:
1166:
1161:
1160:
1159:
1157:FubiniâTonelli
1149:
1144:
1139:
1138:
1137:
1132:
1127:
1119:
1118:
1117:
1112:
1107:
1099:
1093:
1091:
1085:
1084:
1082:
1081:
1076:
1071:
1066:
1061:
1056:
1051:
1045:
1040:
1039:
1038:
1036:in probability
1033:
1023:
1018:
1013:
1007:
1006:
1005:
1000:
995:
984:
982:
978:
977:
975:
974:
969:
964:
959:
954:
949:
948:
947:
937:
932:
931:
930:
920:
915:
910:
905:
900:
895:
890:
885:
879:
877:
871:
870:
868:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
816:
815:
810:
805:
795:
790:
785:
780:
770:
765:
760:
755:
750:
745:
743:Locally finite
740:
730:
725:
720:
715:
710:
705:
695:
690:
685:
680:
675:
670:
665:
660:
655:
649:
647:
640:
639:
637:
636:
631:
626:
621:
616:
615:
614:
604:
599:
591:
586:
585:
584:
574:
569:
568:
567:
557:
552:
547:
546:
545:
535:
530:
525:
519:
517:
513:
512:
510:
509:
500:
499:
498:
488:
483:
475:
470:
460:
458:
457:Basic concepts
454:
453:
450:Measure theory
446:
445:
438:
431:
423:
416:
413:
412:
411:
406:
398:
395:
365:locally finite
298:
297:
286:
283:
280:
277:
274:
271:
266:
262:
258:
254:
251:
220:measure theory
177:diffeomorphism
122:
121:
28:
26:
19:
9:
6:
4:
3:
2:
1352:
1341:
1338:
1336:
1333:
1332:
1330:
1315:
1312:
1310:
1309:Real analysis
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1291:
1289:
1285:
1275:
1272:
1270:
1267:
1265:
1262:
1258:
1255:
1254:
1253:
1250:
1248:
1245:
1244:
1242:
1239:
1233:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1205:
1202:
1201:
1200:
1197:
1196:
1193:
1190:
1188:Other results
1186:
1180:
1177:
1175:
1174:RadonâNikodym
1172:
1170:
1167:
1165:
1162:
1158:
1155:
1154:
1153:
1150:
1148:
1147:Fatou's lemma
1145:
1143:
1140:
1136:
1133:
1131:
1128:
1126:
1123:
1122:
1120:
1116:
1113:
1111:
1108:
1106:
1103:
1102:
1100:
1098:
1095:
1094:
1092:
1090:
1086:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1046:
1044:
1041:
1037:
1034:
1032:
1029:
1028:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:Convergence:
1008:
1004:
1001:
999:
996:
994:
991:
990:
989:
986:
985:
983:
979:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
946:
943:
942:
941:
938:
936:
933:
929:
926:
925:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
880:
878:
876:
872:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
814:
813:Outer regular
811:
809:
808:Inner regular
806:
804:
803:Borel regular
801:
800:
799:
796:
794:
791:
789:
786:
784:
781:
779:
775:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
735:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
700:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
650:
648:
646:
641:
635:
632:
630:
627:
625:
622:
620:
617:
613:
610:
609:
608:
605:
603:
600:
598:
592:
590:
587:
583:
580:
579:
578:
575:
573:
570:
566:
563:
562:
561:
558:
556:
553:
551:
548:
544:
541:
540:
539:
536:
534:
531:
529:
526:
524:
521:
520:
518:
514:
508:
504:
501:
497:
494:
493:
492:
491:Measure space
489:
487:
484:
482:
480:
476:
474:
471:
469:
465:
462:
461:
459:
455:
451:
444:
439:
437:
432:
430:
425:
424:
421:
410:
407:
404:
401:
400:
394:
392:
389:
385:
381:
377:
373:
369:
368:Borel measure
366:
362:
358:
355:
351:
346:
344:
341:
337:
332:
330:
325:
323:
319:
315:
311:
307:
303:
284:
281:
278:
272:
264:
260:
256:
252:
249:
241:
240:
239:
237:
233:
229:
225:
221:
216:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
167:
163:
159:
155:
151:
147:
144:
143:measure space
140:
136:
133:
129:
118:
115:
107:
104:December 2009
96:
93:
89:
86:
82:
79:
75:
72:
68:
65: â
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
1089:Main results
825:Set function
753:Metric outer
733:
708:Decomposable
565:Cylinder set
478:
390:
383:
379:
375:
371:
360:
357:Banach space
349:
347:
342:
333:
326:
321:
317:
313:
309:
306:measure zero
301:
299:
227:
217:
212:
200:
184:
180:
172:
168:
161:
157:
145:
138:
134:
131:
125:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
1049:compact set
1016:of measures
952:Pushforward
945:Projections
935:Logarithmic
778:Probability
768:Pre-measure
550:Borel space
468:of measures
209:pushforward
128:mathematics
1329:Categories
1021:in measure
748:Maximising
718:Equivalent
612:Vitali set
415:References
232:equivalent
74:newspapers
1135:Maharam's
1105:Dominated
918:Intensity
913:Hausdorff
820:Saturated
738:Invariant
643:Types of
602:Ï-algebra
572:đ-system
538:Borel set
533:Baire set
354:separable
282:μ
279:≈
273:μ
265:∗
250:μ
141:, from a
31:does not
1152:Fubini's
1142:Egorov's
1110:Monotone
1069:variable
1047:Random:
998:Strongly
923:Lebesgue
908:Harmonic
898:Gaussian
883:Counting
850:Spectral
845:Singular
835:s-finite
830:Ï-finite
713:Discrete
688:Complete
645:Measures
619:Null set
507:function
397:See also
329:cocycles
253:′
205:Jacobian
1064:process
1059:measure
1054:element
993:Bochner
967:Trivial
962:Tangent
940:Product
798:Regular
776:)
763:Perfect
736:)
701:)
693:Content
683:Complex
624:Support
597:-system
486:Measure
386:is the
150:measure
88:scholar
52:removed
37:sources
1130:Jordan
1115:Vitali
1074:vector
1003:Weakly
865:Vector
840:Signed
793:Random
734:Quasi-
723:Finite
703:Convex
663:Banach
653:Atomic
481:spaces
466:
183:, and
90:
83:
76:
69:
61:
972:Young
893:Euler
888:Dirac
860:Tight
788:Radon
758:Outer
728:Inner
678:Brown
673:Borel
668:Besov
658:Baire
363:is a
352:is a
211:) of
175:is a
164:is a
95:JSTOR
81:books
1236:For
1125:Hahn
981:Maps
903:Haar
774:Sub-
528:Atom
516:Sets
359:and
191:the
130:, a
67:news
35:any
33:cite
370:on
338:on
238:):
195:on
179:of
156:of
126:In
46:by
1331::
324:.
215:.
171:,
772:(
732:(
697:(
595:Ï
505:/
479:L
442:e
435:t
428:v
391:Ό
384:Ό
380:E
376:E
372:E
361:Ό
350:E
343:R
318:T
314:Ό
310:Μ
302:T
285:.
276:)
270:(
261:T
257:=
228:Ό
213:T
201:T
185:Ό
181:M
173:T
169:M
162:X
158:T
146:X
139:T
135:Ό
117:)
111:(
106:)
102:(
92:·
85:·
78:·
71:·
54:.
40:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
