1164:
1242:
1259:
261:
567:
426:
1082:
913:
453:
1074:
1320:
860:
1254:
1211:
1201:
1011:
920:
684:
540:
1249:
1196:
1090:
996:
1115:
1095:
1059:
983:
703:
419:
1237:
1016:
978:
930:
1142:
1110:
1100:
1021:
988:
619:
528:
1159:
1064:
840:
768:
161:
1149:
1232:
678:
609:
545:
1001:
759:
719:
412:
101:
1284:
1184:
1006:
728:
574:
184:
845:
798:
793:
788:
630:
513:
471:
219:
1154:
1120:
1028:
738:
693:
535:
458:
8:
1137:
1127:
973:
937:
763:
492:
449:
132:
105:
815:
1289:
1049:
1034:
733:
614:
592:
372:
1206:
942:
903:
898:
805:
723:
508:
481:
390:
128:
97:
1223:
1132:
908:
893:
883:
868:
835:
830:
820:
698:
673:
488:
382:
302:
291:
29:
1299:
1279:
1054:
947:
925:
783:
748:
668:
562:
276:
195:
151:
1189:
1044:
1039:
850:
825:
778:
708:
688:
648:
638:
435:
80:
21:
386:
1314:
1294:
957:
878:
873:
773:
743:
713:
663:
658:
653:
643:
557:
476:
394:
215:
207:
360:
888:
810:
550:
241:
587:
753:
17:
597:
579:
523:
518:
238:
256:, ÎŁ) that is locally finite and invariant under all translations of
604:
463:
377:
234:
404:
168:, ÎŁ), since every measurable set has zero measure.
64:
denote the trivial measure on some measurable space (
55:
40:
which assigns zero measure to every measurable set:
262:There is no infinite-dimensional Lebesgue measure
1312:
420:
1165:RieszâMarkovâKakutani representation theorem
1260:Vitale's random BrunnâMinkowski inequality
427:
413:
206:trivially satisfies the condition to be a
150:trivially satisfies the condition to be a
376:
183:is always a finite measure, and hence a
1313:
358:
222:of all non-negative Radon measures on
408:
361:"Trivial Measures are not so Trivial"
359:Porter, Christopher P. (2015-04-01).
1273:Applications & related
218:. In fact, it is the vertex of the
13:
434:
325: = {0} and observe that
14:
1332:
198:topological space with its Borel
56:Properties of the trivial measure
1202:Lebesgue differentiation theorem
1083:Carathéodory's extension theorem
352:
1:
345:
7:
1255:PrĂ©kopaâLeindler inequality
365:Theory of Computing Systems
10:
1337:
1197:Lebesgue's density theorem
1321:Measures (measure theory)
1272:
1250:MinkowskiâSteiner formula
1220:
1180:
1173:
1073:
1065:Projection-valued measure
966:
859:
628:
501:
442:
387:10.1007/s00224-015-9614-8
162:strictly positive measure
1233:Isoperimetric inequality
1212:VitaliâHahnâSaks theorem
541:Carathéodory's criterion
252:is the only measure on (
1238:BrunnâMinkowski theorem
1107:Decomposition theorems
102:quasi-invariant measure
76:is the trivial measure
1285:Descriptive set theory
1185:Disintegration theorem
620:Universally measurable
185:locally finite measure
1087:Convergence theorems
546:Cylindrical Ï-algebra
321: \ {0} and
1155:Minkowski inequality
1029:Cylinder set measure
914:Infinite-dimensional
529:equivalence relation
459:Lebesgue integration
36:, ÎŁ) is the measure
1150:Hölder's inequality
1012:of random variables
974:Measurable function
861:Particular measures
450:Absolute continuity
309:: simply decompose
106:measurable function
1290:Probability theory
615:Transverse measure
593:Non-measurable set
575:Locally measurable
260:. See the article
131:and that ÎŁ is the
20:, specifically in
1308:
1307:
1268:
1267:
997:almost everywhere
943:Spherical measure
841:Strictly positive
769:Projection-valued
509:Almost everywhere
482:Probability space
179:) = 0,
164:, regardless of (
129:topological space
98:invariant measure
1328:
1243:Milman's reverse
1226:
1224:Lebesgue measure
1178:
1177:
582:
568:infimum/supremum
489:Measurable space
429:
422:
415:
406:
405:
399:
398:
380:
356:
341:) = 0.
305:with respect to
303:singular measure
292:Lebesgue measure
90:) = 0.
30:measurable space
1336:
1335:
1331:
1330:
1329:
1327:
1326:
1325:
1311:
1310:
1309:
1304:
1300:Spectral theory
1280:Convex analysis
1264:
1221:
1216:
1169:
1069:
1017:in distribution
962:
855:
685:Logarithmically
624:
580:
563:Essential range
497:
438:
433:
403:
402:
357:
353:
348:
282:with its usual
277:Euclidean space
248:-algebra, then
244:with its Borel
202:-algebra, then
152:regular measure
58:
26:trivial measure
12:
11:
5:
1334:
1324:
1323:
1306:
1305:
1303:
1302:
1297:
1292:
1287:
1282:
1276:
1274:
1270:
1269:
1266:
1265:
1263:
1262:
1257:
1252:
1247:
1246:
1245:
1235:
1229:
1227:
1218:
1217:
1215:
1214:
1209:
1207:Sard's theorem
1204:
1199:
1194:
1193:
1192:
1190:Lifting theory
1181:
1175:
1171:
1170:
1168:
1167:
1162:
1157:
1152:
1147:
1146:
1145:
1143:FubiniâTonelli
1135:
1130:
1125:
1124:
1123:
1118:
1113:
1105:
1104:
1103:
1098:
1093:
1085:
1079:
1077:
1071:
1070:
1068:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1031:
1026:
1025:
1024:
1022:in probability
1019:
1009:
1004:
999:
993:
992:
991:
986:
981:
970:
968:
964:
963:
961:
960:
955:
950:
945:
940:
935:
934:
933:
923:
918:
917:
916:
906:
901:
896:
891:
886:
881:
876:
871:
865:
863:
857:
856:
854:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
802:
801:
796:
791:
781:
776:
771:
766:
756:
751:
746:
741:
736:
731:
729:Locally finite
726:
716:
711:
706:
701:
696:
691:
681:
676:
671:
666:
661:
656:
651:
646:
641:
635:
633:
626:
625:
623:
622:
617:
612:
607:
602:
601:
600:
590:
585:
577:
572:
571:
570:
560:
555:
554:
553:
543:
538:
533:
532:
531:
521:
516:
511:
505:
503:
499:
498:
496:
495:
486:
485:
484:
474:
469:
461:
456:
446:
444:
443:Basic concepts
440:
439:
436:Measure theory
432:
431:
424:
417:
409:
401:
400:
371:(3): 487â512.
350:
349:
347:
344:
343:
342:
333:) =
265:
227:
188:
169:
155:
121:
120:
91:
81:if and only if
57:
54:
48:) = 0 for all
22:measure theory
9:
6:
4:
3:
2:
1333:
1322:
1319:
1318:
1316:
1301:
1298:
1296:
1295:Real analysis
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1277:
1275:
1271:
1261:
1258:
1256:
1253:
1251:
1248:
1244:
1241:
1240:
1239:
1236:
1234:
1231:
1230:
1228:
1225:
1219:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1191:
1188:
1187:
1186:
1183:
1182:
1179:
1176:
1174:Other results
1172:
1166:
1163:
1161:
1160:RadonâNikodym
1158:
1156:
1153:
1151:
1148:
1144:
1141:
1140:
1139:
1136:
1134:
1133:Fatou's lemma
1131:
1129:
1126:
1122:
1119:
1117:
1114:
1112:
1109:
1108:
1106:
1102:
1099:
1097:
1094:
1092:
1089:
1088:
1086:
1084:
1081:
1080:
1078:
1076:
1072:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1032:
1030:
1027:
1023:
1020:
1018:
1015:
1014:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:Convergence:
994:
990:
987:
985:
982:
980:
977:
976:
975:
972:
971:
969:
965:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
932:
929:
928:
927:
924:
922:
919:
915:
912:
911:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
870:
867:
866:
864:
862:
858:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
800:
799:Outer regular
797:
795:
794:Inner regular
792:
790:
789:Borel regular
787:
786:
785:
782:
780:
777:
775:
772:
770:
767:
765:
761:
757:
755:
752:
750:
747:
745:
742:
740:
737:
735:
732:
730:
727:
725:
721:
717:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
690:
686:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
636:
634:
632:
627:
621:
618:
616:
613:
611:
608:
606:
603:
599:
596:
595:
594:
591:
589:
586:
584:
578:
576:
573:
569:
566:
565:
564:
561:
559:
556:
552:
549:
548:
547:
544:
542:
539:
537:
534:
530:
527:
526:
525:
522:
520:
517:
515:
512:
510:
507:
506:
504:
500:
494:
490:
487:
483:
480:
479:
478:
477:Measure space
475:
473:
470:
468:
466:
462:
460:
457:
455:
451:
448:
447:
445:
441:
437:
430:
425:
423:
418:
416:
411:
410:
407:
396:
392:
388:
384:
379:
374:
370:
366:
362:
355:
351:
340:
336:
332:
328:
324:
320:
317: =
316:
312:
308:
304:
300:
296:
293:
290:-dimensional
289:
286:-algebra and
285:
281:
278:
275:-dimensional
274:
270:
266:
263:
259:
255:
251:
247:
243:
240:
236:
232:
228:
225:
221:
217:
216:Radon measure
213:
209:
208:tight measure
205:
201:
197:
193:
189:
186:
182:
178:
174:
170:
167:
163:
159:
156:
153:
149:
146:
145:
144:
142:
138:
136:
130:
126:
123:Suppose that
118:
115: â
114:
111: :
110:
107:
103:
100:(and hence a
99:
95:
92:
89:
85:
82:
79:
75:
71:
70:
69:
67:
63:
53:
51:
47:
43:
39:
35:
31:
27:
23:
19:
1075:Main results
952:
811:Set function
739:Metric outer
694:Decomposable
551:Cylinder set
464:
368:
364:
354:
338:
334:
330:
326:
322:
318:
314:
310:
306:
298:
294:
287:
283:
279:
272:
268:
257:
253:
249:
245:
242:Banach space
230:
223:
220:pointed cone
211:
203:
199:
191:
180:
176:
172:
165:
157:
147:
140:
134:
124:
122:
116:
112:
108:
93:
87:
83:
77:
73:
65:
61:
59:
49:
45:
41:
37:
33:
25:
15:
1035:compact set
1002:of measures
938:Pushforward
931:Projections
921:Logarithmic
764:Probability
754:Pre-measure
536:Borel space
454:of measures
239:dimensional
160:is never a
68:, ÎŁ).
18:mathematics
1007:in measure
734:Maximising
704:Equivalent
598:Vitali set
378:1503.06332
346:References
214:is also a
104:) for any
72:A measure
1121:Maharam's
1091:Dominated
904:Intensity
899:Hausdorff
806:Saturated
724:Invariant
629:Types of
588:Ï-algebra
558:đ-system
524:Borel set
519:Baire set
395:1433-0490
210:. Hence,
196:Hausdorff
1315:Category
1138:Fubini's
1128:Egorov's
1096:Monotone
1055:variable
1033:Random:
984:Strongly
909:Lebesgue
894:Harmonic
884:Gaussian
869:Counting
836:Spectral
831:Singular
821:s-finite
816:Ï-finite
699:Discrete
674:Complete
631:Measures
605:Null set
493:function
235:infinite
137:-algebra
1050:process
1045:measure
1040:element
979:Bochner
953:Trivial
948:Tangent
926:Product
784:Regular
762:)
749:Perfect
722:)
687:)
679:Content
669:Complex
610:Support
583:-system
472:Measure
28:on any
1116:Jordan
1101:Vitali
1060:vector
989:Weakly
851:Vector
826:Signed
779:Random
720:Quasi-
709:Finite
689:Convex
649:Banach
639:Atomic
467:spaces
452:
393:
233:is an
171:Since
133:Borel
96:is an
52:in ÎŁ.
24:, the
958:Young
879:Euler
874:Dirac
846:Tight
774:Radon
744:Outer
714:Inner
664:Brown
659:Borel
654:Besov
644:Baire
373:arXiv
301:is a
194:is a
127:is a
1222:For
1111:Hahn
967:Maps
889:Haar
760:Sub-
514:Atom
502:Sets
391:ISSN
60:Let
383:doi
313:as
271:is
267:If
229:If
190:If
139:on
16:In
1317::
389:.
381:.
369:56
367:.
363:.
297:,
143:.
758:(
718:(
683:(
581:Ï
491:/
465:L
428:e
421:t
414:v
397:.
385::
375::
339:B
337:(
335:λ
331:A
329:(
327:Ό
323:B
319:R
315:A
311:R
307:λ
299:Ό
295:λ
288:n
284:Ï
280:R
273:n
269:X
264:.
258:X
254:X
250:Ό
246:Ï
237:-
231:X
226:.
224:X
212:Ό
204:Ό
200:Ï
192:X
187:.
181:Ό
177:X
175:(
173:Ό
166:X
158:Ό
154:.
148:Ό
141:X
135:Ï
125:X
119:.
117:X
113:X
109:f
94:Ό
88:X
86:(
84:Μ
78:Ό
74:Μ
66:X
62:Ό
50:A
46:A
44:(
42:Ό
38:Ό
34:X
32:(
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