853:
1193:
704:
1271:
983:
taking the value of one on the rationals, and zero, otherwise. Clearly the measure of this function should be zero, but how can one find regions that are continuous, given that the rationals are
537:
949:
208:
319:
136:
1067:
636:
450:
424:
376:
1351:
1089:
476:
1387:
Lawrence C. Evans, Ronald F. Gariepy, "Measure Theory and fine properties of functions", CRC Press Taylor & Francis Group, Textbooks in mathematics, Theorem 1.14
1302:
1008:
1322:
744:
724:
656:
597:
577:
557:
496:
981:
859:
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.
871:
and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.
752:
1097:
661:
1204:
1430:
501:
17:
1425:
867:
The proof of Lusin's theorem can be found in many classical books. Intuitively, one expects it as a consequence of
74:
889:
163:
250:
95:
30:
This article is about the theorem of real analysis. For the separation theorem in descriptive set theory, see
987:
in the reals? The requirements for Lusin's theorem can be satisfied with the following construction of a set
31:
1015:
879:
The strength of Lusin's theorem might not be readily apparent, as can be demonstrated by example. Consider
606:
429:
397:
343:
1327:
1072:
455:
42:
1280:
8:
868:
70:
66:
990:
1307:
884:
880:
729:
709:
641:
582:
562:
542:
481:
954:
1384:
M. B. Feldman, "A Proof of Lusin's
Theorem", American Math. Monthly, 88 (1981), 191-2
218:
62:
387:
78:
1405:
600:
848:{\displaystyle \ \sup _{x\in X}|f_{\varepsilon }(x)|\leq \sup _{x\in X}|f(x)|}
1419:
391:
379:
54:
1188:{\displaystyle G_{n}=(x_{n}-\varepsilon /2^{n},x_{n}+\varepsilon /2^{n})}
38:
1304:"knock out" all of the rationals, leaving behind a compact, closed set
984:
236:, defined on the interval and almost-everywhere finite, if for any
1381:
W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990
699:{\displaystyle f_{\varepsilon }:X\rightarrow \mathbb {R} ^{d}}
1324:
which contains no rationals, and has a measure of more than
1266:{\displaystyle E:=\setminus \bigcup _{n=1}^{\infty }G_{n}}
1376:
Real
Analysis: Modern Techniques and Their Applications
1367:
N. Lusin. Sur les propriétés des fonctions mesurables,
1330:
1310:
1283:
1207:
1100:
1075:
1018:
993:
957:
892:
755:
732:
712:
664:
644:
609:
585:
565:
545:
504:
484:
458:
432:
400:
346:
253:
166:
98:
81:, "every measurable function is nearly continuous".
244:, continuous on , such that the measure of the set
1369:Comptes rendus de l'Académie des Sciences de Paris
1345:
1316:
1296:
1265:
1187:
1083:
1061:
1002:
975:
943:
847:
738:
718:
698:
658:to be compact and even find a continuous function
650:
630:
591:
571:
551:
532:{\displaystyle \mu (A\setminus E)<\varepsilon }
531:
490:
470:
444:
418:
370:
313:
202:
130:
1417:
808:
760:
1406:"Luzin criterion - Encyclopedia of Mathematics"
1056:
1019:
944:{\displaystyle 1_{\mathbb {Q} }:\to \{0,1\}}
938:
926:
308:
254:
203:{\displaystyle \mu (E)>b-a-\varepsilon .}
314:{\displaystyle \{x\in :f(x)\neq \phi (x)\}}
145: > 0, there exists a compact
141:be a measurable function. Then, for every
131:{\displaystyle f:\rightarrow \mathbb {C} }
1077:
899:
706:with compact support that coincides with
686:
618:
124:
478:of finite measure there is a closed set
14:
1418:
1062:{\displaystyle \{x_{n};n=1,2,\dots \}}
84:
24:
1248:
631:{\displaystyle Y=\mathbb {R} ^{d}}
465:
390:topological space equipped with a
356:
25:
1442:
1229:
514:
445:{\displaystyle \varepsilon >0}
73:on nearly all its domain. In the
426:be a measurable function. Given
419:{\displaystyle f:X\rightarrow Y}
371:{\displaystyle (X,\Sigma ,\mu )}
229:is defined using this topology.
1346:{\displaystyle 1-2\varepsilon }
1277:Then the sequence of open sets
862:
335:
1398:
1226:
1214:
1182:
1114:
970:
958:
923:
920:
908:
841:
837:
831:
824:
800:
796:
790:
776:
681:
520:
508:
410:
365:
347:
305:
299:
290:
284:
275:
263:
176:
170:
120:
117:
105:
13:
1:
1356:
1084:{\displaystyle \mathbb {Q} }
471:{\displaystyle A\in \Sigma }
7:
10:
1447:
1431:Theorems in measure theory
874:
32:Lusin's separation theorem
29:
1426:Theorems in real analysis
27:Theorem in measure theory
149: ⊆ such that
69:if and only if it is a
1371:154 (1912), 1688–1690.
1347:
1318:
1298:
1267:
1252:
1189:
1085:
1069:be any enumeration of
1063:
1004:
977:
945:
849:
740:
720:
700:
652:
632:
593:
573:
553:
533:
492:
472:
446:
420:
372:
315:
232:Also for any function
204:
132:
89:For an interval , let
1348:
1319:
1299:
1297:{\displaystyle G_{n}}
1268:
1232:
1190:
1086:
1064:
1005:
978:
951:on the unit interval
946:
850:
741:
721:
701:
653:
633:
594:
574:
554:
534:
493:
473:
447:
421:
373:
316:
221:from ; continuity of
205:
133:
43:mathematical analysis
1328:
1308:
1281:
1205:
1098:
1073:
1016:
991:
955:
890:
753:
730:
710:
662:
642:
607:
583:
563:
543:
502:
482:
456:
430:
398:
344:
251:
240:there is a function
164:
96:
75:informal formulation
18:Luzin's theorem
1378:, 2nd ed. Chapter 7
85:Classical statement
71:continuous function
65:finite function is
1343:
1314:
1294:
1263:
1185:
1081:
1059:
1003:{\displaystyle E.}
1000:
973:
941:
885:indicator function
881:Dirichlet function
845:
822:
774:
736:
716:
696:
648:
628:
589:
579:is continuous. If
569:
549:
529:
488:
468:
442:
416:
368:
311:
238:ε > 0
200:
157:is continuous and
128:
1317:{\displaystyle E}
807:
759:
758:
739:{\displaystyle E}
719:{\displaystyle f}
651:{\displaystyle E}
592:{\displaystyle A}
572:{\displaystyle E}
552:{\displaystyle f}
491:{\displaystyle E}
219:subspace topology
63:almost-everywhere
59:Lusin's criterion
16:(Redirected from
1438:
1410:
1409:
1402:
1352:
1350:
1349:
1344:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1293:
1292:
1272:
1270:
1269:
1264:
1262:
1261:
1251:
1246:
1194:
1192:
1191:
1186:
1181:
1180:
1171:
1160:
1159:
1147:
1146:
1137:
1126:
1125:
1110:
1109:
1090:
1088:
1087:
1082:
1080:
1068:
1066:
1065:
1060:
1031:
1030:
1009:
1007:
1006:
1001:
982:
980:
979:
976:{\displaystyle }
974:
950:
948:
947:
942:
904:
903:
902:
869:Egorov's theorem
854:
852:
851:
846:
844:
827:
821:
803:
789:
788:
779:
773:
756:
745:
743:
742:
737:
725:
723:
722:
717:
705:
703:
702:
697:
695:
694:
689:
674:
673:
657:
655:
654:
649:
638:, we can choose
637:
635:
634:
629:
627:
626:
621:
598:
596:
595:
590:
578:
576:
575:
570:
558:
556:
555:
550:
538:
536:
535:
530:
497:
495:
494:
489:
477:
475:
474:
469:
451:
449:
448:
443:
425:
423:
422:
417:
388:second-countable
377:
375:
374:
369:
320:
318:
317:
312:
209:
207:
206:
201:
137:
135:
134:
129:
127:
79:J. E. Littlewood
21:
1446:
1445:
1441:
1440:
1439:
1437:
1436:
1435:
1416:
1415:
1414:
1413:
1404:
1403:
1399:
1359:
1329:
1326:
1325:
1309:
1306:
1305:
1288:
1284:
1282:
1279:
1278:
1257:
1253:
1247:
1236:
1206:
1203:
1202:
1176:
1172:
1167:
1155:
1151:
1142:
1138:
1133:
1121:
1117:
1105:
1101:
1099:
1096:
1095:
1076:
1074:
1071:
1070:
1026:
1022:
1017:
1014:
1013:
992:
989:
988:
956:
953:
952:
898:
897:
893:
891:
888:
887:
877:
865:
840:
823:
811:
799:
784:
780:
775:
763:
754:
751:
750:
746:and such that
731:
728:
727:
711:
708:
707:
690:
685:
684:
669:
665:
663:
660:
659:
643:
640:
639:
622:
617:
616:
608:
605:
604:
601:locally compact
584:
581:
580:
564:
561:
560:
544:
541:
540:
503:
500:
499:
483:
480:
479:
457:
454:
453:
431:
428:
427:
399:
396:
395:
345:
342:
341:
338:
332:is measurable.
252:
249:
248:
165:
162:
161:
123:
97:
94:
93:
87:
61:states that an
51:Luzin's theorem
47:Lusin's theorem
35:
28:
23:
22:
15:
12:
11:
5:
1444:
1434:
1433:
1428:
1412:
1411:
1396:
1395:
1389:
1388:
1385:
1382:
1379:
1372:
1358:
1355:
1342:
1339:
1336:
1333:
1313:
1291:
1287:
1275:
1274:
1260:
1256:
1250:
1245:
1242:
1239:
1235:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1196:
1195:
1184:
1179:
1175:
1170:
1166:
1163:
1158:
1154:
1150:
1145:
1141:
1136:
1132:
1129:
1124:
1120:
1116:
1113:
1108:
1104:
1079:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1029:
1025:
1021:
999:
996:
972:
969:
966:
963:
960:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
901:
896:
883:, that is the
876:
873:
864:
861:
857:
856:
843:
839:
836:
833:
830:
826:
820:
817:
814:
810:
806:
802:
798:
795:
792:
787:
783:
778:
772:
769:
766:
762:
735:
715:
693:
688:
683:
680:
677:
672:
668:
647:
625:
620:
615:
612:
588:
568:
559:restricted to
548:
528:
525:
522:
519:
516:
513:
510:
507:
487:
467:
464:
461:
441:
438:
435:
415:
412:
409:
406:
403:
367:
364:
361:
358:
355:
352:
349:
337:
334:
322:
321:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
225:restricted to
211:
210:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
153:restricted to
139:
138:
126:
122:
119:
116:
113:
110:
107:
104:
101:
86:
83:
26:
9:
6:
4:
3:
2:
1443:
1432:
1429:
1427:
1424:
1423:
1421:
1407:
1401:
1397:
1394:
1393:
1386:
1383:
1380:
1377:
1373:
1370:
1366:
1365:
1364:
1363:
1354:
1340:
1337:
1334:
1331:
1311:
1289:
1285:
1258:
1254:
1243:
1240:
1237:
1233:
1223:
1220:
1217:
1211:
1208:
1201:
1200:
1199:
1177:
1173:
1168:
1164:
1161:
1156:
1152:
1148:
1143:
1139:
1134:
1130:
1127:
1122:
1118:
1111:
1106:
1102:
1094:
1093:
1092:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1027:
1023:
1010:
997:
994:
986:
967:
964:
961:
935:
932:
929:
917:
914:
911:
905:
894:
886:
882:
872:
870:
860:
834:
828:
818:
815:
812:
804:
793:
785:
781:
770:
767:
764:
749:
748:
747:
733:
713:
691:
678:
675:
670:
666:
645:
623:
613:
610:
602:
586:
566:
546:
526:
523:
517:
511:
505:
485:
462:
459:
439:
436:
433:
413:
407:
404:
401:
393:
392:Borel algebra
389:
385:
381:
380:Radon measure
362:
359:
353:
350:
333:
331:
327:
324:is less than
302:
296:
293:
287:
281:
278:
272:
269:
266:
260:
257:
247:
246:
245:
243:
239:
235:
230:
228:
224:
220:
217:inherits the
216:
197:
194:
191:
188:
185:
182:
179:
173:
167:
160:
159:
158:
156:
152:
148:
144:
114:
111:
108:
102:
99:
92:
91:
90:
82:
80:
76:
72:
68:
64:
60:
56:
55:Nikolai Luzin
52:
48:
44:
40:
33:
19:
1400:
1391:
1390:
1375:
1374:G. Folland.
1368:
1361:
1360:
1276:
1197:
1011:
878:
866:
863:On the proof
858:
452:, for every
383:
339:
336:General form
329:
325:
323:
241:
237:
233:
231:
226:
222:
214:
212:
154:
150:
146:
142:
140:
88:
58:
53:, named for
50:
46:
39:mathematical
36:
1420:Categories
1357:References
539:such that
394:, and let
382:space and
213:Note that
67:measurable
1392:Citations
1341:ε
1335:−
1249:∞
1234:⋃
1230:∖
1165:ε
1131:ε
1128:−
1054:…
924:→
816:∈
805:≤
786:ε
768:∈
682:→
671:ε
527:ε
515:∖
506:μ
466:Σ
463:∈
434:ε
411:→
363:μ
357:Σ
297:ϕ
294:≠
261:∈
195:ε
192:−
186:−
168:μ
121:→
41:field of
1362:Sources
875:Example
328:, then
37:In the
1091:. Set
757:
985:dense
498:with
386:be a
378:be a
57:) or
1198:and
1012:Let
603:and
524:<
437:>
340:Let
180:>
49:(or
809:sup
761:sup
726:on
599:is
77:of
1422::
1353:.
1212::=
45:,
1408:.
1338:2
1332:1
1312:E
1290:n
1286:G
1273:.
1259:n
1255:G
1244:1
1241:=
1238:n
1227:]
1224:1
1221:,
1218:0
1215:[
1209:E
1183:)
1178:n
1174:2
1169:/
1162:+
1157:n
1153:x
1149:,
1144:n
1140:2
1135:/
1123:n
1119:x
1115:(
1112:=
1107:n
1103:G
1078:Q
1057:}
1051:,
1048:2
1045:,
1042:1
1039:=
1036:n
1033:;
1028:n
1024:x
1020:{
998:.
995:E
971:]
968:1
965:,
962:0
959:[
939:}
936:1
933:,
930:0
927:{
921:]
918:1
915:,
912:0
909:[
906::
900:Q
895:1
855:.
842:|
838:)
835:x
832:(
829:f
825:|
819:X
813:x
801:|
797:)
794:x
791:(
782:f
777:|
771:X
765:x
734:E
714:f
692:d
687:R
679:X
676::
667:f
646:E
624:d
619:R
614:=
611:Y
587:A
567:E
547:f
521:)
518:E
512:A
509:(
486:E
460:A
440:0
414:Y
408:X
405::
402:f
384:Y
366:)
360:,
354:,
351:X
348:(
330:f
326:ε
309:}
306:)
303:x
300:(
291:)
288:x
285:(
282:f
279::
276:]
273:b
270:,
267:a
264:[
258:x
255:{
242:ϕ
234:f
227:E
223:f
215:E
198:.
189:a
183:b
177:)
174:E
171:(
155:E
151:f
147:E
143:ε
125:C
118:]
115:b
112:,
109:a
106:[
103::
100:f
34:.
20:)
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