577:
1165:
727:
954:
827:
th step of the algorithm, and end up with a Cantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is less than the measure of the initial interval. For instance, suppose the middle intervals of length
75:. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is
602:
959:
573:
Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process.
569:
1359:
of the Smith–Volterra–Cantor set is an example of a bounded function that is not
Riemann integrable on (0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see
949:
17:
1160:{\displaystyle {\begin{aligned}1-\sum _{n=0}^{\infty }2^{n}a^{n+1}&=1-a\sum _{n=0}^{\infty }(2a)^{n}\\&=1-a{\frac {1}{1-2a}}\\&={\frac {1-3a}{1-2a}}\end{aligned}}}
244:
410:
362:
1307:
314:
281:
587:, where the proportion removed from each interval remains constant. Thus, the Smith–Volterra–Cantor set has positive measure while the Cantor set has zero measure.
853:
805:
1253:
583:
Each subsequent iterate in the Smith–Volterra–Cantor set's construction removes proportionally less from the remaining intervals. This stands in contrast to the
1273:
1225:
1205:
1185:
905:
825:
762:
131:
885:
165:
1411:
Ponce
Campuzano, Juan; Maldonado, Miguel (2015). "Vito Volterra's construction of a nonconstant function with a bounded, non Riemann integrable derivative".
764:
showing that the set of the remaining points has a positive measure of 1/2. This makes the Smith–Volterra–Cantor set an example of a closed set whose
595:
By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also the intersection of a sequence of
415:
722:{\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{2^{2n+2}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots ={\frac {1}{2}}\,}
170:
1349:
910:
20:
After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.
1496:
1501:
1398:
1446:
Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets",
1313:
1394:
1317:
367:
64:
1491:
319:
1338:
53:
1278:
167:(the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is
286:
251:
57:
1360:
1467:
831:
783:
8:
1230:
765:
1258:
1428:
1366:
1356:
1210:
1190:
1170:
890:
810:
732:
101:
858:
138:
95:, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the
1486:
1432:
1455:
1420:
769:
1424:
1463:
599:, which means that it is closed. During the process, intervals of total length
1459:
1480:
96:
76:
68:
61:
49:
576:
1413:
BSHM Bulletin
Journal of the British Society for the History of Mathematics
1401:". Proceedings of the London Mathematical Society. First series. 6: 140–153
1325:
1321:
72:
1345:
25:
596:
584:
92:
80:
1312:
Cartesian products of Smith–Volterra–Cantor sets can be used to find
45:
16:
1320:
to a two-dimensional set of this type, it is possible to find an
135:
The process begins by removing the middle 1/4 from the interval
248:
The following steps consist of removing subintervals of width
1385:
Aliprantis and
Burkinshaw (1981), Principles of Real Analysis
1337:
The Smith–Volterra–Cantor set is used in the construction of
1316:
in higher dimensions with nonzero measure. By applying the
316:
remaining intervals. So for the second step the intervals
1369: – List of concrete topologies and topological spaces
1410:
1445:
1328:
such that the points on the curve have positive area.
536:
511:
496:
471:
456:
431:
211:
186:
1281:
1261:
1233:
1213:
1193:
1173:
957:
927:
913:
893:
861:
834:
813:
786:
735:
605:
418:
370:
322:
289:
254:
173:
141:
104:
564:{\displaystyle \left\cup \left\cup \left\cup \left.}
60:. The Smith–Volterra–Cantor set is named after the
1301:
1267:
1247:
1219:
1199:
1179:
1159:
943:
899:
879:
847:
819:
799:
756:
721:
563:
404:
356:
308:
275:
238:
159:
125:
1350:Jordan measure#Extension to more complicated sets
1344:The Smith–Volterra–Cantor set is an example of a
1478:
951:Then, the resulting set has Lebesgue measure
1399:On the integration of discontinuous functions
944:{\displaystyle 0\leq a\leq {\dfrac {1}{3}}.}
718:
775:
44:is an example of a set of points on the
15:
807:from each remaining subinterval at the
1479:
1309:is impossible in this construction.)
1348:that is not Jordan measurable, see
91:Similar to the construction of the
13:
1047:
984:
622:
575:
14:
1513:
239:{\displaystyle \left\cup \left.}
283:from the middle of each of the
86:
1439:
1404:
1388:
1379:
1062:
1052:
874:
862:
748:
736:
399:
371:
351:
323:
154:
142:
117:
105:
52:(in particular it contains no
1:
1448:Georgian Mathematical Journal
1425:10.1080/17498430.2015.1010771
1373:
590:
405:{\displaystyle (25/32,27/32)}
7:
1331:
780:In general, one can remove
357:{\displaystyle (5/32,7/32)}
10:
1518:
1361:Riemann integral#Examples
1314:totally disconnected sets
30:Smith–Volterra–Cantor set
1302:{\displaystyle a>1/3}
81:middle-thirds Cantor set
77:topologically equivalent
1460:10.1023/A:1022102312024
907:th iteration, for some
309:{\displaystyle 2^{n-1}}
276:{\displaystyle 1/4^{n}}
1303:
1269:
1249:
1221:
1201:
1181:
1161:
1051:
988:
945:
901:
881:
849:
821:
801:
758:
723:
626:
580:
565:
412:are removed, leaving
406:
358:
310:
277:
240:
161:
127:
21:
1304:
1270:
1250:
1222:
1202:
1182:
1162:
1031:
968:
946:
902:
882:
850:
848:{\displaystyle a^{n}}
822:
802:
800:{\displaystyle r_{n}}
776:Other fat Cantor sets
759:
724:
606:
579:
566:
407:
359:
311:
278:
241:
162:
128:
19:
1497:Sets of real numbers
1341:(see external link).
1318:Denjoy–Riesz theorem
1279:
1259:
1231:
1211:
1191:
1171:
955:
911:
891:
859:
832:
811:
784:
733:
603:
416:
368:
320:
287:
252:
171:
139:
102:
56:), yet has positive
1339:Volterra's function
1248:{\displaystyle 1/3}
1502:Topological spaces
1367:List of topologies
1357:indicator function
1299:
1268:{\displaystyle 0.}
1265:
1245:
1217:
1197:
1177:
1157:
1155:
941:
936:
897:
877:
845:
817:
797:
754:
719:
581:
561:
545:
520:
505:
480:
465:
440:
402:
354:
306:
273:
236:
220:
195:
157:
123:
22:
1395:Smith, Henry J.S.
1220:{\displaystyle a}
1200:{\displaystyle 1}
1180:{\displaystyle 0}
1151:
1109:
935:
900:{\displaystyle n}
855:are removed from
820:{\displaystyle n}
757:{\displaystyle ,}
729:are removed from
716:
697:
684:
671:
658:
544:
519:
504:
479:
464:
439:
219:
194:
126:{\displaystyle .}
1509:
1472:
1470:
1443:
1437:
1436:
1408:
1402:
1392:
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1383:
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1241:
1226:
1224:
1223:
1218:
1206:
1204:
1203:
1198:
1186:
1184:
1183:
1178:
1167:which goes from
1166:
1164:
1163:
1158:
1156:
1152:
1150:
1136:
1122:
1114:
1110:
1108:
1091:
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998:
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982:
950:
948:
947:
942:
937:
928:
906:
904:
903:
898:
886:
884:
883:
880:{\displaystyle }
878:
854:
852:
851:
846:
844:
843:
826:
824:
823:
818:
806:
804:
803:
798:
796:
795:
770:Lebesgue measure
763:
761:
760:
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728:
726:
725:
720:
717:
709:
698:
690:
685:
677:
672:
664:
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628:
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570:
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562:
557:
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486:
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363:
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282:
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271:
262:
245:
243:
242:
237:
232:
228:
221:
212:
201:
197:
196:
187:
166:
164:
163:
160:{\displaystyle }
158:
132:
130:
129:
124:
1517:
1516:
1512:
1511:
1510:
1508:
1507:
1506:
1477:
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1475:
1444:
1440:
1409:
1405:
1393:
1389:
1384:
1380:
1376:
1334:
1291:
1280:
1277:
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1237:
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1192:
1189:
1188:
1172:
1169:
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1154:
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1137:
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1121:
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1111:
1095:
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1072:
1071:
1065:
1061:
1046:
1035:
1015:
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993:
989:
983:
972:
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956:
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952:
926:
912:
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892:
889:
888:
860:
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856:
839:
835:
833:
830:
829:
812:
809:
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791:
787:
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782:
781:
778:
734:
731:
730:
708:
689:
676:
663:
643:
639:
633:
629:
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621:
610:
604:
601:
600:
593:
535:
534:
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510:
495:
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470:
455:
454:
450:
430:
423:
419:
417:
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413:
391:
377:
369:
366:
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343:
329:
321:
318:
317:
294:
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288:
285:
284:
267:
263:
258:
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249:
210:
209:
205:
185:
178:
174:
172:
169:
168:
140:
137:
136:
103:
100:
99:
89:
12:
11:
5:
1515:
1505:
1504:
1499:
1494:
1492:Measure theory
1489:
1474:
1473:
1454:(3): 201–212,
1438:
1419:(2): 143–152.
1403:
1387:
1377:
1375:
1372:
1371:
1370:
1364:
1353:
1342:
1333:
1330:
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1236:
1216:
1196:
1176:
1149:
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1132:
1129:
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1120:
1117:
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1038:
1034:
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1027:
1024:
1021:
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1012:
1009:
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1002:
996:
992:
986:
981:
978:
975:
971:
967:
964:
961:
960:
940:
934:
931:
925:
922:
919:
916:
896:
876:
873:
870:
867:
864:
842:
838:
816:
794:
790:
777:
774:
753:
750:
747:
744:
741:
738:
715:
712:
707:
704:
701:
696:
693:
688:
683:
680:
675:
670:
667:
662:
655:
652:
649:
646:
642:
636:
632:
624:
619:
616:
613:
609:
592:
589:
560:
556:
552:
549:
543:
540:
533:
529:
525:
518:
515:
509:
503:
500:
493:
489:
485:
478:
475:
469:
463:
460:
453:
449:
445:
438:
435:
429:
426:
422:
401:
398:
394:
390:
387:
384:
380:
376:
373:
353:
350:
346:
342:
339:
336:
332:
328:
325:
303:
300:
297:
293:
270:
266:
261:
257:
235:
231:
227:
224:
218:
215:
208:
204:
200:
193:
190:
184:
181:
177:
156:
153:
150:
147:
144:
122:
119:
116:
113:
110:
107:
88:
85:
62:mathematicians
42:fat Cantor set
9:
6:
4:
3:
2:
1514:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1484:
1482:
1469:
1465:
1461:
1457:
1453:
1449:
1442:
1434:
1430:
1426:
1422:
1418:
1414:
1407:
1400:
1396:
1391:
1382:
1378:
1368:
1365:
1362:
1358:
1354:
1351:
1347:
1343:
1340:
1336:
1335:
1329:
1327:
1323:
1319:
1315:
1310:
1296:
1292:
1288:
1285:
1282:
1262:
1242:
1238:
1234:
1214:
1194:
1174:
1147:
1144:
1141:
1138:
1133:
1130:
1127:
1124:
1118:
1116:
1105:
1102:
1099:
1096:
1092:
1087:
1084:
1081:
1078:
1076:
1066:
1058:
1055:
1042:
1039:
1036:
1032:
1028:
1025:
1022:
1019:
1017:
1010:
1007:
1004:
1000:
994:
990:
979:
976:
973:
969:
965:
962:
938:
932:
929:
923:
920:
917:
914:
894:
871:
868:
865:
840:
836:
814:
792:
788:
773:
771:
768:has positive
767:
751:
745:
742:
739:
713:
710:
705:
702:
699:
694:
691:
686:
681:
678:
673:
668:
665:
660:
653:
650:
647:
644:
640:
634:
630:
617:
614:
611:
607:
598:
588:
586:
578:
574:
571:
558:
554:
550:
547:
541:
538:
531:
527:
523:
516:
513:
507:
501:
498:
491:
487:
483:
476:
473:
467:
461:
458:
451:
447:
443:
436:
433:
427:
424:
420:
396:
392:
388:
385:
382:
378:
374:
348:
344:
340:
337:
334:
330:
326:
301:
298:
295:
291:
268:
264:
259:
255:
246:
233:
229:
225:
222:
216:
213:
206:
202:
198:
191:
188:
182:
179:
175:
151:
148:
145:
133:
120:
114:
111:
108:
98:
97:unit interval
94:
84:
82:
78:
74:
70:
69:Vito Volterra
66:
63:
59:
55:
51:
50:nowhere dense
47:
43:
39:
35:
31:
27:
18:
1451:
1447:
1441:
1416:
1412:
1406:
1390:
1381:
1326:Jordan curve
1322:Osgood curve
1311:
779:
594:
582:
572:
247:
134:
90:
87:Construction
73:Georg Cantor
41:
38:ε-Cantor set
37:
33:
29:
23:
1346:compact set
597:closed sets
65:Henry Smith
26:mathematics
1481:Categories
1374:References
1227:goes from
591:Properties
585:Cantor set
93:Cantor set
1397:(1874). "
1142:−
1128:−
1100:−
1085:−
1048:∞
1033:∑
1026:−
985:∞
970:∑
966:−
924:≤
918:≤
887:for each
703:⋯
623:∞
608:∑
528:∪
488:∪
448:∪
299:−
203:∪
54:intervals
46:real line
1487:Fractals
1433:34546093
1332:See also
766:boundary
48:that is
1468:1679442
79:to the
58:measure
1466:
1431:
28:, the
1429:S2CID
40:, or
1355:The
1324:, a
1286:>
364:and
71:and
1456:doi
1421:doi
1255:to
1207:as
1187:to
36:),
34:SVC
24:In
1483::
1464:MR
1462:,
1450:,
1427:.
1417:30
1415:.
1263:0.
772:.
695:16
542:32
539:27
517:32
514:25
462:32
437:32
397:32
389:27
383:32
375:25
349:32
335:32
83:.
67:,
1471:.
1458::
1452:6
1435:.
1423::
1363:.
1352:.
1297:3
1293:/
1289:1
1283:a
1275:(
1243:3
1239:/
1235:1
1215:a
1195:1
1175:0
1148:a
1145:2
1139:1
1134:a
1131:3
1125:1
1119:=
1106:a
1103:2
1097:1
1093:1
1088:a
1082:1
1079:=
1067:n
1063:)
1059:a
1056:2
1053:(
1043:0
1040:=
1037:n
1029:a
1023:1
1020:=
1011:1
1008:+
1005:n
1001:a
995:n
991:2
980:0
977:=
974:n
963:1
939:.
933:3
930:1
921:a
915:0
895:n
875:]
872:1
869:,
866:0
863:[
841:n
837:a
815:n
793:n
789:r
752:,
749:]
746:1
743:,
740:0
737:[
714:2
711:1
706:=
700:+
692:1
687:+
682:8
679:1
674:+
669:4
666:1
661:=
654:2
651:+
648:n
645:2
641:2
635:n
631:2
618:0
615:=
612:n
559:.
555:]
551:1
548:,
532:[
524:]
508:,
502:8
499:5
492:[
484:]
477:8
474:3
468:,
459:7
452:[
444:]
434:5
428:,
425:0
421:[
400:)
393:/
386:,
379:/
372:(
352:)
345:/
341:7
338:,
331:/
327:5
324:(
302:1
296:n
292:2
269:n
265:4
260:/
256:1
234:.
230:]
226:1
223:,
217:8
214:5
207:[
199:]
192:8
189:3
183:,
180:0
176:[
155:]
152:1
149:,
146:0
143:[
121:.
118:]
115:1
112:,
109:0
106:[
32:(
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