555:
The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure. A variation of this example is a disjoint union of an uncountable number of copies
828:
The space of all ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by open intervals, is a compact
Hausdorff space. The measure that assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinals and assigns 0 to other Borel sets
817:
with the discrete topology has a Borel probability measure such that every compact subset has measure 0, so this measure is outer regular but not inner regular. The existence of measurable cardinals cannot be proved in ZF set theory but (as of 2013) is thought to be consistent with
157:
308:
789:
is an outer regular locally finite Borel measure on a locally compact
Hausdorff space that is not inner regular in the strong sense, though all open sets are inner regular so it is inner regular in the weak sense. The measures
493:
639:
530:
446:
577:
411:
601:
550:
86:
237:
1782:
1860:
1877:
894:
1185:
1044:
849:
352:
318:
if every measurable set is inner regular. Some authors use a different definition: a measure is called inner regular if every
1700:
1531:
379:
probability measure on a locally compact
Hausdorff space with a countable base for its topology, or compact metric space, or
1071:
579:
on a locally compact
Hausdorff space that is inner regular, Ï-finite, and locally finite but not outer regular is given by
162:
This property is sometimes referred to in words as "approximation from within by compact sets." Some authors use the term
1692:
1938:
1478:
1872:
994:
975:
956:
917:
878:
1829:
1819:
451:
1629:
1538:
1302:
1158:
610:
1867:
1814:
1708:
1614:
733:. This measure is inner regular and locally finite, but is not outer regular as any open set containing the
1733:
1713:
1677:
1601:
1321:
1037:
393:
An example of a measure on the real line with its usual topology that is not outer regular is the measure
1855:
1634:
1596:
1548:
498:
416:
1760:
1728:
1718:
1639:
1606:
1237:
1146:
1777:
1682:
1458:
1386:
1767:
1850:
1296:
1227:
1163:
36:
can be approximated from above by open measurable sets and from below by compact measurable sets.
1619:
1377:
1337:
1030:
217:
1902:
1802:
1624:
1346:
1192:
1463:
1416:
1411:
1406:
1248:
1131:
1089:
839:
315:
171:
29:
1772:
1738:
1646:
1356:
1311:
1153:
1076:
1004:
927:
562:
396:
152:{\displaystyle \mu (A)=\sup\{\mu (F)\mid F\subseteq A,F{\text{ compact and measurable}}\}}
8:
1755:
1745:
1591:
1555:
1381:
1110:
1067:
814:
362:
1433:
1907:
1667:
1652:
1351:
1232:
1210:
945:
888:
586:
535:
303:{\displaystyle \mu (A)=\inf\{\mu (G)\mid G\supseteq A,G{\text{ open and measurable}}\}}
1824:
1560:
1521:
1516:
1423:
1341:
1126:
1099:
1011:
990:
971:
952:
913:
874:
25:
1841:
1750:
1526:
1511:
1501:
1486:
1453:
1448:
1438:
1316:
1291:
1106:
344:
1917:
1897:
1672:
1570:
1565:
1543:
1366:
1286:
1180:
1001:
924:
366:
1807:
1662:
1657:
1468:
1443:
1396:
1326:
1306:
1266:
1256:
1053:
829:
is a Borel probability measure that is neither inner regular nor outer regular.
182:
175:
33:
1932:
1912:
1575:
1496:
1491:
1391:
1361:
1331:
1281:
1276:
1271:
1261:
1175:
1094:
844:
376:
359:
190:
53:
1506:
1428:
1168:
325:
A measure is called outer regular if every measurable set is outer regular.
1205:
661:
positive integers. The topology is given as follows. The single points (1/
369:
1371:
583:, Chapter IV, Exercise 5 of section 1) as follows. The topological space
380:
17:
871:
Gradient Flows in Metric Spaces and in the Space of
Probability Measures
868:
1215:
1197:
1141:
1136:
781:) where the inf is taken over all open sets containing the Borel set
348:
328:
A measure is called regular if it is outer regular and inner regular.
822:
741:
387:
1222:
1081:
798:
coincide on all open sets, all compact sets, and all sets on which
1022:
167:
603:
has as underlying set the subset of the real plane given by the
170:
for inner regular. This use of the term is closely related to
673:) are all open sets. A base of neighborhoods of the point (0,
989:. AMS Chelsea Publishing, Providence, RI. p. xii+276.
751:
is the inner regular measure in the previous example, and
713:
is locally compact. The measure Ό is given by letting the
810:-measure though all compact subsets of it have measure 0.
613:
589:
565:
538:
501:
454:
419:
399:
240:
89:
912:. AMS Chelsea Publishing, Providence, RI. xii+276.
944:
677:) is given by wedges consisting of all points in
633:
595:
571:
544:
524:
487:
440:
405:
302:
151:
869:Ambrosio, L., Gigli, N. & Savaré, G. (2005).
823:Measures that are neither inner nor outer regular
742:Outer regular measures that are not inner regular
388:Inner regular measures that are not outer regular
1930:
256:
105:
717:-axis have measure 0 and letting the point (1/
1038:
907:
1783:RieszâMarkovâKakutani representation theorem
893:: CS1 maint: multiple names: authors list (
620:
614:
488:{\displaystyle \mu \left(\{1\}\right)=0\,\,}
469:
463:
297:
259:
146:
108:
216:. This is precisely the condition that the
1878:Vitale's random BrunnâMinkowski inequality
1045:
1031:
627:
521:
520:
484:
483:
965:
951:. New York: John Wiley & Sons, Inc.
873:. Basel: ETH ZĂŒrich, BirkhĂ€user Verlag.
634:{\displaystyle \{0\}\times \mathbb {R} }
580:
52:) be a topological space and let ÎŁ be a
850:Regularity theorem for Lebesgue measure
556:of the real line with Lebesgue measure.
353:regularity theorem for Lebesgue measure
1931:
1026:
987:Probability measures on metric spaces
910:Probability measures on metric spaces
372:Hausdorff space is a regular measure.
1891:Applications & related
947:Convergence of Probability Measures
525:{\displaystyle \mu (A)=\infty \,\,}
337:
13:
1052:
901:
862:
517:
441:{\displaystyle \mu (\emptyset )=0}
426:
14:
1950:
172:tightness of a family of measures
1820:Lebesgue differentiation theorem
1701:Carathéodory's extension theorem
322:measurable set is inner regular.
68:, ÎŁ). A measurable subset
936:
559:An example of a Borel measure
511:
505:
429:
423:
351:is a regular measure: see the
271:
265:
250:
244:
120:
114:
99:
93:
1:
1016:Real Analysis and Probability
985:Parthasarathy, K. R. (2005).
943:Billingsley, Patrick (1999).
908:Parthasarathy, K. R. (2005).
855:
39:
641:together with the points (1/
143: compact and measurable
7:
1873:PrĂ©kopaâLeindler inequality
833:
737:-axis has measure infinity.
332:
10:
1955:
1815:Lebesgue's density theorem
966:Bourbaki, Nicolas (2004).
1939:Measures (measure theory)
1890:
1868:MinkowskiâSteiner formula
1838:
1798:
1791:
1691:
1683:Projection-valued measure
1584:
1477:
1246:
1119:
1060:
294: open and measurable
1851:Isoperimetric inequality
1830:VitaliâHahnâSaks theorem
1159:Carathéodory's criterion
802:has finite measure. The
755:is the measure given by
220:collection of measures {
1856:BrunnâMinkowski theorem
1725:Decomposition theorems
705:for a positive integer
1903:Descriptive set theory
1803:Disintegration theorem
1238:Universally measurable
635:
597:
573:
546:
526:
489:
442:
407:
304:
189:> 0, there is some
153:
1705:Convergence theorems
1164:Cylindrical Ï-algebra
1018:. Chapman & Hall.
840:Borel regular measure
636:
598:
574:
547:
527:
490:
443:
408:
305:
154:
1773:Minkowski inequality
1647:Cylinder set measure
1532:Infinite-dimensional
1147:equivalence relation
1077:Lebesgue integration
611:
587:
572:{\displaystyle \mu }
563:
536:
499:
452:
417:
406:{\displaystyle \mu }
397:
314:A measure is called
238:
87:
1768:Hölder's inequality
1630:of random variables
1592:Measurable function
1479:Particular measures
1068:Absolute continuity
970:. Springer-Verlag.
815:measurable cardinal
806:-axis has infinite
363:probability measure
1908:Probability theory
1233:Transverse measure
1211:Non-measurable set
1193:Locally measurable
631:
593:
569:
542:
532:for any other set
522:
485:
438:
403:
300:
149:
1926:
1925:
1886:
1885:
1615:almost everywhere
1561:Spherical measure
1459:Strictly positive
1387:Projection-valued
1127:Almost everywhere
1100:Probability space
729:) have measure 1/
596:{\displaystyle X}
545:{\displaystyle A}
295:
227:It is said to be
212:) <
181:is inner regular
144:
64:be a measure on (
26:topological space
1946:
1861:Milman's reverse
1844:
1842:Lebesgue measure
1796:
1795:
1200:
1186:infimum/supremum
1107:Measurable space
1047:
1040:
1033:
1024:
1023:
1019:
1000:
981:
962:
950:
930:
923:
905:
899:
898:
892:
884:
866:
701:| †1/
640:
638:
637:
632:
630:
602:
600:
599:
594:
578:
576:
575:
570:
551:
549:
548:
543:
531:
529:
528:
523:
494:
492:
491:
486:
476:
472:
447:
445:
444:
439:
412:
410:
409:
404:
345:Lebesgue measure
338:Regular measures
309:
307:
306:
301:
296:
293:
158:
156:
155:
150:
145:
142:
32:for which every
1954:
1953:
1949:
1948:
1947:
1945:
1944:
1943:
1929:
1928:
1927:
1922:
1918:Spectral theory
1898:Convex analysis
1882:
1839:
1834:
1787:
1687:
1635:in distribution
1580:
1473:
1303:Logarithmically
1242:
1198:
1181:Essential range
1115:
1056:
1051:
1010:
1007:(See chapter 2)
997:
984:
978:
959:
942:
939:
934:
933:
920:
906:
902:
886:
885:
881:
867:
863:
858:
836:
825:
772:
744:
697:| †|
626:
612:
609:
608:
588:
585:
584:
564:
561:
560:
537:
534:
533:
500:
497:
496:
462:
458:
453:
450:
449:
418:
415:
414:
398:
395:
394:
390:
367:locally compact
340:
335:
292:
239:
236:
235:
141:
88:
85:
84:
42:
22:regular measure
12:
11:
5:
1952:
1942:
1941:
1924:
1923:
1921:
1920:
1915:
1910:
1905:
1900:
1894:
1892:
1888:
1887:
1884:
1883:
1881:
1880:
1875:
1870:
1865:
1864:
1863:
1853:
1847:
1845:
1836:
1835:
1833:
1832:
1827:
1825:Sard's theorem
1822:
1817:
1812:
1811:
1810:
1808:Lifting theory
1799:
1793:
1789:
1788:
1786:
1785:
1780:
1775:
1770:
1765:
1764:
1763:
1761:FubiniâTonelli
1753:
1748:
1743:
1742:
1741:
1736:
1731:
1723:
1722:
1721:
1716:
1711:
1703:
1697:
1695:
1689:
1688:
1686:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1649:
1644:
1643:
1642:
1640:in probability
1637:
1627:
1622:
1617:
1611:
1610:
1609:
1604:
1599:
1588:
1586:
1582:
1581:
1579:
1578:
1573:
1568:
1563:
1558:
1553:
1552:
1551:
1541:
1536:
1535:
1534:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1483:
1481:
1475:
1474:
1472:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
1436:
1431:
1426:
1421:
1420:
1419:
1414:
1409:
1399:
1394:
1389:
1384:
1374:
1369:
1364:
1359:
1354:
1349:
1347:Locally finite
1344:
1334:
1329:
1324:
1319:
1314:
1309:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1253:
1251:
1244:
1243:
1241:
1240:
1235:
1230:
1225:
1220:
1219:
1218:
1208:
1203:
1195:
1190:
1189:
1188:
1178:
1173:
1172:
1171:
1161:
1156:
1151:
1150:
1149:
1139:
1134:
1129:
1123:
1121:
1117:
1116:
1114:
1113:
1104:
1103:
1102:
1092:
1087:
1079:
1074:
1064:
1062:
1061:Basic concepts
1058:
1057:
1054:Measure theory
1050:
1049:
1042:
1035:
1027:
1021:
1020:
1008:
995:
982:
976:
963:
957:
938:
935:
932:
931:
918:
900:
879:
860:
859:
857:
854:
853:
852:
847:
842:
835:
832:
831:
830:
824:
821:
820:
819:
811:
764:
743:
740:
739:
738:
629:
625:
622:
619:
616:
592:
581:Bourbaki (2004
568:
557:
553:
541:
519:
516:
513:
510:
507:
504:
482:
479:
475:
471:
468:
465:
461:
457:
437:
434:
431:
428:
425:
422:
402:
389:
386:
385:
384:
373:
356:
339:
336:
334:
331:
330:
329:
326:
323:
311:
310:
299:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
191:compact subset
183:if and only if
176:finite measure
160:
159:
148:
140:
137:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
98:
95:
92:
76:is said to be
54:σ-algebra
41:
38:
34:measurable set
9:
6:
4:
3:
2:
1951:
1940:
1937:
1936:
1934:
1919:
1916:
1914:
1913:Real analysis
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1895:
1893:
1889:
1879:
1876:
1874:
1871:
1869:
1866:
1862:
1859:
1858:
1857:
1854:
1852:
1849:
1848:
1846:
1843:
1837:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1809:
1806:
1805:
1804:
1801:
1800:
1797:
1794:
1792:Other results
1790:
1784:
1781:
1779:
1778:RadonâNikodym
1776:
1774:
1771:
1769:
1766:
1762:
1759:
1758:
1757:
1754:
1752:
1751:Fatou's lemma
1749:
1747:
1744:
1740:
1737:
1735:
1732:
1730:
1727:
1726:
1724:
1720:
1717:
1715:
1712:
1710:
1707:
1706:
1704:
1702:
1699:
1698:
1696:
1694:
1690:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1650:
1648:
1645:
1641:
1638:
1636:
1633:
1632:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:Convergence:
1612:
1608:
1605:
1603:
1600:
1598:
1595:
1594:
1593:
1590:
1589:
1587:
1583:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1550:
1547:
1546:
1545:
1542:
1540:
1537:
1533:
1530:
1529:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1484:
1482:
1480:
1476:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1430:
1427:
1425:
1422:
1418:
1417:Outer regular
1415:
1413:
1412:Inner regular
1410:
1408:
1407:Borel regular
1405:
1404:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1385:
1383:
1379:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1339:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1304:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1254:
1252:
1250:
1245:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1217:
1214:
1213:
1212:
1209:
1207:
1204:
1202:
1196:
1194:
1191:
1187:
1184:
1183:
1182:
1179:
1177:
1174:
1170:
1167:
1166:
1165:
1162:
1160:
1157:
1155:
1152:
1148:
1145:
1144:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1124:
1122:
1118:
1112:
1108:
1105:
1101:
1098:
1097:
1096:
1095:Measure space
1093:
1091:
1088:
1086:
1084:
1080:
1078:
1075:
1073:
1069:
1066:
1065:
1063:
1059:
1055:
1048:
1043:
1041:
1036:
1034:
1029:
1028:
1025:
1017:
1013:
1012:Dudley, R. M.
1009:
1006:
1003:
998:
996:0-8218-3889-X
992:
988:
983:
979:
977:3-540-41129-1
973:
969:
968:Integration I
964:
960:
958:0-471-19745-9
954:
949:
948:
941:
940:
929:
926:
921:
919:0-8218-3889-X
915:
911:
904:
896:
890:
882:
880:3-7643-2428-7
876:
872:
865:
861:
851:
848:
846:
845:Radon measure
843:
841:
838:
837:
827:
826:
816:
812:
809:
805:
801:
797:
793:
788:
784:
780:
776:
771:
767:
762:
758:
754:
750:
746:
745:
736:
732:
728:
724:
720:
716:
712:
709:. This space
708:
704:
700:
696:
693: â
692:
688:
684:
681:of the form (
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
623:
617:
606:
590:
582:
566:
558:
554:
539:
514:
508:
502:
480:
477:
473:
466:
459:
455:
435:
432:
420:
400:
392:
391:
383:, is regular.
382:
378:
374:
371:
368:
364:
361:
357:
354:
350:
346:
342:
341:
327:
324:
321:
317:
316:inner regular
313:
312:
289:
286:
283:
280:
277:
274:
268:
262:
253:
247:
241:
234:
233:
232:
230:
229:outer regular
225:
223:
219:
215:
211:
207:
203:
199:
195:
192:
188:
184:
180:
177:
173:
169:
165:
138:
135:
132:
129:
126:
123:
117:
111:
102:
96:
90:
83:
82:
81:
79:
78:inner regular
75:
71:
67:
63:
59:
55:
51:
47:
37:
35:
31:
27:
23:
19:
1693:Main results
1429:Set function
1401:
1357:Metric outer
1312:Decomposable
1169:Cylinder set
1082:
1015:
986:
967:
946:
937:Bibliography
909:
903:
870:
864:
807:
803:
799:
795:
791:
786:
782:
778:
774:
769:
765:
763:) = inf
760:
756:
752:
748:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
604:
319:
228:
226:
224:} is tight.
221:
213:
209:
205:
201:
197:
193:
186:
178:
163:
161:
77:
73:
69:
65:
61:
57:
49:
45:
43:
21:
15:
1653:compact set
1620:of measures
1556:Pushforward
1549:Projections
1539:Logarithmic
1382:Probability
1372:Pre-measure
1154:Borel space
1072:of measures
381:Radon space
200:such that
18:mathematics
1625:in measure
1352:Maximising
1322:Equivalent
1216:Vitali set
856:References
185:, for all
174:, since a
40:Definition
1739:Maharam's
1709:Dominated
1522:Intensity
1517:Hausdorff
1424:Saturated
1342:Invariant
1247:Types of
1206:Ï-algebra
1176:đ-system
1142:Borel set
1137:Baire set
889:cite book
624:×
567:μ
518:∞
503:μ
456:μ
427:∅
421:μ
401:μ
370:Ï-compact
349:real line
281:⊇
275:∣
263:μ
242:μ
218:singleton
130:⊆
124:∣
112:μ
91:μ
1933:Category
1756:Fubini's
1746:Egorov's
1714:Monotone
1673:variable
1651:Random:
1602:Strongly
1527:Lebesgue
1512:Harmonic
1502:Gaussian
1487:Counting
1454:Spectral
1449:Singular
1439:s-finite
1434:Ï-finite
1317:Discrete
1292:Complete
1249:Measures
1223:Null set
1111:function
1014:(1989).
834:See also
689:) with |
333:Examples
1668:process
1663:measure
1658:element
1597:Bochner
1571:Trivial
1566:Tangent
1544:Product
1402:Regular
1380:)
1367:Perfect
1340:)
1305:)
1297:Content
1287:Complex
1228:Support
1201:-system
1090:Measure
1005:2169627
928:2169627
785:, then
653:) with
365:on any
347:on the
168:synonym
60:. Let
48:,
30:measure
1734:Jordan
1719:Vitali
1678:vector
1607:Weakly
1469:Vector
1444:Signed
1397:Random
1338:Quasi-
1327:Finite
1307:Convex
1267:Banach
1257:Atomic
1085:spaces
1070:
993:
974:
955:
916:
877:
773:
607:-axis
495:, and
413:where
222:μ
214:ε
202:μ
187:ε
179:μ
1576:Young
1497:Euler
1492:Dirac
1464:Tight
1392:Radon
1362:Outer
1332:Inner
1282:Brown
1277:Borel
1272:Besov
1262:Baire
377:Borel
360:Baire
166:as a
164:tight
44:Let (
28:is a
24:on a
1840:For
1729:Hahn
1585:Maps
1507:Haar
1378:Sub-
1132:Atom
1120:Sets
991:ISBN
972:ISBN
953:ISBN
914:ISBN
895:link
875:ISBN
794:and
375:Any
358:Any
343:The
320:open
20:, a
818:it.
747:If
257:inf
231:if
196:of
106:sup
80:if
72:of
56:on
16:In
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1002:MR
925:MR
891:}}
887:{{
813:A
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800:M
796:Ό
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775:Ό
770:S
768:â
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753:M
749:Ό
735:y
731:n
727:n
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721:,
719:n
715:y
711:X
707:n
703:n
699:u
695:y
691:v
687:v
685:,
683:u
679:X
675:y
671:n
669:/
667:m
665:,
663:n
659:n
657:,
655:m
651:n
649:/
647:m
645:,
643:n
628:R
621:}
618:0
615:{
605:y
591:X
552:.
540:A
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509:A
506:(
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474:)
470:}
467:1
464:{
460:(
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298:}
290:G
287:,
284:A
278:G
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269:G
266:(
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248:A
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194:K
147:}
139:F
136:,
133:A
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66:X
62:Ό
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