22:
708:
514:
2001:
865:
828:
773:
2079:
1168:
1203:
1110:
1017:
665:
623:
2096:
735:
598:
390:
885:
793:
643:
316:
294:
224:
203:
467:
1064:
176:
1130:
1084:
1037:
982:
962:
437:
358:
932:
912:
571:
545:
414:
338:
266:
246:
43:
1404:
1263:
1919:
1750:
1290:
1911:
94:
2091:
2157:
1697:
66:
2048:
113:
2038:
73:
1848:
1521:
47:
2086:
1377:
830:
is a strictly positive measure — Wiener measure is an example of a
Gaussian measure on an infinite-dimensional space.
80:
2033:
1927:
1833:
1952:
1932:
1896:
1820:
1540:
1256:
2074:
1853:
1815:
1767:
670:
472:
62:
1979:
32:
1947:
1937:
1858:
1825:
1456:
1365:
1996:
51:
36:
1901:
1605:
1986:
2069:
1515:
1446:
1226:
737:
is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
1382:
1838:
1596:
1556:
1249:
1213:
1209:
2121:
2021:
1843:
1565:
1411:
1757:
1682:
1635:
1630:
1625:
1467:
1350:
1308:
1040:
87:
841:
804:
749:
1991:
1957:
1865:
1575:
1530:
1372:
1295:
1974:
1964:
1810:
1774:
1652:
1600:
1329:
1286:
1135:
297:
1173:
1089:
987:
648:
606:
2126:
1886:
1871:
1570:
1451:
1429:
713:
576:
363:
2043:
1779:
1740:
1735:
1642:
1560:
1345:
1318:
182:
870:
778:
628:
301:
279:
209:
188:
2060:
1969:
1745:
1730:
1720:
1705:
1672:
1667:
1657:
1535:
1510:
1325:
835:
740:
524:
446:
1046:
149:
2136:
2116:
1891:
1789:
1784:
1762:
1620:
1585:
1505:
1399:
1115:
1069:
1022:
967:
947:
891:
744:
422:
343:
179:
2026:
1881:
1876:
1687:
1662:
1615:
1545:
1525:
1485:
1475:
1272:
1230:
917:
897:
798:
556:
530:
440:
399:
323:
273:
251:
231:
131:
2151:
2131:
1794:
1715:
1710:
1610:
1580:
1550:
1500:
1495:
1490:
1480:
1394:
1313:
550:
206:
1725:
1647:
1387:
1424:
1590:
127:
21:
1434:
1416:
1360:
1355:
601:
1441:
1300:
269:
1241:
138:
is one that is "nowhere zero", or that is zero "only on points".
1229: â Concept in mathematics â a measure is strictly positive
1245:
15:
573:
is particularly "coarse" (contains "few" sets). For example,
1086:
is strictly positive as well. The proof is simple: let
1176:
1138:
1118:
1092:
1072:
1049:
1025:
990:
970:
950:
920:
900:
873:
844:
807:
781:
752:
716:
673:
651:
631:
609:
579:
559:
553:
is usually not strictly positive unless the topology
533:
475:
449:
425:
402:
366:
346:
326:
304:
282:
254:
234:
212:
191:
152:
894:
is never strictly positive, regardless of the space
2109:
2057:
2010:
1910:
1803:
1696:
1465:
1338:
1279:
984:are two measures on a measurable topological space
1197:
1162:
1124:
1104:
1078:
1058:
1031:
1011:
976:
956:
926:
906:
879:
859:
822:
787:
767:
729:
702:
659:
637:
617:
592:
565:
539:
508:
461:
431:
408:
384:
352:
332:
310:
288:
260:
240:
218:
197:
170:
703:{\displaystyle T=\{\varnothing ,\mathbb {R} \},}
509:{\displaystyle U\neq \varnothing ,\mu (U)>0.}
645:-algebra is not strictly positive; however, if
1257:
8:
2002:RieszâMarkovâKakutani representation theorem
694:
680:
50:. Unsourced material may be challenged and
2097:Vitale's random BrunnâMinkowski inequality
2014:
1264:
1250:
1242:
1175:
1137:
1117:
1091:
1071:
1048:
1024:
989:
969:
949:
919:
899:
872:
851:
847:
846:
843:
814:
810:
809:
806:
780:
759:
755:
754:
751:
721:
715:
690:
689:
672:
653:
652:
650:
630:
611:
610:
608:
584:
578:
558:
547:(with any topology) is strictly positive.
532:
474:
448:
424:
401:
365:
345:
325:
303:
281:
253:
233:
211:
190:
151:
114:Learn how and when to remove this message
683:
482:
667:is equipped with the trivial topology
7:
2110:Applications & related
801:on the space of continuous paths in
130:, strict positivity is a concept in
48:adding citations to reliable sources
1000:
914:or the topology used, except when
625:with its usual Borel topology and
396:if every non-empty open subset of
376:
283:
192:
14:
2039:Lebesgue differentiation theorem
1920:Carathéodory's extension theorem
1112:be an arbitrary open set; since
860:{\displaystyle \mathbb {R} ^{n}}
823:{\displaystyle \mathbb {R} ^{n}}
795:-algebra) is strictly positive.
768:{\displaystyle \mathbb {R} ^{n}}
20:
1233:its support is the whole space.
1208:Hence, strict positivity is an
887:-algebra) is strictly positive.
416:has strictly positive measure.
1186:
1180:
1148:
1142:
1003:
991:
497:
491:
379:
367:
165:
153:
1:
1163:{\displaystyle \mu (U)>0;}
867:(with its Borel topology and
775:(with its Borel topology and
1198:{\displaystyle \nu (U)>0}
1105:{\displaystyle U\subseteq X}
1012:{\displaystyle (X,\Sigma ),}
660:{\displaystyle \mathbb {R} }
618:{\displaystyle \mathbb {R} }
2092:PrĂ©kopaâLeindler inequality
1039:strictly positive and also
730:{\displaystyle \delta _{0}}
593:{\displaystyle \delta _{0}}
385:{\displaystyle (X,\Sigma )}
296:is at least as fine as the
248:that contains the topology
63:"Strictly positive measure"
2174:
2034:Lebesgue's density theorem
2158:Measures (measure theory)
2087:MinkowskiâSteiner formula
2017:
1902:Projection-valued measure
136:strictly positive measure
2070:Isoperimetric inequality
2049:VitaliâHahnâSaks theorem
1378:Carathéodory's criterion
1227:Support (measure theory)
1170:by absolute continuity,
2075:BrunnâMinkowski theorem
1944:Decomposition theorems
1214:equivalence of measures
880:{\displaystyle \sigma }
788:{\displaystyle \sigma }
638:{\displaystyle \sigma }
311:{\displaystyle \sigma }
289:{\displaystyle \Sigma }
219:{\displaystyle \sigma }
198:{\displaystyle \Sigma }
2122:Descriptive set theory
2022:Disintegration theorem
1457:Universally measurable
1199:
1164:
1132:is strictly positive,
1126:
1106:
1080:
1060:
1033:
1013:
978:
958:
928:
908:
881:
861:
824:
789:
769:
731:
704:
661:
639:
619:
594:
567:
541:
510:
463:
462:{\displaystyle U\in T}
433:
410:
386:
354:
334:
312:
290:
262:
242:
220:
199:
172:
1924:Convergence theorems
1383:Cylindrical Ï-algebra
1200:
1165:
1127:
1107:
1081:
1061:
1059:{\displaystyle \nu ,}
1041:absolutely continuous
1034:
1014:
979:
959:
929:
909:
882:
862:
825:
790:
770:
732:
705:
662:
640:
620:
595:
568:
542:
511:
464:
439:is strictly positive
434:
411:
387:
355:
335:
313:
291:
263:
243:
221:
200:
173:
171:{\displaystyle (X,T)}
1992:Minkowski inequality
1866:Cylinder set measure
1751:Infinite-dimensional
1366:equivalence relation
1296:Lebesgue integration
1174:
1136:
1125:{\displaystyle \mu }
1116:
1090:
1079:{\displaystyle \nu }
1070:
1047:
1032:{\displaystyle \mu }
1023:
988:
977:{\displaystyle \nu }
968:
957:{\displaystyle \mu }
948:
918:
898:
871:
842:
805:
779:
750:
714:
671:
649:
629:
607:
577:
557:
531:
473:
447:
432:{\displaystyle \mu }
423:
400:
364:
353:{\displaystyle \mu }
344:
324:
302:
280:
252:
232:
210:
189:
150:
44:improve this article
1987:Hölder's inequality
1849:of random variables
1811:Measurable function
1698:Particular measures
1287:Absolute continuity
2127:Probability theory
1452:Transverse measure
1430:Non-measurable set
1412:Locally measurable
1195:
1160:
1122:
1102:
1076:
1056:
1029:
1009:
974:
954:
924:
904:
877:
857:
820:
785:
765:
727:
700:
657:
635:
615:
590:
563:
537:
506:
459:
429:
406:
382:
350:
340:). Then a measure
330:
308:
286:
258:
238:
216:
195:
168:
2145:
2144:
2105:
2104:
1834:almost everywhere
1780:Spherical measure
1678:Strictly positive
1606:Projection-valued
1346:Almost everywhere
1319:Probability space
927:{\displaystyle X}
907:{\displaystyle X}
566:{\displaystyle T}
540:{\displaystyle X}
409:{\displaystyle X}
394:strictly positive
333:{\displaystyle X}
261:{\displaystyle T}
241:{\displaystyle X}
183:topological space
134:. Intuitively, a
124:
123:
116:
98:
2165:
2080:Milman's reverse
2063:
2061:Lebesgue measure
2015:
1419:
1405:infimum/supremum
1326:Measurable space
1266:
1259:
1252:
1243:
1212:with respect to
1204:
1202:
1201:
1196:
1169:
1167:
1166:
1161:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1085:
1083:
1082:
1077:
1065:
1063:
1062:
1057:
1043:with respect to
1038:
1036:
1035:
1030:
1018:
1016:
1015:
1010:
983:
981:
980:
975:
963:
961:
960:
955:
933:
931:
930:
925:
913:
911:
910:
905:
886:
884:
883:
878:
866:
864:
863:
858:
856:
855:
850:
836:Lebesgue measure
829:
827:
826:
821:
819:
818:
813:
794:
792:
791:
786:
774:
772:
771:
766:
764:
763:
758:
741:Gaussian measure
736:
734:
733:
728:
726:
725:
709:
707:
706:
701:
693:
666:
664:
663:
658:
656:
644:
642:
641:
636:
624:
622:
621:
616:
614:
599:
597:
596:
591:
589:
588:
572:
570:
569:
564:
546:
544:
543:
538:
525:Counting measure
515:
513:
512:
507:
468:
466:
465:
460:
438:
436:
435:
430:
419:More concisely,
415:
413:
412:
407:
391:
389:
388:
383:
359:
357:
356:
351:
339:
337:
336:
331:
317:
315:
314:
309:
295:
293:
292:
287:
267:
265:
264:
259:
247:
245:
244:
239:
225:
223:
222:
217:
204:
202:
201:
196:
177:
175:
174:
169:
119:
112:
108:
105:
99:
97:
56:
24:
16:
2173:
2172:
2168:
2167:
2166:
2164:
2163:
2162:
2148:
2147:
2146:
2141:
2137:Spectral theory
2117:Convex analysis
2101:
2058:
2053:
2006:
1906:
1854:in distribution
1799:
1692:
1522:Logarithmically
1461:
1417:
1400:Essential range
1334:
1275:
1270:
1240:
1223:
1172:
1171:
1134:
1133:
1114:
1113:
1088:
1087:
1068:
1067:
1045:
1044:
1021:
1020:
986:
985:
966:
965:
946:
945:
941:
916:
915:
896:
895:
892:trivial measure
869:
868:
845:
840:
839:
808:
803:
802:
777:
776:
753:
748:
747:
745:Euclidean space
717:
712:
711:
669:
668:
647:
646:
627:
626:
605:
604:
580:
575:
574:
555:
554:
529:
528:
521:
471:
470:
445:
444:
421:
420:
398:
397:
362:
361:
342:
341:
322:
321:
300:
299:
278:
277:
268:(so that every
250:
249:
230:
229:
208:
207:
187:
186:
148:
147:
144:
120:
109:
103:
100:
57:
55:
41:
25:
12:
11:
5:
2171:
2169:
2161:
2160:
2150:
2149:
2143:
2142:
2140:
2139:
2134:
2129:
2124:
2119:
2113:
2111:
2107:
2106:
2103:
2102:
2100:
2099:
2094:
2089:
2084:
2083:
2082:
2072:
2066:
2064:
2055:
2054:
2052:
2051:
2046:
2044:Sard's theorem
2041:
2036:
2031:
2030:
2029:
2027:Lifting theory
2018:
2012:
2008:
2007:
2005:
2004:
1999:
1994:
1989:
1984:
1983:
1982:
1980:FubiniâTonelli
1972:
1967:
1962:
1961:
1960:
1955:
1950:
1942:
1941:
1940:
1935:
1930:
1922:
1916:
1914:
1908:
1907:
1905:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1868:
1863:
1862:
1861:
1859:in probability
1856:
1846:
1841:
1836:
1830:
1829:
1828:
1823:
1818:
1807:
1805:
1801:
1800:
1798:
1797:
1792:
1787:
1782:
1777:
1772:
1771:
1770:
1760:
1755:
1754:
1753:
1743:
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1702:
1700:
1694:
1693:
1691:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1639:
1638:
1633:
1628:
1618:
1613:
1608:
1603:
1593:
1588:
1583:
1578:
1573:
1568:
1566:Locally finite
1563:
1553:
1548:
1543:
1538:
1533:
1528:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1483:
1478:
1472:
1470:
1463:
1462:
1460:
1459:
1454:
1449:
1444:
1439:
1438:
1437:
1427:
1422:
1414:
1409:
1408:
1407:
1397:
1392:
1391:
1390:
1380:
1375:
1370:
1369:
1368:
1358:
1353:
1348:
1342:
1340:
1336:
1335:
1333:
1332:
1323:
1322:
1321:
1311:
1306:
1298:
1293:
1283:
1281:
1280:Basic concepts
1277:
1276:
1273:Measure theory
1271:
1269:
1268:
1261:
1254:
1246:
1239:
1236:
1235:
1234:
1231:if and only if
1222:
1219:
1218:
1217:
1206:
1194:
1191:
1188:
1185:
1182:
1179:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1121:
1101:
1098:
1095:
1075:
1055:
1052:
1028:
1008:
1005:
1002:
999:
996:
993:
973:
953:
940:
937:
936:
935:
923:
903:
888:
876:
854:
849:
833:
832:
831:
817:
812:
799:Wiener measure
784:
762:
757:
738:
724:
720:
699:
696:
692:
688:
685:
682:
679:
676:
655:
634:
613:
587:
583:
562:
548:
536:
520:
517:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
458:
455:
452:
441:if and only if
428:
405:
381:
378:
375:
372:
369:
349:
329:
307:
285:
274:measurable set
257:
237:
215:
194:
167:
164:
161:
158:
155:
143:
140:
132:measure theory
122:
121:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
2170:
2159:
2156:
2155:
2153:
2138:
2135:
2133:
2132:Real analysis
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2114:
2112:
2108:
2098:
2095:
2093:
2090:
2088:
2085:
2081:
2078:
2077:
2076:
2073:
2071:
2068:
2067:
2065:
2062:
2056:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2028:
2025:
2024:
2023:
2020:
2019:
2016:
2013:
2011:Other results
2009:
2003:
2000:
1998:
1997:RadonâNikodym
1995:
1993:
1990:
1988:
1985:
1981:
1978:
1977:
1976:
1973:
1971:
1970:Fatou's lemma
1968:
1966:
1963:
1959:
1956:
1954:
1951:
1949:
1946:
1945:
1943:
1939:
1936:
1934:
1931:
1929:
1926:
1925:
1923:
1921:
1918:
1917:
1915:
1913:
1909:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1869:
1867:
1864:
1860:
1857:
1855:
1852:
1851:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:Convergence:
1831:
1827:
1824:
1822:
1819:
1817:
1814:
1813:
1812:
1809:
1808:
1806:
1802:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1769:
1766:
1765:
1764:
1761:
1759:
1756:
1752:
1749:
1748:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1703:
1701:
1699:
1695:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1637:
1636:Outer regular
1634:
1632:
1631:Inner regular
1629:
1627:
1626:Borel regular
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1314:Measure space
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551:Dirac measure
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104:December 2009
96:
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75:
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68:
65: â
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
1912:Main results
1677:
1648:Set function
1576:Metric outer
1531:Decomposable
1388:Cylinder set
1301:
418:
393:
145:
135:
125:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
1872:compact set
1839:of measures
1775:Pushforward
1768:Projections
1758:Logarithmic
1601:Probability
1591:Pre-measure
1373:Borel space
1291:of measures
527:on any set
128:mathematics
1844:in measure
1571:Maximising
1541:Equivalent
1435:Vitali set
1238:References
939:Properties
469:such that
392:is called
142:Definition
74:newspapers
1958:Maharam's
1928:Dominated
1741:Intensity
1736:Hausdorff
1643:Saturated
1561:Invariant
1466:Types of
1425:Ï-algebra
1395:đ-system
1361:Borel set
1356:Baire set
1210:invariant
1178:ν
1140:μ
1120:μ
1097:⊆
1074:ν
1051:ν
1027:μ
1001:Σ
972:ν
952:μ
934:is empty.
875:σ
783:σ
719:δ
684:∅
633:σ
602:real line
582:δ
489:μ
483:∅
480:≠
454:∈
427:μ
377:Σ
348:μ
306:σ
284:Σ
214:σ
193:Σ
180:Hausdorff
31:does not
2152:Category
1975:Fubini's
1965:Egorov's
1933:Monotone
1892:variable
1870:Random:
1821:Strongly
1746:Lebesgue
1731:Harmonic
1721:Gaussian
1706:Counting
1673:Spectral
1668:Singular
1658:s-finite
1653:Ï-finite
1536:Discrete
1511:Complete
1468:Measures
1442:Null set
1330:function
1221:See also
1205:as well.
519:Examples
443:for all
318:-algebra
270:open set
226:-algebra
185:and let
1887:process
1882:measure
1877:element
1816:Bochner
1790:Trivial
1785:Tangent
1763:Product
1621:Regular
1599:)
1586:Perfect
1559:)
1524:)
1516:Content
1506:Complex
1447:Support
1420:-system
1309:Measure
600:on the
88:scholar
52:removed
37:sources
1953:Jordan
1938:Vitali
1897:vector
1826:Weakly
1688:Vector
1663:Signed
1616:Random
1557:Quasi-
1546:Finite
1526:Convex
1486:Banach
1476:Atomic
1304:spaces
1289:
298:Borel
276:, and
90:
83:
76:
69:
61:
1795:Young
1716:Euler
1711:Dirac
1683:Tight
1611:Radon
1581:Outer
1551:Inner
1501:Brown
1496:Borel
1491:Besov
1481:Baire
1066:then
1019:with
710:then
272:is a
205:be a
178:be a
95:JSTOR
81:books
2059:For
1948:Hahn
1804:Maps
1726:Haar
1597:Sub-
1351:Atom
1339:Sets
1190:>
1152:>
964:and
890:The
501:>
146:Let
67:news
35:any
33:cite
944:If
838:on
743:on
360:on
320:on
228:on
126:In
46:by
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504:0.
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163:T
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154:(
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102:(
92:·
85:·
78:·
71:·
54:.
40:.
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