36:
2305:
536:
2003:
652:
1935:
1479:
1326:
1853:
2115:
1722:
2224:
1802:
902:
2169:
1567:
396:
2196:
1660:
1596:
1541:
426:
1351:
1261:
1188:
1752:
1386:
1215:
472:
1071:
1873:
342:
270:
178:
569:
154:
1045:
922:
715:
695:
593:
2216:
2139:
1628:
1406:
1239:
1147:
1123:
1099:
982:
955:
867:
847:
816:
796:
768:
748:
675:
446:
362:
314:
294:
246:
222:
202:
1514:
1021:
3148:
3226:
3243:
65:
480:
1940:
2551:
2410:
2048:
605:
1878:
1414:
3066:
1269:
2897:
2437:
3058:
3309:
2844:
2300:{\displaystyle \limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(X\setminus K_{\varepsilon })<-\varepsilon .}
1810:
3238:
2372:
2349:
2326:
2074:
1671:
87:
58:
3195:
3185:
2995:
2904:
2668:
2028:
1765:
2524:
3233:
3180:
3074:
2980:
872:
17:
3099:
3079:
3043:
2967:
2687:
2403:
2043:
2014:
2038:
3221:
3000:
2962:
2914:
1150:
3126:
3094:
3084:
3005:
2972:
2603:
2512:
2148:
1546:
375:
2174:
2053:
1636:
1572:
1519:
924:. It can be shown that a family of measure is tight if and only if it is sequentially weakly compact.
904:
from the family, there is a subsequence of measures that converges weakly to some probability measure
404:
3143:
3048:
2824:
2752:
48:
3133:
3216:
2662:
2593:
1599:
775:
52:
44:
2529:
1334:
1244:
1171:
3304:
2985:
2743:
2703:
2396:
2065:
1730:
1364:
1193:
1154:
572:
2064:
A strengthening of tightness is the concept of exponential tightness, which has applications in
3268:
3168:
2990:
2712:
2558:
2033:
1126:
451:
69:
1050:
2782:
2777:
2772:
2614:
2497:
2455:
1858:
723:
327:
255:
163:
544:
3138:
3104:
3012:
2722:
2677:
2519:
2442:
2382:
127:
1030:
907:
700:
680:
578:
8:
3121:
3111:
2957:
2921:
2747:
2476:
2433:
2069:
596:
273:
2799:
3273:
3033:
3018:
2717:
2598:
2576:
2338:
2201:
2124:
1613:
1391:
1224:
1132:
1108:
1084:
967:
940:
852:
832:
801:
781:
753:
733:
660:
431:
347:
299:
279:
231:
207:
187:
1487:
987:
3190:
2926:
2887:
2882:
2789:
2707:
2492:
2465:
2368:
2345:
2322:
1805:
3207:
3116:
2892:
2877:
2867:
2852:
2819:
2814:
2804:
2682:
2657:
2472:
2360:
1663:
984:
is tight. This is not necessarily so for non-metrisable compact spaces. If we take
3283:
3263:
3038:
2936:
2931:
2909:
2767:
2732:
2652:
2546:
2379:
2118:
1631:
771:
321:
157:
112:. The intuitive idea is that a given collection of measures does not "escape to
3173:
3028:
3023:
2834:
2809:
2762:
2692:
2672:
2632:
2622:
2419:
1755:
1024:
317:
249:
109:
3298:
3278:
2941:
2862:
2857:
2757:
2727:
2697:
2647:
2642:
2637:
2627:
2541:
2460:
2017:
of a sequence of probability measures, especially when the measure space has
1218:
961:
869:
of probability measures is sequentially weakly compact if for every sequence
399:
181:
2872:
2794:
2534:
1102:
2571:
2737:
1358:
101:
2581:
1354:
958:
1662:
with its usual Borel topology and Ï-algebra. Consider a collection of
2563:
2507:
2502:
2021:
1166:
531:{\displaystyle |\mu |(X\setminus K_{\varepsilon })<\varepsilon .}
2588:
2447:
2018:
225:
113:
1998:{\displaystyle \{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}}
2388:
829:
Another equivalent criterion of the tightness of a collection
1759:
647:{\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,}
1930:{\displaystyle \{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}}
1474:{\displaystyle M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}}
1047:
on it that is not inner regular. Therefore, the singleton
964:, then every collection of (possibly complex) measures on
2013:
Tightness is often a necessary criterion for proving the
1321:{\displaystyle M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}}
1569:. In general, a collection of Dirac delta measures on
2227:
2204:
2177:
2151:
2127:
2077:
1943:
1881:
1861:
1813:
1768:
1733:
1674:
1639:
1616:
1575:
1549:
1522:
1490:
1417:
1394:
1367:
1337:
1272:
1247:
1227:
1196:
1174:
1135:
1111:
1087:
1053:
1033:
990:
970:
943:
910:
875:
855:
835:
804:
784:
756:
736:
703:
683:
663:
608:
581:
547:
483:
454:
434:
407:
378:
350:
330:
302:
282:
258:
234:
210:
190:
166:
130:
1605:
1361:
subsets, and any such set, since it is bounded, has
2337:
2299:
2210:
2190:
2163:
2133:
2109:
1997:
1929:
1867:
1847:
1796:
1746:
1716:
1654:
1622:
1598:is tight if, and only if, the collection of their
1590:
1561:
1535:
1508:
1473:
1400:
1380:
1345:
1320:
1255:
1233:
1209:
1182:
1141:
1117:
1093:
1065:
1039:
1015:
976:
949:
916:
896:
861:
849:is sequentially weakly compact. We say the family
841:
810:
790:
762:
742:
709:
689:
669:
646:
587:
563:
530:
466:
440:
420:
390:
356:
336:
308:
288:
264:
240:
216:
196:
172:
148:
1848:{\displaystyle C_{i}\in \mathbb {R} ^{n\times n}}
3296:
2229:
57:but its sources remain unclear because it lacks
2358:
2110:{\displaystyle (\mu _{\delta })_{\delta >0}}
1717:{\displaystyle \Gamma =\{\gamma _{i}|i\in I\},}
1160:
2404:
2367:. Berlin: Springer-Verlag. pp. xii+480.
3149:RieszâMarkovâKakutani representation theorem
2344:. New York, NY: John Wiley & Sons, Inc.
2321:. New York, NY: John Wiley & Sons, Inc.
2008:
1971:
1944:
1909:
1882:
1708:
1681:
1468:
1431:
1315:
1286:
1060:
1054:
595:. Very often, the measures in question are
2335:
2316:
1331:is not tight, since the compact subsets of
3244:Vitale's random BrunnâMinkowski inequality
2411:
2397:
1875:is tight if, and only if, the collections
1129:, a collection of probability measures on
1979:
1917:
1829:
1797:{\displaystyle m_{i}\in \mathbb {R} ^{n}}
1784:
1642:
1578:
1464:
1339:
1311:
1249:
1176:
643:
88:Learn how and when to remove this message
2059:
897:{\displaystyle \left\{\mu _{n}\right\}}
14:
3297:
2392:
599:, so the last part can be written as
3257:Applications & related
1408:. On the other hand, the collection
1105:, then every probability measure on
29:
2340:Convergence of Probability Measures
2049:Tightness in classical Wiener space
1190:with its usual Borel topology. Let
697:, then (depending upon the author)
24:
2418:
1862:
1675:
331:
259:
167:
25:
3321:
2269:
2164:{\displaystyle \varepsilon >0}
1606:A collection of Gaussian measures
1562:{\displaystyle \varepsilon >0}
932:
503:
391:{\displaystyle \varepsilon >0}
3186:Lebesgue differentiation theorem
3067:Carathéodory's extension theorem
2191:{\displaystyle K_{\varepsilon }}
1655:{\displaystyle \mathbb {R} ^{n}}
1591:{\displaystyle \mathbb {R} ^{n}}
1536:{\displaystyle K_{\varepsilon }}
1076:
421:{\displaystyle K_{\varepsilon }}
34:
2029:Finite-dimensional distribution
1484:is tight: the compact interval
1388:-measure zero for large enough
2282:
2263:
2236:
2092:
2078:
1958:
1896:
1695:
1503:
1491:
1453:
1300:
1149:is tight if and only if it is
1027:, then there exists a measure
1010:
991:
625:
612:
557:
549:
516:
497:
493:
485:
143:
131:
119:
13:
1:
2336:Billingsley, Patrick (1999).
2317:Billingsley, Patrick (1995).
2310:
677:consists of a single measure
316:be a collection of (possibly
2365:Probability in Banach spaces
2171:, there is a compact subset
2054:Tightness in Skorokhod space
2044:Weak convergence of measures
1346:{\displaystyle \mathbb {R} }
1256:{\displaystyle \mathbb {R} }
1183:{\displaystyle \mathbb {R} }
1161:A collection of point masses
448:such that, for all measures
7:
3239:PrĂ©kopaâLeindler inequality
1747:{\displaystyle \gamma _{i}}
1381:{\displaystyle \delta _{n}}
1221:, a unit mass at the point
1210:{\displaystyle \delta _{x}}
927:
717:may either be said to be a
272:is at least as fine as the
204:that contains the topology
10:
3326:
3181:Lebesgue's density theorem
1125:is tight. Furthermore, by
3310:Measures (measure theory)
3256:
3234:MinkowskiâSteiner formula
3204:
3164:
3157:
3057:
3049:Projection-valued measure
2950:
2843:
2612:
2485:
2426:
2009:Tightness and convergence
820:separable random variable
467:{\displaystyle \mu \in M}
27:Concept in measure theory
3217:Isoperimetric inequality
3196:VitaliâHahnâSaks theorem
2525:Carathéodory's criterion
1066:{\displaystyle \{\mu \}}
798:is a tight measure then
776:probability distribution
43:This article includes a
3222:BrunnâMinkowski theorem
3091:Decomposition theorems
2319:Probability and Measure
2066:large deviations theory
1868:{\displaystyle \Gamma }
573:total variation measure
337:{\displaystyle \Sigma }
265:{\displaystyle \Sigma }
173:{\displaystyle \Sigma }
72:more precise citations.
3269:Descriptive set theory
3169:Disintegration theorem
2604:Universally measurable
2301:
2212:
2192:
2165:
2135:
2111:
1999:
1931:
1869:
1855:. Then the collection
1849:
1798:
1748:
1718:
1656:
1624:
1592:
1563:
1537:
1510:
1475:
1402:
1382:
1347:
1322:
1257:
1235:
1211:
1184:
1143:
1119:
1095:
1067:
1041:
1017:
978:
951:
918:
898:
863:
843:
812:
792:
764:
744:
711:
691:
671:
657:If a tight collection
648:
589:
565:
564:{\displaystyle |\mu |}
532:
468:
442:
422:
392:
358:
338:
324:) measures defined on
310:
290:
266:
242:
218:
198:
174:
150:
3071:Convergence theorems
2530:Cylindrical Ï-algebra
2302:
2213:
2193:
2166:
2136:
2112:
2060:Exponential tightness
2039:LĂ©vyâProkhorov metric
2000:
1932:
1870:
1850:
1799:
1749:
1719:
1657:
1625:
1593:
1564:
1538:
1511:
1476:
1403:
1383:
1348:
1323:
1258:
1236:
1212:
1185:
1144:
1120:
1096:
1068:
1042:
1018:
979:
952:
919:
899:
864:
844:
824:Radon random variable
813:
793:
765:
745:
724:inner regular measure
712:
692:
672:
649:
590:
566:
533:
469:
443:
423:
393:
359:
339:
311:
291:
267:
243:
219:
199:
175:
151:
149:{\displaystyle (X,T)}
3139:Minkowski inequality
3013:Cylinder set measure
2898:Infinite-dimensional
2513:equivalence relation
2443:Lebesgue integration
2225:
2202:
2175:
2149:
2125:
2075:
2070:probability measures
1941:
1879:
1859:
1811:
1766:
1731:
1672:
1637:
1614:
1573:
1547:
1520:
1488:
1415:
1392:
1365:
1335:
1270:
1245:
1225:
1194:
1172:
1133:
1109:
1085:
1051:
1040:{\displaystyle \mu }
1031:
988:
968:
941:
917:{\displaystyle \mu }
908:
873:
853:
833:
802:
782:
754:
734:
710:{\displaystyle \mu }
701:
690:{\displaystyle \mu }
681:
661:
606:
597:probability measures
588:{\displaystyle \mu }
579:
545:
481:
452:
432:
405:
376:
348:
328:
300:
280:
274:Borel σ-algebra
256:
232:
208:
188:
164:
128:
3134:Hölder's inequality
2996:of random variables
2958:Measurable function
2845:Particular measures
2434:Absolute continuity
2143:exponentially tight
2034:Prokhorov's theorem
1153:in the topology of
1127:Prokhorov's theorem
3274:Probability theory
2599:Transverse measure
2577:Non-measurable set
2559:Locally measurable
2297:
2243:
2208:
2188:
2161:
2131:
2121:topological space
2107:
2005:are both bounded.
1995:
1927:
1865:
1845:
1794:
1744:
1727:where the measure
1714:
1652:
1620:
1588:
1559:
1533:
1506:
1471:
1398:
1378:
1353:are precisely the
1343:
1318:
1253:
1231:
1207:
1180:
1139:
1115:
1091:
1063:
1037:
1013:
974:
947:
914:
894:
859:
839:
808:
788:
760:
740:
707:
687:
667:
644:
585:
561:
528:
464:
438:
418:
388:
354:
334:
306:
286:
262:
238:
214:
194:
170:
146:
45:list of references
3292:
3291:
3252:
3251:
2981:almost everywhere
2927:Spherical measure
2825:Strictly positive
2753:Projection-valued
2493:Almost everywhere
2466:Probability space
2361:Talagrand, Michel
2228:
2211:{\displaystyle X}
2134:{\displaystyle X}
1806:covariance matrix
1664:Gaussian measures
1623:{\displaystyle n}
1401:{\displaystyle n}
1263:. The collection
1234:{\displaystyle x}
1142:{\displaystyle X}
1118:{\displaystyle X}
1094:{\displaystyle X}
977:{\displaystyle X}
950:{\displaystyle X}
862:{\displaystyle M}
842:{\displaystyle M}
811:{\displaystyle Y}
791:{\displaystyle X}
763:{\displaystyle X}
743:{\displaystyle Y}
670:{\displaystyle M}
441:{\displaystyle X}
357:{\displaystyle M}
344:. The collection
309:{\displaystyle M}
289:{\displaystyle X}
241:{\displaystyle X}
217:{\displaystyle T}
197:{\displaystyle X}
98:
97:
90:
16:(Redirected from
3317:
3227:Milman's reverse
3210:
3208:Lebesgue measure
3162:
3161:
2566:
2552:infimum/supremum
2473:Measurable space
2413:
2406:
2399:
2390:
2389:
2378:
2359:Ledoux, Michel;
2355:
2343:
2332:
2306:
2304:
2303:
2298:
2281:
2280:
2262:
2261:
2242:
2217:
2215:
2214:
2209:
2197:
2195:
2194:
2189:
2187:
2186:
2170:
2168:
2167:
2162:
2140:
2138:
2137:
2132:
2116:
2114:
2113:
2108:
2106:
2105:
2090:
2089:
2015:weak convergence
2004:
2002:
2001:
1996:
1994:
1993:
1982:
1961:
1956:
1955:
1936:
1934:
1933:
1928:
1926:
1925:
1920:
1899:
1894:
1893:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1844:
1843:
1832:
1823:
1822:
1803:
1801:
1800:
1795:
1793:
1792:
1787:
1778:
1777:
1753:
1751:
1750:
1745:
1743:
1742:
1723:
1721:
1720:
1715:
1698:
1693:
1692:
1661:
1659:
1658:
1653:
1651:
1650:
1645:
1629:
1627:
1626:
1621:
1597:
1595:
1594:
1589:
1587:
1586:
1581:
1568:
1566:
1565:
1560:
1542:
1540:
1539:
1534:
1532:
1531:
1515:
1513:
1512:
1509:{\displaystyle }
1507:
1480:
1478:
1477:
1472:
1467:
1456:
1451:
1450:
1446:
1427:
1426:
1407:
1405:
1404:
1399:
1387:
1385:
1384:
1379:
1377:
1376:
1352:
1350:
1349:
1344:
1342:
1327:
1325:
1324:
1319:
1314:
1303:
1298:
1297:
1282:
1281:
1262:
1260:
1259:
1254:
1252:
1240:
1238:
1237:
1232:
1216:
1214:
1213:
1208:
1206:
1205:
1189:
1187:
1186:
1181:
1179:
1155:weak convergence
1148:
1146:
1145:
1140:
1124:
1122:
1121:
1116:
1100:
1098:
1097:
1092:
1072:
1070:
1069:
1064:
1046:
1044:
1043:
1038:
1022:
1020:
1019:
1016:{\displaystyle }
1014:
1009:
1008:
983:
981:
980:
975:
956:
954:
953:
948:
923:
921:
920:
915:
903:
901:
900:
895:
893:
889:
888:
868:
866:
865:
860:
848:
846:
845:
840:
818:is said to be a
817:
815:
814:
809:
797:
795:
794:
789:
769:
767:
766:
761:
749:
747:
746:
741:
716:
714:
713:
708:
696:
694:
693:
688:
676:
674:
673:
668:
653:
651:
650:
645:
624:
623:
594:
592:
591:
586:
570:
568:
567:
562:
560:
552:
537:
535:
534:
529:
515:
514:
496:
488:
473:
471:
470:
465:
447:
445:
444:
439:
427:
425:
424:
419:
417:
416:
397:
395:
394:
389:
363:
361:
360:
355:
343:
341:
340:
335:
315:
313:
312:
307:
295:
293:
292:
287:
271:
269:
268:
263:
247:
245:
244:
239:
223:
221:
220:
215:
203:
201:
200:
195:
179:
177:
176:
171:
155:
153:
152:
147:
108:is a concept in
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
3325:
3324:
3320:
3319:
3318:
3316:
3315:
3314:
3295:
3294:
3293:
3288:
3284:Spectral theory
3264:Convex analysis
3248:
3205:
3200:
3153:
3053:
3001:in distribution
2946:
2839:
2669:Logarithmically
2608:
2564:
2547:Essential range
2481:
2422:
2417:
2385:(See chapter 2)
2375:
2352:
2329:
2313:
2276:
2272:
2257:
2253:
2232:
2226:
2223:
2222:
2203:
2200:
2199:
2182:
2178:
2176:
2173:
2172:
2150:
2147:
2146:
2126:
2123:
2122:
2095:
2091:
2085:
2081:
2076:
2073:
2072:
2062:
2011:
1983:
1978:
1977:
1957:
1951:
1947:
1942:
1939:
1938:
1921:
1916:
1915:
1895:
1889:
1885:
1880:
1877:
1876:
1860:
1857:
1856:
1833:
1828:
1827:
1818:
1814:
1812:
1809:
1808:
1788:
1783:
1782:
1773:
1769:
1767:
1764:
1763:
1738:
1734:
1732:
1729:
1728:
1694:
1688:
1684:
1673:
1670:
1669:
1646:
1641:
1640:
1638:
1635:
1634:
1632:Euclidean space
1615:
1612:
1611:
1608:
1582:
1577:
1576:
1574:
1571:
1570:
1548:
1545:
1544:
1527:
1523:
1521:
1518:
1517:
1489:
1486:
1485:
1463:
1452:
1442:
1438:
1434:
1422:
1418:
1416:
1413:
1412:
1393:
1390:
1389:
1372:
1368:
1366:
1363:
1362:
1338:
1336:
1333:
1332:
1310:
1299:
1293:
1289:
1277:
1273:
1271:
1268:
1267:
1248:
1246:
1243:
1242:
1226:
1223:
1222:
1201:
1197:
1195:
1192:
1191:
1175:
1173:
1170:
1169:
1163:
1134:
1131:
1130:
1110:
1107:
1106:
1086:
1083:
1082:
1079:
1052:
1049:
1048:
1032:
1029:
1028:
1004:
1000:
989:
986:
985:
969:
966:
965:
942:
939:
938:
935:
930:
909:
906:
905:
884:
880:
876:
874:
871:
870:
854:
851:
850:
834:
831:
830:
803:
800:
799:
783:
780:
779:
772:random variable
755:
752:
751:
735:
732:
731:
702:
699:
698:
682:
679:
678:
662:
659:
658:
619:
615:
607:
604:
603:
580:
577:
576:
556:
548:
546:
543:
542:
510:
506:
492:
484:
482:
479:
478:
453:
450:
449:
433:
430:
429:
412:
408:
406:
403:
402:
377:
374:
373:
370:uniformly tight
349:
346:
345:
329:
326:
325:
301:
298:
297:
281:
278:
277:
257:
254:
253:
233:
230:
229:
224:. (Thus, every
209:
206:
205:
189:
186:
185:
165:
162:
161:
158:Hausdorff space
129:
126:
125:
122:
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
3323:
3313:
3312:
3307:
3305:Measure theory
3290:
3289:
3287:
3286:
3281:
3276:
3271:
3266:
3260:
3258:
3254:
3253:
3250:
3249:
3247:
3246:
3241:
3236:
3231:
3230:
3229:
3219:
3213:
3211:
3202:
3201:
3199:
3198:
3193:
3191:Sard's theorem
3188:
3183:
3178:
3177:
3176:
3174:Lifting theory
3165:
3159:
3155:
3154:
3152:
3151:
3146:
3141:
3136:
3131:
3130:
3129:
3127:FubiniâTonelli
3119:
3114:
3109:
3108:
3107:
3102:
3097:
3089:
3088:
3087:
3082:
3077:
3069:
3063:
3061:
3055:
3054:
3052:
3051:
3046:
3041:
3036:
3031:
3026:
3021:
3015:
3010:
3009:
3008:
3006:in probability
3003:
2993:
2988:
2983:
2977:
2976:
2975:
2970:
2965:
2954:
2952:
2948:
2947:
2945:
2944:
2939:
2934:
2929:
2924:
2919:
2918:
2917:
2907:
2902:
2901:
2900:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2855:
2849:
2847:
2841:
2840:
2838:
2837:
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2786:
2785:
2780:
2775:
2765:
2760:
2755:
2750:
2740:
2735:
2730:
2725:
2720:
2715:
2713:Locally finite
2710:
2700:
2695:
2690:
2685:
2680:
2675:
2665:
2660:
2655:
2650:
2645:
2640:
2635:
2630:
2625:
2619:
2617:
2610:
2609:
2607:
2606:
2601:
2596:
2591:
2586:
2585:
2584:
2574:
2569:
2561:
2556:
2555:
2554:
2544:
2539:
2538:
2537:
2527:
2522:
2517:
2516:
2515:
2505:
2500:
2495:
2489:
2487:
2483:
2482:
2480:
2479:
2470:
2469:
2468:
2458:
2453:
2445:
2440:
2430:
2428:
2427:Basic concepts
2424:
2423:
2420:Measure theory
2416:
2415:
2408:
2401:
2393:
2387:
2386:
2373:
2356:
2350:
2333:
2327:
2312:
2309:
2308:
2307:
2296:
2293:
2290:
2287:
2284:
2279:
2275:
2271:
2268:
2265:
2260:
2256:
2252:
2249:
2246:
2241:
2238:
2235:
2231:
2230:lim sup
2207:
2185:
2181:
2160:
2157:
2154:
2141:is said to be
2130:
2104:
2101:
2098:
2094:
2088:
2084:
2080:
2068:. A family of
2061:
2058:
2057:
2056:
2051:
2046:
2041:
2036:
2031:
2010:
2007:
1992:
1989:
1986:
1981:
1976:
1973:
1970:
1967:
1964:
1960:
1954:
1950:
1946:
1924:
1919:
1914:
1911:
1908:
1905:
1902:
1898:
1892:
1888:
1884:
1864:
1842:
1839:
1836:
1831:
1826:
1821:
1817:
1791:
1786:
1781:
1776:
1772:
1756:expected value
1741:
1737:
1725:
1724:
1713:
1710:
1707:
1704:
1701:
1697:
1691:
1687:
1683:
1680:
1677:
1649:
1644:
1619:
1607:
1604:
1585:
1580:
1558:
1555:
1552:
1530:
1526:
1505:
1502:
1499:
1496:
1493:
1482:
1481:
1470:
1466:
1462:
1459:
1455:
1449:
1445:
1441:
1437:
1433:
1430:
1425:
1421:
1397:
1375:
1371:
1341:
1329:
1328:
1317:
1313:
1309:
1306:
1302:
1296:
1292:
1288:
1285:
1280:
1276:
1251:
1230:
1204:
1200:
1178:
1162:
1159:
1138:
1114:
1090:
1078:
1075:
1073:is not tight.
1062:
1059:
1056:
1036:
1025:order topology
1012:
1007:
1003:
999:
996:
993:
973:
946:
934:
933:Compact spaces
931:
929:
926:
913:
892:
887:
883:
879:
858:
838:
807:
787:
759:
739:
706:
686:
666:
655:
654:
642:
639:
636:
633:
630:
627:
622:
618:
614:
611:
584:
559:
555:
551:
539:
538:
527:
524:
521:
518:
513:
509:
505:
502:
499:
495:
491:
487:
463:
460:
457:
437:
415:
411:
400:compact subset
387:
384:
381:
372:) if, for any
368:(or sometimes
353:
333:
305:
285:
261:
250:measurable set
237:
213:
193:
182:σ-algebra
169:
145:
142:
139:
136:
133:
121:
118:
110:measure theory
96:
95:
53:external links
42:
40:
33:
26:
9:
6:
4:
3:
2:
3322:
3311:
3308:
3306:
3303:
3302:
3300:
3285:
3282:
3280:
3279:Real analysis
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3259:
3255:
3245:
3242:
3240:
3237:
3235:
3232:
3228:
3225:
3224:
3223:
3220:
3218:
3215:
3214:
3212:
3209:
3203:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3175:
3172:
3171:
3170:
3167:
3166:
3163:
3160:
3158:Other results
3156:
3150:
3147:
3145:
3144:RadonâNikodym
3142:
3140:
3137:
3135:
3132:
3128:
3125:
3124:
3123:
3120:
3118:
3117:Fatou's lemma
3115:
3113:
3110:
3106:
3103:
3101:
3098:
3096:
3093:
3092:
3090:
3086:
3083:
3081:
3078:
3076:
3073:
3072:
3070:
3068:
3065:
3064:
3062:
3060:
3056:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3016:
3014:
3011:
3007:
3004:
3002:
2999:
2998:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:Convergence:
2978:
2974:
2971:
2969:
2966:
2964:
2961:
2960:
2959:
2956:
2955:
2953:
2949:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2916:
2913:
2912:
2911:
2908:
2906:
2903:
2899:
2896:
2895:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2850:
2848:
2846:
2842:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2784:
2783:Outer regular
2781:
2779:
2778:Inner regular
2776:
2774:
2773:Borel regular
2771:
2770:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2745:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2705:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2670:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2620:
2618:
2616:
2611:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2583:
2580:
2579:
2578:
2575:
2573:
2570:
2568:
2562:
2560:
2557:
2553:
2550:
2549:
2548:
2545:
2543:
2540:
2536:
2533:
2532:
2531:
2528:
2526:
2523:
2521:
2518:
2514:
2511:
2510:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2490:
2488:
2484:
2478:
2474:
2471:
2467:
2464:
2463:
2462:
2461:Measure space
2459:
2457:
2454:
2452:
2450:
2446:
2444:
2441:
2439:
2435:
2432:
2431:
2429:
2425:
2421:
2414:
2409:
2407:
2402:
2400:
2395:
2394:
2391:
2384:
2381:
2376:
2374:3-540-52013-9
2370:
2366:
2362:
2357:
2353:
2351:0-471-19745-9
2347:
2342:
2341:
2334:
2330:
2328:0-471-00710-2
2324:
2320:
2315:
2314:
2294:
2291:
2288:
2285:
2277:
2273:
2266:
2258:
2254:
2250:
2247:
2244:
2239:
2233:
2221:
2220:
2219:
2205:
2183:
2179:
2158:
2155:
2152:
2144:
2128:
2120:
2102:
2099:
2096:
2086:
2082:
2071:
2067:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2026:
2025:
2023:
2020:
2016:
2006:
1990:
1987:
1984:
1974:
1968:
1965:
1962:
1952:
1948:
1922:
1912:
1906:
1903:
1900:
1890:
1886:
1840:
1837:
1834:
1824:
1819:
1815:
1807:
1789:
1779:
1774:
1770:
1761:
1757:
1739:
1735:
1711:
1705:
1702:
1699:
1689:
1685:
1678:
1668:
1667:
1666:
1665:
1647:
1633:
1630:-dimensional
1617:
1603:
1601:
1583:
1556:
1553:
1550:
1528:
1524:
1516:will work as
1500:
1497:
1494:
1460:
1457:
1447:
1443:
1439:
1435:
1428:
1423:
1419:
1411:
1410:
1409:
1395:
1373:
1369:
1360:
1356:
1307:
1304:
1294:
1290:
1283:
1278:
1274:
1266:
1265:
1264:
1228:
1220:
1219:Dirac measure
1202:
1198:
1168:
1165:Consider the
1158:
1156:
1152:
1136:
1128:
1112:
1104:
1088:
1077:Polish spaces
1074:
1057:
1034:
1026:
1005:
1001:
997:
994:
971:
963:
962:compact space
960:
944:
925:
911:
890:
885:
881:
877:
856:
836:
827:
825:
821:
805:
785:
777:
773:
757:
737:
728:
726:
725:
720:
719:tight measure
704:
684:
664:
640:
637:
634:
631:
628:
620:
616:
609:
602:
601:
600:
598:
582:
574:
553:
525:
522:
519:
511:
507:
500:
489:
477:
476:
475:
461:
458:
455:
435:
413:
409:
401:
398:, there is a
385:
382:
379:
371:
367:
351:
323:
319:
303:
283:
275:
251:
235:
227:
211:
191:
183:
159:
140:
137:
134:
117:
115:
111:
107:
103:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
18:Tight measure
3059:Main results
2829:
2795:Set function
2723:Metric outer
2678:Decomposable
2535:Cylinder set
2448:
2364:
2339:
2318:
2145:if, for any
2142:
2063:
2012:
1726:
1609:
1602:is bounded.
1483:
1330:
1164:
1103:Polish space
1080:
936:
828:
823:
819:
729:
722:
721:or to be an
718:
656:
540:
369:
365:
123:
105:
99:
84:
75:
64:Please help
56:
3019:compact set
2986:of measures
2922:Pushforward
2915:Projections
2905:Logarithmic
2748:Probability
2738:Pre-measure
2520:Borel space
2438:of measures
1217:denote the
226:open subset
120:Definitions
102:mathematics
70:introducing
3299:Categories
2991:in measure
2718:Maximising
2688:Equivalent
2582:Vitali set
2311:References
2218:such that
1151:precompact
959:metrizable
364:is called
160:, and let
78:March 2016
3105:Maharam's
3075:Dominated
2888:Intensity
2883:Hausdorff
2790:Saturated
2708:Invariant
2613:Types of
2572:Ï-algebra
2542:đ-system
2508:Borel set
2503:Baire set
2292:ε
2289:−
2278:ε
2270:∖
2259:δ
2255:μ
2251:
2245:δ
2237:↓
2234:δ
2184:ε
2153:ε
2119:Hausdorff
2097:δ
2087:δ
2083:μ
2022:dimension
1988:×
1975:⊆
1966:∈
1913:⊆
1904:∈
1863:Γ
1838:×
1825:∈
1780:∈
1736:γ
1703:∈
1686:γ
1676:Γ
1610:Consider
1551:ε
1529:ε
1461:∈
1436:δ
1370:δ
1308:∈
1291:δ
1199:δ
1167:real line
1058:μ
1035:μ
1023:with its
1002:ω
912:μ
882:μ
705:μ
685:μ
638:ε
635:−
621:ε
610:μ
583:μ
554:μ
523:ε
512:ε
504:∖
490:μ
459:∈
456:μ
414:ε
380:ε
332:Σ
260:Σ
168:Σ
106:tightness
3122:Fubini's
3112:Egorov's
3080:Monotone
3039:variable
3017:Random:
2968:Strongly
2893:Lebesgue
2878:Harmonic
2868:Gaussian
2853:Counting
2820:Spectral
2815:Singular
2805:s-finite
2800:Ï-finite
2683:Discrete
2658:Complete
2615:Measures
2589:Null set
2477:function
2363:(1991).
2019:infinite
1600:supports
1543:for any
928:Examples
770:-valued
114:infinity
3034:process
3029:measure
3024:element
2963:Bochner
2937:Trivial
2932:Tangent
2910:Product
2768:Regular
2746:)
2733:Perfect
2706:)
2671:)
2663:Content
2653:Complex
2594:Support
2567:-system
2456:Measure
2383:1102015
1359:bounded
571:is the
322:complex
296:.) Let
66:improve
3100:Jordan
3085:Vitali
3044:vector
2973:Weakly
2835:Vector
2810:Signed
2763:Random
2704:Quasi-
2693:Finite
2673:Convex
2633:Banach
2623:Atomic
2451:spaces
2436:
2371:
2348:
2325:
2024:. See
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