1419:
419:
1140:
990:
223:
1642:
681:
857:
1130:
1145:
1895:
1813:
537:
840:
494:
2037:
1963:
1045:
1414:{\displaystyle {\begin{aligned}C_{t}&=C_{t}\,\cap \,C_{t+1}\,\cap \cdots \cap \,C_{t+m}\\&=\{x\in S^{\mathbb {Z} }:x_{t}=a_{0},\ldots ,x_{t+m}=a_{m}\}\end{aligned}}.}
118:
1675:
569:
190:
148:
1926:
1733:
1706:
755:
728:
1427:. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but the complement of a cylinder set is a
1837:
1757:
1483:
1463:
1012:
701:
592:
439:
210:
73:
414:{\displaystyle \bigcap _{i=1}^{n}p_{Y_{i}}^{-1}\left(A_{i}\right)=\left\{\left(x\right)\in X\mid p_{Y_{1}}(x)\in A_{1},\dots ,p_{Y_{n}}(x)\in A_{n}\right\}}
2938:
3016:
3033:
1050:
605:
2341:
2200:
2856:
1018:. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on
2687:
2227:
2848:
762:
2634:
3028:
2985:
2975:
2096:
1762:
985:{\displaystyle S^{\mathbb {Z} }=\{x=(\ldots ,x_{-1},x_{0},x_{1},\ldots ):x_{k}\in S\;\forall k\in \mathbb {Z} \}.}
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17:
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2013:
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1970:
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2502:
2348:
151:
2121: â Use of filters to describe and characterize all basic topological notions and results.
772:
number of open cylinders is important; allowing infinite intersections generally results in a
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by cylinder sets corresponding to the components' open sets. That is cylinders of the form
2079:
Cylinder sets over topological vector spaces are the core ingredient in the definition of
8:
2911:
2901:
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2711:
2537:
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2050:-adic numbers can be applied to cylinder sets, and in particular, the definition of
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40:
36:
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2141: â one of two closely related types of functions or operations in set theory
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2813:
2624:
2599:
2552:
2482:
2462:
2422:
2412:
2209:
2051:
1974:
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2064:
apply to cylinder sets. These types of measure spaces appear in the theory of
1936:
Cylinder sets are often used to define a topology on sets that are subsets of
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2731:
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2361:
1637:{\displaystyle C_{A}=\{x\in V:(f_{1}(x),f_{2}(x),\ldots ,f_{n}(x))\in A\}}
2527:
1473:
1431:
of cylinders, and so cylinder sets are also closed, and are thus clopen.
28:
2371:
1424:
2353:
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2292:
1678:
2378:
2237:
765:
by cylinder sets corresponding to the components' measurable sets.
213:
2178:
1134:
The intersections of a finite number of open cylinders are the
768:
The restriction that the cylinder set be the intersection of a
676:{\textstyle \bigcap _{i=1}^{n}p_{Y_{i}}^{-1}\left(U_{i}\right)}
497:
43:
on a product of sets; they are also a generating family of the
787:
776:
topology. In the latter case, the resulting topology is the
2003:
on the space: for example, one says that two strings are
1125:{\displaystyle C_{t}=\{x\in S^{\mathbb {Z} }:x_{t}=a\}.}
757:. In the same manner, in case of measurable spaces, the
220:
of such preimages. Explicitly, it is a set of the form,
2007:
if a fraction 1âΔ of the letters in the strings match.
1981:; for example, the measure of a cylinder set of length
608:
84:
2127: â way to generate a measure over product spaces
2016:
1942:
1907:
1849:
1825:
1765:
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61:
2143:
Pages displaying wikidata descriptions as a fallback
2129:
Pages displaying wikidata descriptions as a fallback
2031:
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67:
3086:
2115: â Family of sets representing "large" sets
1928:are taken to be continuous linear functionals.
1434:
1843:, the definition is made instead for elements
192:that maps every element of the product to its
2194:
2083:, which provide the formal definition of the
2939:RieszâMarkovâKakutani representation theorem
1808:{\displaystyle f_{j}\in (V^{*})^{\otimes n}}
1631:
1535:
1401:
1316:
1116:
1076:
976:
876:
829:
805:
2160:
2072:. A generalization of these systems is the
1973:. Cylinder sets are often used to define a
3034:Vitale's random BrunnâMinkowski inequality
2201:
2187:
961:
788:Cylinder sets in products of discrete sets
2023:
1949:
1890:{\displaystyle f_{j}\in (V')^{\otimes n}}
1331:
1273:
1263:
1230:
1226:
1091:
1031:
972:
867:
1480:), the cylinder sets may be defined as
14:
3087:
1999:Cylinder sets may be used to define a
2182:
1965:and occur frequently in the study of
50:
3047:Applications & related
532:{\displaystyle A_{i}\subseteq Y_{i}}
216:of a canonical projection or finite
835:{\displaystyle S=\{1,2,\ldots ,n\}}
489:{\displaystyle Y_{1},...Y_{n}\in S}
120:of all sets in the collection. The
24:
2208:
962:
25:
3111:
2150: â Mathematical construction
2976:Lebesgue differentiation theorem
2857:Carathéodory's extension theorem
2032:{\displaystyle S^{\mathbb {Z} }}
1958:{\displaystyle S^{\mathbb {Z} }}
1040:{\displaystyle S^{\mathbb {Z} }}
1931:
854:in these letters is denoted by
212:component. A cylinder set is a
113:{\textstyle X=\prod _{Y\in S}Y}
1875:
1863:
1793:
1779:
1670:{\displaystyle A\subset K^{n}}
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13:
1:
2154:
564:{\displaystyle 1\leq i\leq n}
1979:Kolmogorov extension theorem
1435:Definition for vector spaces
842:be a finite set, containing
185:{\displaystyle p_{Y}:X\to Y}
7:
3029:PrĂ©kopaâLeindler inequality
2168:Encyclopedia of Mathematics
2106:
1901:. That is, the functionals
1439:Given a finite or infinite-
10:
3116:
2971:Lebesgue's density theorem
780:; cylinder sets are never
598:, the product topology is
441:, finite sequence of sets
3046:
3024:MinkowskiâSteiner formula
2994:
2954:
2947:
2847:
2839:Projection-valued measure
2740:
2633:
2402:
2275:
2216:
1841:topological vector spaces
3007:Isoperimetric inequality
2986:VitaliâHahnâSaks theorem
2315:Carathéodory's criterion
2046:, some of the theory of
2039:can be considered to be
994:The natural topology on
850:. The collection of all
3012:BrunnâMinkowski theorem
2881:Decomposition theorems
2139:Projection (set theory)
1971:subshift of finite type
574:Then, when all sets in
3059:Descriptive set theory
2959:Disintegration theorem
2394:Universally measurable
2081:abstract Wiener spaces
2033:
1959:
1922:
1891:
1833:
1809:
1753:
1729:
1702:
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629:
588:
565:
533:
490:
435:
415:
247:
206:
186:
144:
143:{\displaystyle Y\in S}
124:corresponding to some
114:
75:of sets, consider the
69:
2861:Convergence theorems
2320:Cylindrical Ï-algebra
2161:R.A. Minlos (2001) ,
2134:Cylindrical Ï-algebra
2101:statistical mechanics
2085:Feynman path integral
2070:nonsingular odometers
2034:
1960:
1923:
1921:{\displaystyle f_{j}}
1899:continuous dual space
1892:
1834:
1810:
1754:
1730:
1728:{\displaystyle f_{j}}
1703:
1701:{\displaystyle K^{n}}
1672:
1639:
1460:
1416:
1127:
1042:
1009:
987:
837:
752:
750:{\displaystyle Y_{i}}
725:
723:{\displaystyle U_{i}}
698:
678:
609:
589:
566:
534:
491:
436:
416:
227:
207:
187:
145:
115:
70:
2929:Minkowski inequality
2803:Cylinder set measure
2688:Infinite-dimensional
2303:equivalence relation
2233:Lebesgue integration
2125:Cylinder set measure
2093:quantum field theory
2014:
1969:; see, for example,
1940:
1905:
1847:
1839:. When dealing with
1823:
1817:algebraic dual space
1763:
1743:
1712:
1685:
1648:
1484:
1449:
1141:
1051:
1022:
998:
858:
796:
761:is the one which is
734:
707:
687:
606:
578:
543:
503:
445:
425:
224:
196:
157:
128:
122:canonical projection
82:
59:
2924:Hölder's inequality
2786:of random variables
2748:Measurable function
2635:Particular measures
2224:Absolute continuity
2119:Filters in topology
2113:Filter (set theory)
2089:functional integral
852:bi-infinite strings
654:
272:
55:Given a collection
3064:Probability theory
2389:Transverse measure
2367:Non-measurable set
2349:Locally measurable
2097:partition function
2029:
1985:might be given by
1955:
1918:
1887:
1829:
1805:
1749:
1725:
1698:
1667:
1634:
1455:
1423:Cylinder sets are
1411:
1406:
1122:
1037:
1004:
982:
832:
759:cylinder Ï-algebra
747:
720:
693:
673:
630:
596:topological spaces
584:
561:
529:
486:
431:
421:for any choice of
411:
248:
202:
182:
140:
110:
106:
65:
51:General definition
45:cylinder Ï-algebra
3082:
3081:
3042:
3041:
2771:almost everywhere
2717:Spherical measure
2615:Strictly positive
2543:Projection-valued
2283:Almost everywhere
2256:Probability space
2066:dynamical systems
2010:Since strings in
1967:symbolic dynamics
1832:{\displaystyle V}
1752:{\displaystyle V}
1737:linear functional
1458:{\displaystyle V}
1016:discrete topology
1007:{\displaystyle S}
696:{\displaystyle i}
587:{\displaystyle S}
434:{\displaystyle n}
205:{\displaystyle Y}
91:
77:Cartesian product
68:{\displaystyle S}
16:(Redirected from
3107:
3095:General topology
3017:Milman's reverse
3000:
2998:Lebesgue measure
2952:
2951:
2356:
2342:infimum/supremum
2263:Measurable space
2203:
2196:
2189:
2180:
2179:
2175:
2144:
2130:
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2028:
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2026:
1995:
1991:
1964:
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1509:
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1496:
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1464:
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1461:
1456:
1420:
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1407:
1400:
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1387:
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1362:
1361:
1349:
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1334:
1309:
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1301:
1289:
1288:
1259:
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1209:
1208:
1189:
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1170:
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1157:
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1131:
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1128:
1123:
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1094:
1063:
1062:
1046:
1044:
1043:
1038:
1036:
1035:
1034:
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1010:
1005:
991:
989:
988:
983:
975:
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953:
932:
931:
919:
918:
906:
905:
872:
871:
870:
841:
839:
838:
833:
756:
754:
753:
748:
746:
745:
729:
727:
726:
721:
719:
718:
702:
700:
699:
694:
682:
680:
679:
674:
672:
668:
667:
653:
645:
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643:
628:
623:
593:
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590:
585:
570:
568:
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562:
538:
536:
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528:
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457:
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440:
438:
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404:
383:
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357:
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335:
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309:
290:
286:
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271:
263:
262:
261:
246:
241:
211:
209:
208:
203:
191:
189:
188:
183:
169:
168:
149:
147:
146:
141:
119:
117:
116:
111:
105:
74:
72:
71:
66:
41:product topology
21:
3115:
3114:
3110:
3109:
3108:
3106:
3105:
3104:
3085:
3084:
3083:
3078:
3074:Spectral theory
3054:Convex analysis
3038:
2995:
2990:
2943:
2843:
2791:in distribution
2736:
2629:
2459:Logarithmically
2398:
2354:
2337:Essential range
2271:
2212:
2207:
2157:
2142:
2128:
2109:
2074:Markov odometer
2068:and are called
2022:
2021:
2017:
2015:
2012:
2011:
1993:
1986:
1948:
1947:
1943:
1941:
1938:
1937:
1934:
1912:
1908:
1906:
1903:
1902:
1878:
1874:
1866:
1854:
1850:
1848:
1845:
1844:
1824:
1821:
1820:
1796:
1792:
1786:
1782:
1770:
1766:
1764:
1761:
1760:
1744:
1741:
1740:
1719:
1715:
1713:
1710:
1709:
1692:
1688:
1686:
1683:
1682:
1661:
1657:
1649:
1646:
1645:
1607:
1603:
1579:
1575:
1557:
1553:
1523:
1519:
1504:
1500:
1491:
1487:
1485:
1482:
1481:
1478:complex numbers
1450:
1447:
1446:
1437:
1405:
1404:
1395:
1391:
1376:
1372:
1357:
1353:
1344:
1340:
1330:
1329:
1325:
1307:
1306:
1297:
1293:
1278:
1274:
1254:
1250:
1235:
1231:
1217:
1213:
1204:
1200:
1193:
1184:
1180:
1165:
1161:
1152:
1148:
1144:
1142:
1139:
1138:
1104:
1100:
1090:
1089:
1085:
1058:
1054:
1052:
1049:
1048:
1030:
1029:
1025:
1023:
1020:
1019:
999:
996:
995:
971:
949:
945:
927:
923:
914:
910:
898:
894:
866:
865:
861:
859:
856:
855:
797:
794:
793:
790:
741:
737:
735:
732:
731:
714:
710:
708:
705:
704:
688:
685:
684:
683:where for each
663:
659:
655:
646:
639:
635:
634:
624:
613:
607:
604:
603:
579:
576:
575:
544:
541:
540:
523:
519:
510:
506:
504:
501:
500:
474:
470:
452:
448:
446:
443:
442:
426:
423:
422:
400:
396:
376:
372:
371:
367:
352:
348:
328:
324:
323:
319:
299:
298:
294:
281:
277:
273:
264:
257:
253:
252:
242:
231:
225:
222:
221:
197:
194:
193:
164:
160:
158:
155:
154:
129:
126:
125:
95:
83:
80:
79:
60:
57:
56:
53:
23:
22:
15:
12:
11:
5:
3113:
3103:
3102:
3100:Measure theory
3097:
3080:
3079:
3077:
3076:
3071:
3066:
3061:
3056:
3050:
3048:
3044:
3043:
3040:
3039:
3037:
3036:
3031:
3026:
3021:
3020:
3019:
3009:
3003:
3001:
2992:
2991:
2989:
2988:
2983:
2981:Sard's theorem
2978:
2973:
2968:
2967:
2966:
2964:Lifting theory
2955:
2949:
2945:
2944:
2942:
2941:
2936:
2931:
2926:
2921:
2920:
2919:
2917:FubiniâTonelli
2909:
2904:
2899:
2898:
2897:
2892:
2887:
2879:
2878:
2877:
2872:
2867:
2859:
2853:
2851:
2845:
2844:
2842:
2841:
2836:
2831:
2826:
2821:
2816:
2811:
2805:
2800:
2799:
2798:
2796:in probability
2793:
2783:
2778:
2773:
2767:
2766:
2765:
2760:
2755:
2744:
2742:
2738:
2737:
2735:
2734:
2729:
2724:
2719:
2714:
2709:
2708:
2707:
2697:
2692:
2691:
2690:
2680:
2675:
2670:
2665:
2660:
2655:
2650:
2645:
2639:
2637:
2631:
2630:
2628:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2576:
2575:
2570:
2565:
2555:
2550:
2545:
2540:
2530:
2525:
2520:
2515:
2510:
2505:
2503:Locally finite
2500:
2490:
2485:
2480:
2475:
2470:
2465:
2455:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2409:
2407:
2400:
2399:
2397:
2396:
2391:
2386:
2381:
2376:
2375:
2374:
2364:
2359:
2351:
2346:
2345:
2344:
2334:
2329:
2328:
2327:
2317:
2312:
2307:
2306:
2305:
2295:
2290:
2285:
2279:
2277:
2273:
2272:
2270:
2269:
2260:
2259:
2258:
2248:
2243:
2235:
2230:
2220:
2218:
2217:Basic concepts
2214:
2213:
2210:Measure theory
2206:
2205:
2198:
2191:
2183:
2177:
2176:
2163:"Cylinder Set"
2156:
2153:
2152:
2151:
2145:
2136:
2131:
2122:
2116:
2108:
2105:
2055:-adic measures
2025:
2020:
1951:
1946:
1933:
1930:
1915:
1911:
1884:
1881:
1877:
1872:
1869:
1865:
1862:
1857:
1853:
1828:
1802:
1799:
1795:
1789:
1785:
1781:
1778:
1773:
1769:
1748:
1722:
1718:
1695:
1691:
1664:
1660:
1656:
1653:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1610:
1606:
1602:
1599:
1596:
1593:
1590:
1587:
1582:
1578:
1574:
1571:
1568:
1565:
1560:
1556:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1526:
1522:
1518:
1515:
1512:
1507:
1503:
1499:
1494:
1490:
1454:
1436:
1433:
1410:
1403:
1398:
1394:
1390:
1385:
1382:
1379:
1375:
1371:
1368:
1365:
1360:
1356:
1352:
1347:
1343:
1339:
1333:
1328:
1324:
1321:
1318:
1315:
1312:
1310:
1308:
1305:
1300:
1296:
1292:
1287:
1284:
1281:
1277:
1272:
1269:
1266:
1262:
1257:
1253:
1249:
1244:
1241:
1238:
1234:
1229:
1225:
1220:
1216:
1212:
1207:
1203:
1199:
1196:
1194:
1192:
1187:
1183:
1179:
1176:
1173:
1168:
1164:
1160:
1155:
1151:
1147:
1146:
1121:
1118:
1115:
1112:
1107:
1103:
1099:
1093:
1088:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1061:
1057:
1033:
1028:
1003:
981:
978:
974:
970:
967:
964:
960:
957:
952:
948:
944:
941:
938:
935:
930:
926:
922:
917:
913:
909:
904:
901:
897:
893:
890:
887:
884:
881:
878:
875:
869:
864:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
789:
786:
744:
740:
717:
713:
692:
671:
666:
662:
658:
652:
649:
642:
638:
633:
627:
622:
619:
616:
612:
583:
560:
557:
554:
551:
548:
526:
522:
518:
513:
509:
485:
482:
477:
473:
469:
466:
463:
460:
455:
451:
430:
409:
403:
399:
395:
392:
389:
386:
379:
375:
370:
366:
363:
360:
355:
351:
347:
344:
341:
338:
331:
327:
322:
318:
315:
312:
308:
305:
302:
297:
293:
289:
284:
280:
276:
270:
267:
260:
256:
251:
245:
240:
237:
234:
230:
201:
181:
178:
175:
172:
167:
163:
139:
136:
133:
109:
104:
101:
98:
94:
90:
87:
64:
52:
49:
9:
6:
4:
3:
2:
3112:
3101:
3098:
3096:
3093:
3092:
3090:
3075:
3072:
3070:
3069:Real analysis
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3051:
3049:
3045:
3035:
3032:
3030:
3027:
3025:
3022:
3018:
3015:
3014:
3013:
3010:
3008:
3005:
3004:
3002:
2999:
2993:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2965:
2962:
2961:
2960:
2957:
2956:
2953:
2950:
2948:Other results
2946:
2940:
2937:
2935:
2934:RadonâNikodym
2932:
2930:
2927:
2925:
2922:
2918:
2915:
2914:
2913:
2910:
2908:
2907:Fatou's lemma
2905:
2903:
2900:
2896:
2893:
2891:
2888:
2886:
2883:
2882:
2880:
2876:
2873:
2871:
2868:
2866:
2863:
2862:
2860:
2858:
2855:
2854:
2852:
2850:
2846:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2806:
2804:
2801:
2797:
2794:
2792:
2789:
2788:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:Convergence:
2768:
2764:
2761:
2759:
2756:
2754:
2751:
2750:
2749:
2746:
2745:
2743:
2739:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2706:
2703:
2702:
2701:
2698:
2696:
2693:
2689:
2686:
2685:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2640:
2638:
2636:
2632:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2574:
2573:Outer regular
2571:
2569:
2568:Inner regular
2566:
2564:
2563:Borel regular
2561:
2560:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2535:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2495:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2460:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2410:
2408:
2406:
2401:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2373:
2370:
2369:
2368:
2365:
2363:
2360:
2358:
2352:
2350:
2347:
2343:
2340:
2339:
2338:
2335:
2333:
2330:
2326:
2323:
2322:
2321:
2318:
2316:
2313:
2311:
2308:
2304:
2301:
2300:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2280:
2278:
2274:
2268:
2264:
2261:
2257:
2254:
2253:
2252:
2251:Measure space
2249:
2247:
2244:
2242:
2240:
2236:
2234:
2231:
2229:
2225:
2222:
2221:
2219:
2215:
2211:
2204:
2199:
2197:
2192:
2190:
2185:
2184:
2181:
2174:
2170:
2169:
2164:
2159:
2158:
2149:
2146:
2140:
2137:
2135:
2132:
2126:
2123:
2120:
2117:
2114:
2111:
2110:
2104:
2102:
2098:
2094:
2090:
2086:
2082:
2077:
2075:
2071:
2067:
2063:
2062:-adic metrics
2061:
2056:
2054:
2049:
2045:
2044:-adic numbers
2043:
2018:
2008:
2006:
2002:
1997:
1990:
1984:
1980:
1976:
1972:
1968:
1944:
1929:
1913:
1909:
1900:
1882:
1879:
1870:
1867:
1860:
1855:
1851:
1842:
1826:
1818:
1800:
1797:
1787:
1783:
1776:
1771:
1767:
1746:
1738:
1720:
1716:
1693:
1689:
1680:
1662:
1658:
1654:
1651:
1628:
1625:
1616:
1608:
1604:
1600:
1597:
1594:
1588:
1580:
1576:
1572:
1566:
1558:
1554:
1547:
1544:
1541:
1538:
1532:
1524:
1520:
1516:
1513:
1510:
1505:
1501:
1492:
1488:
1479:
1475:
1472:(such as the
1471:
1468:
1452:
1445:
1442:
1432:
1430:
1426:
1421:
1408:
1396:
1392:
1388:
1383:
1380:
1377:
1373:
1369:
1366:
1363:
1358:
1354:
1350:
1345:
1341:
1337:
1326:
1322:
1319:
1313:
1311:
1298:
1294:
1285:
1282:
1279:
1275:
1270:
1267:
1264:
1255:
1251:
1242:
1239:
1236:
1232:
1227:
1218:
1214:
1205:
1201:
1197:
1195:
1185:
1181:
1177:
1174:
1171:
1166:
1162:
1153:
1149:
1137:
1136:cylinder sets
1132:
1119:
1113:
1110:
1105:
1101:
1097:
1086:
1082:
1079:
1073:
1067:
1059:
1055:
1026:
1017:
1001:
992:
979:
968:
965:
958:
955:
950:
946:
942:
936:
933:
928:
924:
920:
915:
911:
907:
902:
899:
895:
891:
888:
882:
879:
873:
862:
853:
849:
845:
826:
823:
820:
817:
814:
811:
808:
802:
799:
785:
783:
782:Hilbert cubes
779:
775:
771:
766:
764:
760:
742:
738:
715:
711:
690:
669:
664:
660:
656:
650:
647:
640:
636:
631:
625:
620:
617:
614:
610:
601:
597:
581:
572:
558:
555:
552:
549:
546:
524:
520:
516:
511:
507:
499:
483:
480:
475:
471:
467:
464:
461:
458:
453:
449:
428:
407:
401:
397:
393:
387:
377:
373:
368:
364:
361:
358:
353:
349:
345:
339:
329:
325:
320:
316:
313:
310:
306:
303:
300:
295:
291:
287:
282:
278:
274:
268:
265:
258:
254:
249:
243:
238:
235:
232:
228:
219:
215:
199:
179:
173:
170:
165:
161:
153:
137:
134:
131:
123:
107:
102:
99:
96:
92:
88:
85:
78:
62:
48:
46:
42:
38:
34:
33:cylinder sets
30:
19:
18:Open cylinder
2849:Main results
2585:Set function
2513:Metric outer
2468:Decomposable
2325:Cylinder set
2324:
2238:
2166:
2148:Ultraproduct
2078:
2059:
2052:
2047:
2041:
2009:
2004:
1998:
1988:
1982:
1977:, using the
1935:
1932:Applications
1469:
1444:vector space
1438:
1422:
1135:
1133:
993:
847:
843:
791:
778:box topology
769:
767:
573:
218:intersection
121:
54:
32:
26:
2809:compact set
2776:of measures
2712:Pushforward
2705:Projections
2695:Logarithmic
2538:Probability
2528:Pre-measure
2310:Borel space
2228:of measures
1759:; that is,
1708:, and each
1441:dimensional
1425:clopen sets
846:objects or
730:is open in
29:mathematics
3089:Categories
2781:in measure
2508:Maximising
2478:Equivalent
2372:Vitali set
2155:References
2095:, and the
2895:Maharam's
2865:Dominated
2678:Intensity
2673:Hausdorff
2580:Saturated
2498:Invariant
2403:Types of
2362:Ï-algebra
2332:đ-system
2298:Borel set
2293:Baire set
2173:EMS Press
1880:⊗
1861:∈
1798:⊗
1788:∗
1777:∈
1679:Borel set
1655:⊂
1626:∈
1598:…
1542:∈
1514:…
1367:…
1323:∈
1271:∩
1268:⋯
1265:∩
1228:∩
1175:…
1083:∈
969:∈
963:∀
956:∈
937:…
900:−
889:…
821:…
763:generated
648:−
611:⋂
600:generated
556:≤
550:≤
517:⊆
481:∈
394:∈
362:…
346:∈
317:∣
311:∈
266:−
229:⋂
177:→
135:∈
100:∈
93:∏
2912:Fubini's
2902:Egorov's
2870:Monotone
2829:variable
2807:Random:
2758:Strongly
2683:Lebesgue
2668:Harmonic
2658:Gaussian
2643:Counting
2610:Spectral
2605:Singular
2595:s-finite
2590:Ï-finite
2473:Discrete
2448:Complete
2405:Measures
2379:Null set
2267:function
2107:See also
1871:′
214:preimage
152:function
2824:process
2819:measure
2814:element
2753:Bochner
2727:Trivial
2722:Tangent
2700:Product
2558:Regular
2536:)
2523:Perfect
2496:)
2461:)
2453:Content
2443:Complex
2384:Support
2357:-system
2246:Measure
2005:Δ-close
1975:measure
1465:over a
1014:is the
848:letters
498:subsets
150:is the
39:of the
35:form a
2890:Jordan
2875:Vitali
2834:vector
2763:Weakly
2625:Vector
2600:Signed
2553:Random
2494:Quasi-
2483:Finite
2463:Convex
2423:Banach
2413:Atomic
2241:spaces
2226:
2001:metric
1992:or by
1897:, the
1815:, the
1644:where
770:finite
31:, the
2732:Young
2653:Euler
2648:Dirac
2620:Tight
2548:Radon
2518:Outer
2488:Inner
2438:Brown
2433:Borel
2428:Besov
2418:Baire
1735:is a
1677:is a
1467:field
1429:union
774:finer
37:basis
2996:For
2885:Hahn
2741:Maps
2663:Haar
2534:Sub-
2288:Atom
2276:Sets
2057:and
1474:real
1047:are
792:Let
594:are
539:for
496:and
2099:of
2091:of
2087:or
2076:.
1996:.
1994:1/2
1819:to
1739:on
1681:in
1476:or
27:In
3091::
2171:,
2165:,
2103:.
1987:1/
784:.
703:,
571:.
47:.
2532:(
2492:(
2457:(
2355:Ï
2265:/
2239:L
2202:e
2195:t
2188:v
2060:p
2053:p
2048:p
2042:p
2024:Z
2019:S
1989:m
1983:m
1950:Z
1945:S
1914:j
1910:f
1883:n
1876:)
1868:V
1864:(
1856:j
1852:f
1827:V
1801:n
1794:)
1784:V
1780:(
1772:j
1768:f
1747:V
1721:j
1717:f
1694:n
1690:K
1663:n
1659:K
1652:A
1632:}
1629:A
1623:)
1620:)
1617:x
1614:(
1609:n
1605:f
1601:,
1595:,
1592:)
1589:x
1586:(
1581:2
1577:f
1573:,
1570:)
1567:x
1564:(
1559:1
1555:f
1551:(
1548::
1545:V
1539:x
1536:{
1533:=
1530:]
1525:n
1521:f
1517:,
1511:,
1506:1
1502:f
1498:[
1493:A
1489:C
1470:K
1453:V
1409:.
1402:}
1397:m
1393:a
1389:=
1384:m
1381:+
1378:t
1374:x
1370:,
1364:,
1359:0
1355:a
1351:=
1346:t
1342:x
1338::
1332:Z
1327:S
1320:x
1317:{
1314:=
1304:]
1299:m
1295:a
1291:[
1286:m
1283:+
1280:t
1276:C
1261:]
1256:1
1252:a
1248:[
1243:1
1240:+
1237:t
1233:C
1224:]
1219:0
1215:a
1211:[
1206:t
1202:C
1198:=
1191:]
1186:m
1182:a
1178:,
1172:,
1167:0
1163:a
1159:[
1154:t
1150:C
1120:.
1117:}
1114:a
1111:=
1106:t
1102:x
1098::
1092:Z
1087:S
1080:x
1077:{
1074:=
1071:]
1068:a
1065:[
1060:t
1056:C
1032:Z
1027:S
1002:S
980:.
977:}
973:Z
966:k
959:S
951:k
947:x
943::
940:)
934:,
929:1
925:x
921:,
916:0
912:x
908:,
903:1
896:x
892:,
886:(
883:=
880:x
877:{
874:=
868:Z
863:S
844:n
830:}
827:n
824:,
818:,
815:2
812:,
809:1
806:{
803:=
800:S
743:i
739:Y
716:i
712:U
691:i
670:)
665:i
661:U
657:(
651:1
641:i
637:Y
632:p
626:n
621:1
618:=
615:i
582:S
559:n
553:i
547:1
525:i
521:Y
512:i
508:A
484:S
476:n
472:Y
468:.
465:.
462:.
459:,
454:1
450:Y
429:n
408:}
402:n
398:A
391:)
388:x
385:(
378:n
374:Y
369:p
365:,
359:,
354:1
350:A
343:)
340:x
337:(
330:1
326:Y
321:p
314:X
307:)
304:x
301:(
296:{
292:=
288:)
283:i
279:A
275:(
269:1
259:i
255:Y
250:p
244:n
239:1
236:=
233:i
200:Y
180:Y
174:X
171::
166:Y
162:p
138:S
132:Y
108:Y
103:S
97:Y
89:=
86:X
63:S
20:)
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