1307:
1081:
For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. See
2201:
856:
defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets." However, modern usage is to call the distinguished sub-algebra the
759:
304:
1748:
103:, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a
507:
2198:
1513:
565:
442:
1444:
806:
1133:
920:
1640:
1618:
240:
208:
2028:
975:
1815:
1682:
1368:
1337:
354:
1883:
1859:
1839:
1795:
1596:
1576:
1537:
1392:
1078:. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum.
4163:
693:
4241:
570:
To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps
4258:
1551:(see References), especially Exercise (27.2) on page 209, Definition (22.9) on page 169, Exercise (3.4)(ii) on page 14, and on page 196.
3566:
3425:
453:
531:
250:
2250:
4081:
382:
3912:
2012:
3452:
1690:
4073:
875:σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.
3859:
2190:
4253:
763:
In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of
4210:
4200:
1452:
622:
will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is
4324:
4010:
3919:
3683:
3539:
657:
4248:
4195:
4089:
3995:
2183:
17:
4114:
4094:
4058:
3982:
3702:
3418:
1302:{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}}
687:
1397:
4236:
4015:
3977:
3929:
2178:
1555:
770:
4141:
4109:
4099:
4020:
3987:
3618:
3527:
2243:
1094:
134:
893:
4158:
4063:
3839:
3767:
2964:
2895:
2399:
1543:(all Borel sets are also analytic), and complete in the class of analytic sets. For more details see
521:
58:
4148:
4231:
3677:
3608:
2953:
1685:
665:
3544:
1909:
1623:
1601:
4000:
3758:
3718:
3411:
2033:
218:
186:
122:
of the topological space, rather than the open sets. The two definitions are equivalent for many
4283:
4183:
4005:
3727:
3573:
2173:
1914:
1544:
941:
879:
676:
112:
3844:
3797:
3792:
3787:
3629:
3512:
3470:
2236:
1800:
310:
2095:, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150
1750:. However, this is a proof of existence (via the axiom of choice), not an explicit example.
1648:
871:
sets (of a topological space), whereas Mackey's definition refers to a set equipped with an
4153:
4119:
4027:
3737:
3692:
3534:
3457:
3388:
3307:
2155:
1548:
1346:
1315:
1070:
1047:
996:
817:
332:
1113:
cannot be exhibited, although the existence of such a set is implied, for example, by the
8:
4136:
4126:
3972:
3936:
3762:
3491:
3448:
2263:
1090:
923:
887:
764:
62:
3814:
867:. The reason for this distinction is that the Borel sets are the σ-algebra generated by
96:
is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
4288:
4048:
4033:
3732:
3613:
3591:
2269:
1971:
1868:
1844:
1824:
1780:
1581:
1561:
1522:
1377:
1125:
1110:
645:
54:
682:
In the construction by transfinite induction, it can be shown that, in each step, the
130:
4319:
4205:
3941:
3902:
3897:
3804:
3722:
3507:
3480:
2824:
2806:
2213:
2122:
2008:
1975:
1963:
1899:
1121:
661:
39:
4222:
4131:
3907:
3892:
3882:
3867:
3834:
3829:
3819:
3697:
3672:
3487:
2707:
2042:
1955:
1865:
or if every compact saturated subset is closed (which is the case in particular if
1862:
862:
4298:
4278:
4053:
3951:
3946:
3924:
3782:
3747:
3667:
3561:
2205:
2108:
2004:
1894:
1763:
1114:
1015:
633:
127:
108:
4188:
4043:
4038:
3849:
3824:
3777:
3707:
3687:
3647:
3637:
3434:
3081:
2365:
322:
100:
2216:
66:
4313:
4293:
3956:
3877:
3872:
3772:
3742:
3712:
3662:
3657:
3652:
3642:
3556:
3475:
2145:
1967:
1941:
1818:
1106:
853:
653:
368:
104:
81:
3887:
3809:
3549:
3064:
2194:
1920:
1540:
1083:
1075:
992:
150:
123:
3586:
1988:
Jochen
Wengenroth, Is every sigma-algebra the Borel algebra of a topology?
3752:
2133:
2061:
2047:
1759:
1447:
683:
649:
119:
31:
3596:
2453:
2382:
2115:, Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition of
2091:
Tommy
Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in:
1959:
1946:
1771:
1023:
47:
1777:
Norberg and
Vervaat redefine the Borel algebra of a topological space
3578:
3517:
2221:
1904:
169:
51:
2542:
2520:
1821:. This definition is well-suited for applications in the case where
3603:
3462:
2432:
1987:
927:
883:
754:{\displaystyle \aleph _{1}\cdot 2^{\aleph _{0}}\,=2^{\aleph _{0}}.}
43:
1394:
be the set of all irrational numbers that correspond to sequences
3403:
1516:
1340:
811:
118:
In some contexts, Borel sets are defined to be generated by the
1620:
is a countable union of countable sets, so that any subset of
1105:
An example of a subset of the reals that is non-Borel, due to
690:. So, the total number of Borel sets is less than or equal to
1841:
is not
Hausdorff. It coincides with the usual definition if
671:
The Borel algebra on the reals is the smallest σ-algebra on
2162:, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)
299:{\displaystyle T_{\delta \sigma }=(T_{\delta })_{\sigma }.}
1753:
153:, the Borel algebra in the first sense may be described
1817:-algebra generated by its open subsets and its compact
1270:
1258:
1237:
1225:
1204:
1192:
1171:
1159:
767:
sets that exist, which is strictly larger and equal to
528:
from the class of open sets by iterating the operation
1446:
with the following property: there exists an infinite
1273:
1261:
1240:
1228:
1207:
1195:
1174:
1162:
668:
is by definition also a measure on the Borel algebra.
1871:
1847:
1827:
1803:
1783:
1693:
1651:
1626:
1604:
1598:
is non-Borel. In fact, it is consistent with ZF that
1584:
1564:
1558:(ZF) are sufficient to formalize the construction of
1525:
1455:
1400:
1380:
1349:
1318:
1136:
944:
896:
773:
696:
534:
456:
385:
335:
253:
221:
189:
2142:
See especially Sect. 51 "Borel sets and Baire sets".
1743:{\displaystyle f\colon \{0,1\}^{\omega }\to \{0,1\}}
995:, that is, a topological space such that there is a
1109:, is described below. In contrast, an example of a
2211:
1877:
1853:
1833:
1809:
1789:
1742:
1676:
1634:
1612:
1590:
1570:
1531:
1507:
1438:
1386:
1362:
1331:
1301:
969:
914:
800:
753:
559:
501:
436:
348:
298:
234:
202:
4311:
604:can be obtained by iterating the operation over
930:measurable sets, i.e., for all measurable sets
140:
242:be all countable intersections of elements of
3419:
2244:
1944:(1966), "Ergodic Theory and Virtual Groups",
812:Standard Borel spaces and Kuratowski theorems
632:The resulting sequence of sets is termed the
502:{\displaystyle G^{i}=\bigcup _{j<i}G^{j}.}
4164:Riesz–Markov–Kakutani representation theorem
1737:
1725:
1713:
1700:
1508:{\displaystyle (a_{k_{0}},a_{k_{1}},\dots )}
560:{\displaystyle G\mapsto G_{\delta \sigma }.}
1124:has a unique representation by an infinite
4259:Vitale's random Brunn–Minkowski inequality
3426:
3412:
2251:
2237:
1998:
1645:Another non-Borel set is an inverse image
1093:on a standard Borel space turns it into a
1053:Considered as Borel spaces, the real line
2046:
1628:
1606:
1278:
727:
329:For the base case of the definition, let
2208:that have been formally proved about it.
1766:Hausdorff topological space is called a
644:An important example, especially in the
437:{\displaystyle G^{i}=_{\delta \sigma }.}
1578:, it cannot be proven in ZF alone that
512:The claim is that the Borel algebra is
210:be all countable unions of elements of
14:
4312:
2132:
2079:
1940:
1917: – Subfield of mathematical logic
1754:Alternative non-equivalent definitions
890:between measurable spaces. A function
76:, the collection of all Borel sets on
3407:
2212:
2026:
648:, is the Borel algebra on the set of
375:has an immediately preceding ordinal
356:be the collection of open subsets of
4272:Applications & related
2060:
1439:{\displaystyle (a_{0},a_{1},\dots )}
524:. That is, the Borel algebra can be
2068:. Courier Corporation. p. 142.
1554:It's important to note, that while
1074:is the Borel space associated to a
801:{\displaystyle 2^{2^{\aleph _{0}}}}
574:into itself for any limit ordinal
24:
3433:
2129:. Wadsworth, Brooks and Cole, 1989
846:is the σ-algebra of Borel sets of
785:
737:
716:
698:
593:, there is some countable ordinal
586:is closed under countable unions.
567:to the first uncountable ordinal.
25:
4336:
2166:
1100:
652:. It is the algebra on which the
629:, the first uncountable ordinal.
582:is an uncountable limit ordinal,
107:. Borel sets and the associated
4201:Lebesgue differentiation theorem
4082:Carathéodory's extension theorem
2160:Classical Descriptive Set Theory
915:{\displaystyle f:X\rightarrow Y}
522:first uncountable ordinal number
168:(that is, for any subset of the
111:also play a fundamental role in
2029:"Sur les ensembles analytiques"
1046:(This result is reminiscent of
133:, but can be different in more
2085:
2072:
2054:
2020:
1992:
1981:
1934:
1770:if it belongs to the smallest
1722:
1671:
1665:
1519:of the next element. This set
1502:
1456:
1433:
1401:
964:
958:
906:
538:
419:
399:
284:
270:
65:. Borel sets are named after
13:
1:
2127:Real Analysis and Probability
2102:
1774:containing all compact sets.
1539:is not Borel. However, it is
1006:that defines the topology of
2413:= co-recursively enumerable
2113:An Invitation to C*-algebras
1635:{\displaystyle \mathbb {R} }
1613:{\displaystyle \mathbb {R} }
1515:such that each element is a
826:be a topological space. The
688:cardinality of the continuum
615:varies over all Borel sets,
141:Generating the Borel algebra
99:Borel sets are important in
50:) through the operations of
7:
4254:Prékopa–Leindler inequality
2179:Encyclopedia of Mathematics
1923: – Concept in topology
1888:
325:, in the following manner:
235:{\displaystyle T_{\delta }}
203:{\displaystyle T_{\sigma }}
10:
4341:
4196:Lebesgue's density theorem
1343:and all the other numbers
1095:standard probability space
1061:with a countable set, and
815:
639:
27:Class of mathematical sets
4271:
4249:Minkowski–Steiner formula
4219:
4179:
4172:
4072:
4064:Projection-valued measure
3965:
3858:
3627:
3500:
3441:
3397:
3394:
3311:
3251:
3246:
3243:
3176:
3165:
3098:
3087:
2957:
2889:
2884:
2881:
2811:
2795:
2790:
2787:
2711:
2651:
2646:
2643:
2564:
2553:
2470:
2459:
2369:
2354:
2314:
2275:
2268:
2261:
2041:: Sect. 62, pages 76–78,
1999:Srivastava, S.M. (1991),
970:{\displaystyle f^{-1}(B)}
878:Measurable spaces form a
686:of sets is, at most, the
4232:Isoperimetric inequality
4211:Vitali–Hahn–Saks theorem
3540:Carathéodory's criterion
2783:= boldface arithmetical
2303:(sometimes the same as Δ
2093:Probability and Lattices
1927:
1686:infinite parity function
666:probability distribution
450:is a limit ordinal, set
92:. The Borel algebra on
72:For a topological space
46:(or, equivalently, from
42:that can be formed from
4237:Brunn–Minkowski theorem
4106:Decomposition theorems
3049:= lightface coanalytic
2034:Fundamenta Mathematicae
2027:Lusin, Nicolas (1927),
1810:{\displaystyle \sigma }
1556:Zermelo–Fraenkel axioms
4325:Descriptive set theory
4284:Descriptive set theory
4184:Disintegration theorem
3619:Universally measurable
2400:recursively enumerable
2001:A Course on Borel Sets
1915:Descriptive set theory
1879:
1855:
1835:
1811:
1791:
1744:
1678:
1677:{\displaystyle f^{-1}}
1636:
1614:
1592:
1572:
1545:descriptive set theory
1533:
1509:
1440:
1388:
1364:
1333:
1303:
971:
916:
802:
755:
675:that contains all the
561:
503:
438:
350:
300:
236:
204:
126:spaces, including all
113:descriptive set theory
4086:Convergence theorems
3545:Cylindrical σ-algebra
3037:= lightface analytic
2193:of Borel Sets in the
2152:, Prentice Hall, 1988
2140:. D. van Nostrand Co.
1910:Cylindrical σ-algebra
1880:
1856:
1836:
1812:
1792:
1745:
1679:
1637:
1615:
1593:
1573:
1534:
1510:
1441:
1389:
1365:
1363:{\displaystyle a_{k}}
1334:
1332:{\displaystyle a_{0}}
1304:
1022:as a Borel space is
972:
917:
803:
756:
656:is defined. Given a
646:theory of probability
562:
504:
439:
351:
349:{\displaystyle G^{0}}
311:transfinite induction
301:
237:
205:
4154:Minkowski inequality
4028:Cylinder set measure
3913:Infinite-dimensional
3528:equivalence relation
3458:Lebesgue integration
2156:Alexander S. Kechris
2048:10.4064/fm-10-1-1-95
1869:
1845:
1825:
1801:
1781:
1691:
1649:
1624:
1602:
1582:
1562:
1523:
1453:
1398:
1378:
1347:
1316:
1134:
1071:standard Borel space
1018:metric space. Then
942:
894:
888:measurable functions
818:Standard Borel space
771:
694:
658:real random variable
532:
454:
383:
333:
251:
219:
187:
90:Borel σ-algebra
4149:Hölder's inequality
4011:of random variables
3973:Measurable function
3860:Particular measures
3449:Absolute continuity
2066:The Axiom of Choice
1272:
1260:
1239:
1227:
1206:
1194:
1173:
1161:
1091:probability measure
765:Lebesgue measurable
589:For each Borel set
63:relative complement
4289:Probability theory
3614:Transverse measure
3592:Non-measurable set
3574:Locally measurable
2214:Weisstein, Eric W.
2204:2020-06-01 at the
1960:10.1007/BF01361167
1875:
1851:
1831:
1807:
1787:
1740:
1674:
1632:
1610:
1588:
1568:
1529:
1505:
1436:
1384:
1360:
1329:
1299:
1295:
1290:
1285:
1280:
1267:
1234:
1201:
1168:
1126:continued fraction
1111:non-measurable set
967:
912:
798:
751:
557:
499:
485:
434:
346:
296:
232:
200:
4307:
4306:
4267:
4266:
3996:almost everywhere
3942:Spherical measure
3840:Strictly positive
3768:Projection-valued
3508:Almost everywhere
3481:Probability space
3402:
3401:
2954:hyperarithmetical
2191:Formal definition
2014:978-0-387-98412-4
1900:Borel isomorphism
1878:{\displaystyle X}
1854:{\displaystyle X}
1834:{\displaystyle X}
1819:saturated subsets
1790:{\displaystyle X}
1591:{\displaystyle A}
1571:{\displaystyle A}
1532:{\displaystyle A}
1387:{\displaystyle A}
1297:
1292:
1287:
1282:
1271:
1259:
1238:
1226:
1205:
1193:
1172:
1160:
1122:irrational number
1048:Maharam's theorem
977:is measurable in
864:measurable spaces
662:probability space
470:
160:For a collection
145:In the case that
40:topological space
16:(Redirected from
4332:
4242:Milman's reverse
4225:
4223:Lebesgue measure
4177:
4176:
3581:
3567:infimum/supremum
3488:Measurable space
3428:
3421:
3414:
3405:
3404:
3381:
3380:
3369:
3368:
3357:
3356:
3345:
3344:
3333:
3332:
3321:
3320:
3305:
3304:
3296:
3295:
3287:
3286:
3278:
3277:
3269:
3268:
3260:
3259:
3237:
3236:
3223:
3222:
3211:
3210:
3200:
3199:
3186:
3185:
3174:
3173:
3159:
3158:
3145:
3144:
3133:
3132:
3122:
3121:
3108:
3107:
3096:
3095:
3078:
3077:
3061:
3060:
3048:
3047:
3036:
3035:
3014:
3013:
3002:
3001:
2987:
2986:
2972:
2971:
2951:
2950:
2942:
2941:
2939:
2938:
2925:
2924:
2922:
2921:
2908:
2907:
2904:
2903:
2876:
2875:
2863:
2862:
2851:
2850:
2840:
2839:
2821:
2820:
2804:
2803:
2781:
2780:
2769:
2768:
2757:
2756:
2745:
2744:
2733:
2732:
2721:
2720:
2705:
2704:
2696:
2695:
2687:
2686:
2678:
2677:
2669:
2668:
2660:
2659:
2631:
2630:
2611:
2610:
2599:
2598:
2588:
2587:
2574:
2573:
2562:
2561:
2539:
2538:
2517:
2516:
2505:
2504:
2494:
2493:
2480:
2479:
2468:
2467:
2446:
2445:
2425:
2424:
2412:
2411:
2397:
2396:
2379:
2378:
2363:
2362:
2348:
2347:
2336:
2335:
2324:
2323:
2311:
2310:
2302:
2301:
2293:
2292:
2284:
2283:
2266:
2259:
2258:
2253:
2246:
2239:
2233:
2227:
2226:
2199:list of theorems
2187:
2141:
2096:
2089:
2083:
2076:
2070:
2069:
2058:
2052:
2051:
2050:
2024:
2018:
2017:
1996:
1990:
1985:
1979:
1978:
1938:
1884:
1882:
1881:
1876:
1863:second countable
1860:
1858:
1857:
1852:
1840:
1838:
1837:
1832:
1816:
1814:
1813:
1808:
1796:
1794:
1793:
1788:
1762:, a subset of a
1749:
1747:
1746:
1741:
1721:
1720:
1683:
1681:
1680:
1675:
1664:
1663:
1642:is a Borel set.
1641:
1639:
1638:
1633:
1631:
1619:
1617:
1616:
1611:
1609:
1597:
1595:
1594:
1589:
1577:
1575:
1574:
1569:
1547:and the book by
1538:
1536:
1535:
1530:
1514:
1512:
1511:
1506:
1495:
1494:
1493:
1492:
1475:
1474:
1473:
1472:
1445:
1443:
1442:
1437:
1426:
1425:
1413:
1412:
1393:
1391:
1390:
1385:
1369:
1367:
1366:
1361:
1359:
1358:
1338:
1336:
1335:
1330:
1328:
1327:
1308:
1306:
1305:
1300:
1298:
1296:
1294:
1293:
1291:
1289:
1288:
1286:
1284:
1283:
1281:
1279:
1268:
1266:
1256:
1251:
1250:
1235:
1233:
1223:
1218:
1217:
1202:
1200:
1190:
1185:
1184:
1169:
1167:
1157:
1152:
1151:
1065:are isomorphic.
976:
974:
973:
968:
957:
956:
921:
919:
918:
913:
861:and such spaces
807:
805:
804:
799:
797:
796:
795:
794:
793:
792:
760:
758:
757:
752:
747:
746:
745:
744:
726:
725:
724:
723:
706:
705:
566:
564:
563:
558:
553:
552:
508:
506:
505:
500:
495:
494:
484:
466:
465:
443:
441:
440:
435:
430:
429:
417:
416:
395:
394:
355:
353:
352:
347:
345:
344:
305:
303:
302:
297:
292:
291:
282:
281:
266:
265:
241:
239:
238:
233:
231:
230:
209:
207:
206:
201:
199:
198:
131:σ-compact spaces
38:is any set in a
21:
4340:
4339:
4335:
4334:
4333:
4331:
4330:
4329:
4310:
4309:
4308:
4303:
4299:Spectral theory
4279:Convex analysis
4263:
4220:
4215:
4168:
4068:
4016:in distribution
3961:
3854:
3684:Logarithmically
3623:
3579:
3562:Essential range
3496:
3437:
3432:
3379:
3376:
3375:
3374:
3367:
3364:
3363:
3362:
3355:
3352:
3351:
3350:
3343:
3340:
3339:
3338:
3331:
3328:
3327:
3326:
3319:
3316:
3315:
3314:
3303:
3300:
3299:
3298:
3294:
3291:
3290:
3289:
3285:
3282:
3281:
3280:
3276:
3273:
3272:
3271:
3267:
3264:
3263:
3262:
3258:
3255:
3254:
3253:
3235:
3232:
3231:
3230:
3221:
3218:
3217:
3216:
3209:
3206:
3205:
3204:
3198:
3195:
3194:
3193:
3184:
3181:
3180:
3179:
3172:
3169:
3168:
3167:
3157:
3154:
3153:
3152:
3143:
3140:
3139:
3138:
3131:
3128:
3127:
3126:
3120:
3117:
3116:
3115:
3106:
3103:
3102:
3101:
3094:
3091:
3090:
3089:
3076:
3073:
3072:
3071:
3059:
3056:
3055:
3054:
3046:
3043:
3042:
3041:
3034:
3031:
3030:
3029:
3012:
3009:
3008:
3007:
3000:
2999:
2994:
2993:
2992:
2985:
2984:
2979:
2978:
2977:
2970:
2968:
2962:
2961:
2960:
2949:
2946:
2945:
2944:
2940:
2937:
2934:
2933:
2932:
2929:
2928:
2927:
2923:
2920:
2917:
2916:
2915:
2912:
2911:
2910:
2906:
2902:
2899:
2898:
2897:
2893:
2892:
2891:
2874:
2871:
2870:
2869:
2861:
2858:
2857:
2856:
2849:
2846:
2845:
2844:
2838:
2835:
2834:
2833:
2819:
2816:
2815:
2814:
2802:
2799:
2798:
2797:
2779:
2776:
2775:
2774:
2767:
2764:
2763:
2762:
2755:
2752:
2751:
2750:
2743:
2740:
2739:
2738:
2731:
2728:
2727:
2726:
2719:
2716:
2715:
2714:
2703:
2700:
2699:
2698:
2694:
2691:
2690:
2689:
2685:
2682:
2681:
2680:
2676:
2673:
2672:
2671:
2667:
2664:
2663:
2662:
2658:
2655:
2654:
2653:
2639:
2629:
2626:
2625:
2624:
2619:
2609:
2606:
2605:
2604:
2597:
2594:
2593:
2592:
2586:
2583:
2582:
2581:
2572:
2569:
2568:
2567:
2560:
2557:
2556:
2555:
2548:
2537:
2534:
2533:
2532:
2526:
2515:
2512:
2511:
2510:
2503:
2500:
2499:
2498:
2492:
2489:
2488:
2487:
2478:
2475:
2474:
2473:
2466:
2463:
2462:
2461:
2444:
2441:
2440:
2439:
2423:
2420:
2419:
2418:
2410:
2407:
2406:
2405:
2395:
2392:
2391:
2390:
2377:
2374:
2373:
2372:
2361:
2358:
2357:
2356:
2346:
2343:
2342:
2341:
2334:
2331:
2330:
2329:
2322:
2319:
2318:
2317:
2309:
2306:
2305:
2304:
2300:
2297:
2296:
2295:
2291:
2288:
2287:
2286:
2282:
2279:
2278:
2277:
2262:
2257:
2231:
2206:Wayback Machine
2172:
2169:
2134:Halmos, Paul R.
2117:Polish topology
2109:William Arveson
2105:
2100:
2099:
2090:
2086:
2077:
2073:
2059:
2055:
2025:
2021:
2015:
2005:Springer Verlag
1997:
1993:
1986:
1982:
1939:
1935:
1930:
1895:Borel hierarchy
1891:
1885:is Hausdorff).
1870:
1867:
1866:
1846:
1843:
1842:
1826:
1823:
1822:
1802:
1799:
1798:
1782:
1779:
1778:
1764:locally compact
1756:
1716:
1712:
1692:
1689:
1688:
1656:
1652:
1650:
1647:
1646:
1627:
1625:
1622:
1621:
1605:
1603:
1600:
1599:
1583:
1580:
1579:
1563:
1560:
1559:
1524:
1521:
1520:
1488:
1484:
1483:
1479:
1468:
1464:
1463:
1459:
1454:
1451:
1450:
1421:
1417:
1408:
1404:
1399:
1396:
1395:
1379:
1376:
1375:
1354:
1350:
1348:
1345:
1344:
1323:
1319:
1317:
1314:
1313:
1274:
1269:
1262:
1257:
1255:
1246:
1242:
1241:
1236:
1229:
1224:
1222:
1213:
1209:
1208:
1203:
1196:
1191:
1189:
1180:
1176:
1175:
1170:
1163:
1158:
1156:
1147:
1143:
1135:
1132:
1131:
1115:axiom of choice
1103:
1057:, the union of
1042:a finite space.
1010:and that makes
949:
945:
943:
940:
939:
895:
892:
891:
859:measurable sets
820:
814:
788:
784:
783:
779:
778:
774:
772:
769:
768:
740:
736:
735:
731:
719:
715:
714:
710:
701:
697:
695:
692:
691:
642:
634:Borel hierarchy
628:
620:
609:
598:
545:
541:
533:
530:
529:
519:
490:
486:
474:
461:
457:
455:
452:
451:
422:
418:
406:
402:
390:
386:
384:
381:
380:
340:
336:
334:
331:
330:
287:
283:
277:
273:
258:
254:
252:
249:
248:
226:
222:
220:
217:
216:
194:
190:
188:
185:
184:
143:
109:Borel hierarchy
84:, known as the
28:
23:
22:
15:
12:
11:
5:
4338:
4328:
4327:
4322:
4305:
4304:
4302:
4301:
4296:
4291:
4286:
4281:
4275:
4273:
4269:
4268:
4265:
4264:
4262:
4261:
4256:
4251:
4246:
4245:
4244:
4234:
4228:
4226:
4217:
4216:
4214:
4213:
4208:
4206:Sard's theorem
4203:
4198:
4193:
4192:
4191:
4189:Lifting theory
4180:
4174:
4170:
4169:
4167:
4166:
4161:
4156:
4151:
4146:
4145:
4144:
4142:Fubini–Tonelli
4134:
4129:
4124:
4123:
4122:
4117:
4112:
4104:
4103:
4102:
4097:
4092:
4084:
4078:
4076:
4070:
4069:
4067:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4030:
4025:
4024:
4023:
4021:in probability
4018:
4008:
4003:
3998:
3992:
3991:
3990:
3985:
3980:
3969:
3967:
3963:
3962:
3960:
3959:
3954:
3949:
3944:
3939:
3934:
3933:
3932:
3922:
3917:
3916:
3915:
3905:
3900:
3895:
3890:
3885:
3880:
3875:
3870:
3864:
3862:
3856:
3855:
3853:
3852:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3801:
3800:
3795:
3790:
3780:
3775:
3770:
3765:
3755:
3750:
3745:
3740:
3735:
3730:
3728:Locally finite
3725:
3715:
3710:
3705:
3700:
3695:
3690:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3634:
3632:
3625:
3624:
3622:
3621:
3616:
3611:
3606:
3601:
3600:
3599:
3589:
3584:
3576:
3571:
3570:
3569:
3559:
3554:
3553:
3552:
3542:
3537:
3532:
3531:
3530:
3520:
3515:
3510:
3504:
3502:
3498:
3497:
3495:
3494:
3485:
3484:
3483:
3473:
3468:
3460:
3455:
3445:
3443:
3442:Basic concepts
3439:
3438:
3435:Measure theory
3431:
3430:
3423:
3416:
3408:
3400:
3399:
3396:
3392:
3391:
3377:
3365:
3353:
3341:
3329:
3317:
3310:
3301:
3292:
3283:
3274:
3265:
3256:
3249:
3248:
3245:
3241:
3240:
3233:
3226:
3219:
3212:
3207:
3201:
3196:
3189:
3188:
3182:
3175:
3170:
3163:
3162:
3155:
3148:
3141:
3134:
3129:
3123:
3118:
3111:
3110:
3104:
3097:
3092:
3085:
3084:
3074:
3067:
3057:
3050:
3044:
3038:
3032:
3025:
3024:
3010:
2997:
2995:
2982:
2980:
2966:
2963:
2956:
2947:
2935:
2930:
2918:
2913:
2900:
2894:
2887:
2886:
2883:
2879:
2878:
2872:
2865:
2859:
2852:
2847:
2841:
2836:
2829:
2828:
2817:
2810:
2800:
2793:
2792:
2789:
2785:
2784:
2777:
2765:
2753:
2741:
2729:
2717:
2710:
2701:
2692:
2683:
2674:
2665:
2656:
2649:
2648:
2645:
2641:
2640:
2637:
2627:
2620:
2617:
2607:
2600:
2595:
2589:
2584:
2577:
2576:
2570:
2563:
2558:
2551:
2550:
2546:
2535:
2528:
2524:
2513:
2506:
2501:
2495:
2490:
2483:
2482:
2476:
2469:
2464:
2457:
2456:
2442:
2435:
2421:
2414:
2408:
2402:
2393:
2386:
2385:
2375:
2368:
2359:
2352:
2351:
2344:
2332:
2320:
2313:
2307:
2298:
2289:
2280:
2273:
2272:
2267:
2256:
2255:
2248:
2241:
2230:
2229:
2228:
2209:
2188:
2168:
2167:External links
2165:
2164:
2163:
2153:
2143:
2138:Measure theory
2130:
2123:Richard Dudley
2120:
2104:
2101:
2098:
2097:
2084:
2071:
2053:
2019:
2013:
1991:
1980:
1954:(3): 187–207,
1932:
1931:
1929:
1926:
1925:
1924:
1918:
1912:
1907:
1902:
1897:
1890:
1887:
1874:
1850:
1830:
1806:
1786:
1755:
1752:
1739:
1736:
1733:
1730:
1727:
1724:
1719:
1715:
1711:
1708:
1705:
1702:
1699:
1696:
1673:
1670:
1667:
1662:
1659:
1655:
1630:
1608:
1587:
1567:
1528:
1504:
1501:
1498:
1491:
1487:
1482:
1478:
1471:
1467:
1462:
1458:
1435:
1432:
1429:
1424:
1420:
1416:
1411:
1407:
1403:
1383:
1374:integers. Let
1357:
1353:
1326:
1322:
1310:
1309:
1277:
1265:
1254:
1249:
1245:
1232:
1221:
1216:
1212:
1199:
1188:
1183:
1179:
1166:
1155:
1150:
1146:
1142:
1139:
1102:
1101:Non-Borel sets
1099:
1044:
1043:
1040:
1034:
966:
963:
960:
955:
952:
948:
911:
908:
905:
902:
899:
830:associated to
813:
810:
791:
787:
782:
777:
750:
743:
739:
734:
730:
722:
718:
713:
709:
704:
700:
641:
638:
626:
618:
611:. However, as
607:
596:
578:; moreover if
556:
551:
548:
544:
540:
537:
517:
510:
509:
498:
493:
489:
483:
480:
477:
473:
469:
464:
460:
444:
433:
428:
425:
421:
415:
412:
409:
405:
401:
398:
393:
389:
361:
343:
339:
323:ordinal number
309:Now define by
307:
306:
295:
290:
286:
280:
276:
272:
269:
264:
261:
257:
246:
229:
225:
214:
197:
193:
164:of subsets of
142:
139:
101:measure theory
82:σ-algebra
26:
9:
6:
4:
3:
2:
4337:
4326:
4323:
4321:
4318:
4317:
4315:
4300:
4297:
4295:
4294:Real analysis
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4276:
4274:
4270:
4260:
4257:
4255:
4252:
4250:
4247:
4243:
4240:
4239:
4238:
4235:
4233:
4230:
4229:
4227:
4224:
4218:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4190:
4187:
4186:
4185:
4182:
4181:
4178:
4175:
4173:Other results
4171:
4165:
4162:
4160:
4159:Radon–Nikodym
4157:
4155:
4152:
4150:
4147:
4143:
4140:
4139:
4138:
4135:
4133:
4132:Fatou's lemma
4130:
4128:
4125:
4121:
4118:
4116:
4113:
4111:
4108:
4107:
4105:
4101:
4098:
4096:
4093:
4091:
4088:
4087:
4085:
4083:
4080:
4079:
4077:
4075:
4071:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4031:
4029:
4026:
4022:
4019:
4017:
4014:
4013:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:Convergence:
3993:
3989:
3986:
3984:
3981:
3979:
3976:
3975:
3974:
3971:
3970:
3968:
3964:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3931:
3928:
3927:
3926:
3923:
3921:
3918:
3914:
3911:
3910:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3865:
3863:
3861:
3857:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3799:
3798:Outer regular
3796:
3794:
3793:Inner regular
3791:
3789:
3788:Borel regular
3786:
3785:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3760:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3720:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3685:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3635:
3633:
3631:
3626:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3598:
3595:
3594:
3593:
3590:
3588:
3585:
3583:
3577:
3575:
3572:
3568:
3565:
3564:
3563:
3560:
3558:
3555:
3551:
3548:
3547:
3546:
3543:
3541:
3538:
3536:
3533:
3529:
3526:
3525:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3505:
3503:
3499:
3493:
3489:
3486:
3482:
3479:
3478:
3477:
3476:Measure space
3474:
3472:
3469:
3467:
3465:
3461:
3459:
3456:
3454:
3450:
3447:
3446:
3444:
3440:
3436:
3429:
3424:
3422:
3417:
3415:
3410:
3409:
3406:
3393:
3390:
3386:
3382:
3370:
3358:
3346:
3334:
3322:
3309:
3250:
3242:
3238:
3227:
3224:
3213:
3202:
3191:
3190:
3187:
3164:
3160:
3149:
3146:
3135:
3124:
3113:
3112:
3109:
3086:
3083:
3079:
3068:
3066:
3062:
3051:
3039:
3027:
3026:
3023:
3019:
3015:
3003:
2988:
2973:
2969:
2955:
2905:
2888:
2880:
2877:
2866:
2864:
2853:
2842:
2831:
2830:
2826:
2822:
2808:
2794:
2786:
2782:
2770:
2758:
2746:
2734:
2722:
2709:
2650:
2642:
2636:
2632:
2621:
2616:
2612:
2601:
2590:
2579:
2578:
2575:
2552:
2549:
2545:
2540:
2529:
2527:
2523:
2518:
2507:
2496:
2485:
2484:
2481:
2458:
2455:
2451:
2447:
2436:
2434:
2430:
2426:
2415:
2403:
2401:
2388:
2387:
2384:
2380:
2367:
2353:
2350:(if defined)
2349:
2337:
2325:
2274:
2271:
2265:
2260:
2254:
2249:
2247:
2242:
2240:
2235:
2234:
2224:
2223:
2218:
2215:
2210:
2207:
2203:
2200:
2196:
2192:
2189:
2185:
2181:
2180:
2175:
2171:
2170:
2161:
2157:
2154:
2151:
2150:Real Analysis
2147:
2146:Halsey Royden
2144:
2139:
2135:
2131:
2128:
2124:
2121:
2118:
2114:
2110:
2107:
2106:
2094:
2088:
2081:
2075:
2067:
2063:
2057:
2049:
2044:
2040:
2037:(in French),
2036:
2035:
2030:
2023:
2016:
2010:
2006:
2002:
1995:
1989:
1984:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1948:
1943:
1937:
1933:
1922:
1919:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1892:
1886:
1872:
1864:
1848:
1828:
1820:
1804:
1784:
1775:
1773:
1769:
1765:
1761:
1758:According to
1751:
1734:
1731:
1728:
1717:
1709:
1706:
1703:
1697:
1694:
1687:
1668:
1660:
1657:
1653:
1643:
1585:
1565:
1557:
1552:
1550:
1549:A. S. Kechris
1546:
1542:
1526:
1518:
1499:
1496:
1489:
1485:
1480:
1476:
1469:
1465:
1460:
1449:
1430:
1427:
1422:
1418:
1414:
1409:
1405:
1381:
1373:
1355:
1351:
1342:
1324:
1320:
1275:
1263:
1252:
1247:
1243:
1230:
1219:
1214:
1210:
1197:
1186:
1181:
1177:
1164:
1153:
1148:
1144:
1140:
1137:
1130:
1129:
1128:
1127:
1123:
1118:
1116:
1112:
1108:
1098:
1096:
1092:
1087:
1085:
1079:
1077:
1073:
1072:
1066:
1064:
1060:
1056:
1051:
1049:
1041:
1038:
1035:
1032:
1029:
1028:
1027:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
998:
994:
990:
986:
982:
980:
961:
953:
950:
946:
937:
933:
929:
925:
909:
903:
900:
897:
889:
885:
882:in which the
881:
876:
874:
870:
866:
865:
860:
855:
854:George Mackey
851:
849:
845:
841:
837:
834:is the pair (
833:
829:
825:
819:
809:
789:
780:
775:
766:
761:
748:
741:
732:
728:
720:
711:
707:
702:
689:
685:
680:
678:
674:
669:
667:
663:
660:defined on a
659:
655:
654:Borel measure
651:
647:
637:
635:
630:
625:
621:
614:
610:
603:
599:
592:
587:
585:
581:
577:
573:
568:
554:
549:
546:
542:
535:
527:
523:
515:
496:
491:
487:
481:
478:
475:
471:
467:
462:
458:
449:
445:
431:
426:
423:
413:
410:
407:
403:
396:
391:
387:
378:
374:
370:
369:limit ordinal
366:
362:
359:
341:
337:
328:
327:
326:
324:
320:
316:
312:
293:
288:
278:
274:
267:
262:
259:
255:
247:
245:
227:
223:
215:
213:
195:
191:
183:
182:
181:
179:
175:
171:
167:
163:
158:
156:
152:
148:
138:
136:
132:
129:
125:
121:
116:
114:
110:
106:
105:Borel measure
102:
97:
95:
91:
87:
86:Borel algebra
83:
79:
75:
70:
68:
64:
60:
56:
53:
49:
45:
41:
37:
33:
19:
18:Borel algebra
4074:Main results
3810:Set function
3738:Metric outer
3693:Decomposable
3550:Cylinder set
3522:
3463:
3384:
3372:
3360:
3348:
3336:
3324:
3312:
3228:
3214:
3177:
3150:
3136:
3099:
3069:
3052:
3021:
3017:
3005:
2990:
2975:
2958:
2867:
2854:
2812:
2772:
2760:
2748:
2736:
2724:
2712:
2708:arithmetical
2634:
2622:
2614:
2602:
2565:
2543:
2530:
2521:
2508:
2471:
2449:
2437:
2428:
2416:
2370:
2339:
2327:
2315:
2220:
2195:Mizar system
2177:
2159:
2149:
2137:
2126:
2116:
2112:
2092:
2087:
2074:
2065:
2062:Jech, Thomas
2056:
2038:
2032:
2022:
2000:
1994:
1983:
1951:
1945:
1942:Mackey, G.W.
1936:
1921:Polish space
1776:
1767:
1757:
1644:
1553:
1371:
1311:
1119:
1104:
1088:
1084:analytic set
1080:
1076:Polish space
1069:
1067:
1062:
1058:
1054:
1052:
1045:
1036:
1030:
1019:
1011:
1007:
1003:
999:
993:Polish space
988:
984:
983:
978:
935:
931:
877:
872:
868:
863:
858:
852:
847:
843:
839:
835:
831:
827:
823:
821:
762:
681:
672:
670:
650:real numbers
643:
631:
623:
616:
612:
605:
601:
594:
590:
588:
583:
579:
575:
571:
569:
525:
513:
511:
447:
376:
372:
364:
357:
318:
314:
308:
243:
211:
177:
173:
165:
161:
159:
157:as follows.
155:generatively
154:
151:metric space
146:
144:
135:pathological
124:well-behaved
120:compact sets
117:
98:
93:
89:
85:
77:
73:
71:
59:intersection
57:, countable
35:
29:
4034:compact set
4001:of measures
3937:Pushforward
3930:Projections
3920:Logarithmic
3763:Probability
3753:Pre-measure
3535:Borel space
3453:of measures
2232:This box:
2217:"Borel Set"
2174:"Borel set"
2082:, page 219)
2080:Halmos 1950
1760:Paul Halmos
1448:subsequence
1014:a complete
828:Borel space
313:a sequence
67:Émile Borel
48:closed sets
32:mathematics
4314:Categories
4006:in measure
3733:Maximising
3703:Equivalent
3597:Vitali set
3389:projective
3308:analytical
3082:coanalytic
2197:, and the
2103:References
1947:Math. Ann.
1026:to one of
1024:isomorphic
938:, the set
928:pulls back
924:measurable
816:See also:
600:such that
4120:Maharam's
4090:Dominated
3903:Intensity
3898:Hausdorff
3805:Saturated
3723:Invariant
3628:Types of
3587:σ-algebra
3557:𝜆-system
3523:Borel set
3518:Baire set
3239:= CPCPCA
2825:countable
2807:recursive
2366:recursive
2264:Lightface
2222:MathWorld
2184:EMS Press
1976:119738592
1968:0025-5831
1905:Baire set
1805:σ
1768:Borel set
1723:→
1718:ω
1698::
1658:−
1500:…
1431:…
1276:⋱
1016:separable
951:−
907:→
884:morphisms
873:arbitrary
842:), where
786:ℵ
738:ℵ
717:ℵ
708:⋅
699:ℵ
677:intervals
550:σ
547:δ
539:↦
526:generated
516:, where ω
472:⋃
427:σ
424:δ
411:−
379:− 1. Let
367:is not a
289:σ
279:δ
263:σ
260:δ
228:δ
196:σ
170:power set
128:Hausdorff
52:countable
44:open sets
36:Borel set
4320:Topology
4137:Fubini's
4127:Egorov's
4095:Monotone
4054:variable
4032:Random:
3983:Strongly
3908:Lebesgue
3893:Harmonic
3883:Gaussian
3868:Counting
3835:Spectral
3830:Singular
3820:s-finite
3815:σ-finite
3698:Discrete
3673:Complete
3630:Measures
3604:Null set
3492:function
3225:= PCPCA
3065:analytic
2270:Boldface
2202:Archived
2136:(1950).
2064:(2008).
1889:See also
1541:analytic
1372:positive
1339:is some
880:category
317:, where
137:spaces.
80:forms a
4049:process
4044:measure
4039:element
3978:Bochner
3952:Trivial
3947:Tangent
3925:Product
3783:Regular
3761:)
3748:Perfect
3721:)
3686:)
3678:Content
3668:Complex
3609:Support
3582:-system
3471:Measure
3161:= CPCA
3080:= CA =
2186:, 2001
1797:as the
1517:divisor
1341:integer
987:. Let
985:Theorem
664:, its
640:Example
520:is the
371:, then
180:), let
4115:Jordan
4100:Vitali
4059:vector
3988:Weakly
3850:Vector
3825:Signed
3778:Random
3719:Quasi-
3708:Finite
3688:Convex
3648:Banach
3638:Atomic
3466:spaces
3451:
3147:= PCA
3063:= A =
2454:closed
2383:clopen
2011:
1974:
1966:
1772:σ-ring
1684:of an
1312:where
1120:Every
1089:Every
997:metric
926:if it
684:number
321:is an
61:, and
3957:Young
3878:Euler
3873:Dirac
3845:Tight
3773:Radon
3743:Outer
3713:Inner
3663:Brown
3658:Borel
3653:Besov
3643:Baire
3342:<ω
3330:<ω
3318:<ω
3275:<ω
3266:<ω
3257:<ω
3022:Borel
2742:<ω
2730:<ω
2718:<ω
2675:<ω
2666:<ω
2657:<ω
1972:S2CID
1928:Notes
1107:Lusin
991:be a
176:) of
149:is a
55:union
4221:For
4110:Hahn
3966:Maps
3888:Haar
3759:Sub-
3513:Atom
3501:Sets
3279:= Σ
2433:open
2252:edit
2245:talk
2238:view
2009:ISBN
1964:ISSN
1370:are
886:are
869:open
822:Let
479:<
34:, a
3347:=
3297:= Δ
3288:= Π
3270:= Δ
3261:= Π
2943:= Δ
2926:= Δ
2909:= Π
2823:(α
2805:(α
2697:= Δ
2688:= Π
2679:= Σ
2670:= Δ
2661:= Π
2294:= Δ
2285:= Π
2043:doi
1956:doi
1952:166
1861:is
1050:.)
1002:on
934:in
922:is
808:).
446:If
363:If
88:or
30:In
4316::
3398:⋮
3395:⋮
3387:=
3383:=
3371:=
3359:=
3335:=
3323:=
3306:=
3247:⋮
3244:⋮
3020:=
3016:=
3004:=
2989:=
2974:=
2952:=
2885:⋮
2882:⋮
2827:)
2809:)
2791:⋮
2788:⋮
2771:=
2759:=
2747:=
2735:=
2723:=
2706:=
2647:⋮
2644:⋮
2638:σδ
2633:=
2618:δσ
2613:=
2541:=
2519:=
2452:=
2448:=
2431:=
2427:=
2398:=
2381:=
2364:=
2338:=
2326:=
2312:)
2219:.
2182:,
2176:,
2158:,
2148:,
2125:,
2111:,
2039:10
2031:,
2007:,
2003:,
1970:,
1962:,
1950:,
1117:.
1097:.
1086:.
1068:A
981:.
850:.
679:.
636:.
172:P(
115:.
69:.
3757:(
3717:(
3682:(
3580:π
3490:/
3464:L
3427:e
3420:t
3413:v
3385:P
3378:0
3373:Δ
3366:0
3361:Π
3354:0
3349:Σ
3337:Δ
3325:Π
3313:Σ
3302:0
3293:0
3284:0
3252:Σ
3234:3
3229:Π
3220:3
3215:Σ
3208:3
3203:Π
3197:3
3192:Σ
3183:3
3178:Δ
3171:3
3166:Δ
3156:2
3151:Π
3142:2
3137:Σ
3130:2
3125:Π
3119:2
3114:Σ
3105:2
3100:Δ
3093:2
3088:Δ
3075:1
3070:Π
3058:1
3053:Σ
3045:1
3040:Π
3033:1
3028:Σ
3018:B
3011:1
3006:Δ
2998:1
2996:ω
2991:Δ
2983:1
2981:ω
2976:Π
2967:1
2965:ω
2959:Σ
2948:1
2936:1
2931:ω
2919:1
2914:ω
2901:1
2896:ω
2890:Σ
2873:α
2868:Π
2860:α
2855:Σ
2848:α
2843:Π
2837:α
2832:Σ
2818:α
2813:Δ
2801:α
2796:Δ
2778:0
2773:Δ
2766:0
2761:Π
2754:0
2749:Σ
2737:Δ
2725:Π
2713:Σ
2702:0
2693:0
2684:0
2652:Σ
2635:F
2628:3
2623:Π
2615:G
2608:3
2603:Σ
2596:3
2591:Π
2585:3
2580:Σ
2571:3
2566:Δ
2559:3
2554:Δ
2547:δ
2544:G
2536:2
2531:Π
2525:σ
2522:F
2514:2
2509:Σ
2502:2
2497:Π
2491:2
2486:Σ
2477:2
2472:Δ
2465:2
2460:Δ
2450:F
2443:1
2438:Π
2429:G
2422:1
2417:Σ
2409:1
2404:Π
2394:1
2389:Σ
2376:1
2371:Δ
2360:1
2355:Δ
2345:0
2340:Δ
2333:0
2328:Π
2321:0
2316:Σ
2308:1
2299:0
2290:0
2281:0
2276:Σ
2225:.
2119:)
2078:(
2045::
1958::
1873:X
1849:X
1829:X
1785:X
1738:}
1735:1
1732:,
1729:0
1726:{
1714:}
1710:1
1707:,
1704:0
1701:{
1695:f
1672:]
1669:0
1666:[
1661:1
1654:f
1629:R
1607:R
1586:A
1566:A
1527:A
1503:)
1497:,
1490:1
1486:k
1481:a
1477:,
1470:0
1466:k
1461:a
1457:(
1434:)
1428:,
1423:1
1419:a
1415:,
1410:0
1406:a
1402:(
1382:A
1356:k
1352:a
1325:0
1321:a
1264:1
1253:+
1248:3
1244:a
1231:1
1220:+
1215:2
1211:a
1198:1
1187:+
1182:1
1178:a
1165:1
1154:+
1149:0
1145:a
1141:=
1138:x
1063:R
1059:R
1055:R
1039:,
1037:Z
1033:,
1031:R
1020:X
1012:X
1008:X
1004:X
1000:d
989:X
979:X
965:)
962:B
959:(
954:1
947:f
936:Y
932:B
910:Y
904:X
901::
898:f
848:X
844:B
840:B
838:,
836:X
832:X
824:X
790:0
781:2
776:2
749:.
742:0
733:2
729:=
721:0
712:2
703:1
673:R
627:1
624:ω
619:B
617:α
613:B
608:B
606:α
602:B
597:B
595:α
591:B
584:G
580:m
576:m
572:G
555:.
543:G
536:G
518:1
514:G
497:.
492:j
488:G
482:i
476:j
468:=
463:i
459:G
448:i
432:.
420:]
414:1
408:i
404:G
400:[
397:=
392:i
388:G
377:i
373:i
365:i
360:.
358:X
342:0
338:G
319:m
315:G
294:.
285:)
275:T
271:(
268:=
256:T
244:T
224:T
212:T
192:T
178:X
174:X
166:X
162:T
147:X
94:X
78:X
74:X
20:)
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