4169:
3697:
4722:
3856:
3222:
2958:
3475:
1640:, that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other.
3125:
4830:
4284:
in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about it from arbitrarily often repeating the experiment until the random time
4065:
3959:
3278:
2838:
4578:
2588:
4008:
3612:
460:
2797:
1638:
3900:
3529:
2727:
770:
721:
4762:
4532:
4389:
4334:
4258:
4204:
1037:
672:
632:
270:
4641:
3775:
3141:
1358:
4446:
3580:
3361:
1762:. This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups.
105:
4358:
3758:, and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.
3604:
3322:
355:
56:
3383:
892:
4478:
1555:
1418:
4632:
4605:
302:, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure
4415:
4225:
4057:
4034:
3549:
3298:
2747:
2685:
2626:
210:
2870:
1941:
1195:
1169:
4501:
4303:
4282:
2420:
1760:
1525:
1388:
1292:
1087:
947:
557:
502:
393:
327:
3396:
3756:
3728:
3495:
3003:
2982:
2858:
2504:
2480:
2460:
2440:
2393:
2369:
2349:
2329:
2301:
2279:
2257:
2237:
2217:
2193:
2173:
2153:
2130:
2108:
2087:
2067:
2047:
2025:
2002:
1981:
1961:
1911:
1891:
1871:
1851:
1827:
1807:
1783:
1733:
1709:
1686:
1666:
1595:
1575:
1498:
1478:
1458:
1438:
1312:
1258:
1238:
1215:
1139:
1111:
1060:
991:
971:
920:
577:
522:
296:
230:
178:
152:
132:
3019:
4770:
820:
772:). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the
824:
583:. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. This article does
4164:{\displaystyle {\mathcal {F}}_{\tau }:=\{A\in {\mathcal {F}}\vert \forall t\geq 0\colon A\cap \{\tau \leq t\}\in {\mathcal {F}}_{t}\}}
3730:". Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of
3912:
3231:
2802:
4537:
2525:
5028:
4879:
3964:
3692:{\displaystyle {\mathcal {F}}_{\infty }=\sigma \left(\bigcup _{t\geq 0}{\mathcal {F}}_{t}\right)\subseteq {\mathcal {F}}.}
406:
4896:
2752:
4937:
4845:
2645:
4954:
3861:
3503:
2696:
726:
677:
5057:
4727:
17:
4717:{\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}
4506:
4363:
4308:
4232:
4178:
3851:{\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}
3217:{\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}
996:
1600:
2636:(a filtration on a vector space), considering a set to be a vector space over the field with one element.
637:
597:
235:
1321:
4420:
3554:
3335:
396:
3710:
context is equivalent to events that can be discriminated, or "questions that can be answered at time
64:
5047:
4980:
Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras".
4339:
3585:
3303:
336:
37:
3767:
3366:
875:
5052:
4451:
4610:
4583:
2953:{\displaystyle t_{1}\leq t_{2}\implies {\mathcal {F}}_{t_{1}}\subseteq {\mathcal {F}}_{t_{2}}.}
2663:
2629:
773:
524:, or (in more general cases, when the notion of union does not make sense) that the canonical
4394:
4210:
4042:
4013:
3534:
3283:
2732:
2670:
2633:
2593:
2507:
189:
1916:
1530:
1393:
1174:
1144:
4850:
4486:
4288:
4267:
3735:
3470:{\displaystyle {\mathcal {F}}_{t}={\mathcal {F}}_{t+}:=\bigcap _{s>t}{\mathcal {F}}_{s}}
2398:
1738:
1689:
1503:
1366:
1270:
1065:
1039:). Note that this use of the word "filtration" corresponds to our "descending filtration".
925:
535:
480:
371:
305:
8:
812:
792:
155:
112:
3738:, where a filtration represents the information available up to and including each time
4989:
4840:
3741:
3713:
3480:
2988:
2967:
2843:
2659:
2489:
2465:
2445:
2425:
2378:
2354:
2334:
2314:
2286:
2264:
2260:
2242:
2222:
2202:
2178:
2158:
2138:
2115:
2093:
2072:
2052:
2032:
2010:
1987:
1966:
1946:
1896:
1876:
1856:
1836:
1812:
1792:
1768:
1718:
1694:
1671:
1651:
1580:
1560:
1483:
1463:
1443:
1423:
1297:
1243:
1223:
1200:
1124:
1096:
1045:
976:
956:
905:
816:
804:
562:
507:
474:
299:
281:
215:
163:
137:
117:
5024:
4933:
4875:
3225:
2655:
1261:
1090:
895:
796:
5016:
403:), but not with respect to other operations (say, multiplication) that satisfy only
4999:
3390:
3329:
2690:
2196:
1830:
856:
788:
473:
Sometimes, filtrations are supposed to satisfy the additional requirement that the
365:
3120:{\displaystyle t\in \{0,1,\dots ,N\},\mathbb {N} _{0},{\mbox{ or }}[0,+\infty ).}
2483:
1315:
950:
867:
777:
400:
330:
4825:{\displaystyle {\mathcal {F}}_{\tau _{1}}\subseteq {\mathcal {F}}_{\tau _{2}}.}
2651:
840:
800:
467:
463:
59:
5003:
5041:
4764:
4635:
4207:
3903:
2667:
1786:
808:
4903:
3761:
3006:
529:
525:
3731:
3707:
3010:
2861:
2519:
580:
28:
4336:. In particular, if the underlying probability space is finite (i.e.
781:
158:
108:
5021:
Stochastic
Differential Equations: An Introduction with Applications
3706:-algebra defines the set of events that can be measured, which in a
3500:
It is also useful (in the case of an unbounded index set) to define
3386:
4994:
4391:(with respect to set inclusion) are given by the union over all
1121:
of subgroups appearing in the filtration, that is, a subset of
1117:
to the filtration. A basis for this topology is the set of all
2518:
A maximal filtration of a set is equivalent to an ordering (a
4534:-measurable. However, simple examples show that, in general,
1118:
1643:
3954:{\displaystyle \left\{{\mathcal {F}}_{t}\right\}_{t\geq 0}}
3273:{\displaystyle \left\{{\mathcal {F}}_{t}\right\}_{t\geq 0}}
2306:
1141:
is defined to be open if it is a union of sets of the form
2833:{\displaystyle {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}
329:
gaining in complexity with time. Hence, a process that is
4573:{\displaystyle \sigma (\tau )\neq {\mathcal {F}}_{\tau }}
2583:{\displaystyle \{0\}\subseteq \{0,1\}\subseteq \{0,1,2\}}
3762:
Relation to stopping times: stopping time sigma-algebras
2049:
is then the topology associated to this filtration. If
3324:. A filtered probability space is said to satisfy the
3090:
2985:
will usually depend on context: the set of values for
4773:
4730:
4644:
4613:
4586:
4540:
4509:
4489:
4454:
4423:
4397:
4366:
4342:
4311:
4291:
4270:
4235:
4213:
4181:
4068:
4045:
4016:
4003:{\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t}}
3967:
3915:
3864:
3778:
3744:
3716:
3615:
3588:
3557:
3537:
3506:
3483:
3399:
3369:
3338:
3306:
3286:
3234:
3144:
3022:
2991:
2970:
2873:
2846:
2805:
2755:
2735:
2699:
2673:
2596:
2528:
2492:
2468:
2448:
2428:
2401:
2381:
2357:
2337:
2317:
2289:
2267:
2245:
2225:
2205:
2181:
2161:
2141:
2118:
2096:
2075:
2055:
2035:
2013:
1990:
1969:
1949:
1919:
1899:
1879:
1859:
1839:
1815:
1795:
1771:
1741:
1721:
1697:
1674:
1654:
1603:
1583:
1563:
1533:
1506:
1486:
1466:
1446:
1426:
1396:
1369:
1324:
1300:
1273:
1246:
1226:
1203:
1177:
1147:
1127:
1099:
1068:
1048:
999:
979:
959:
928:
908:
878:
729:
680:
640:
600:
565:
538:
510:
483:
409:
374:
339:
308:
284:
238:
218:
192:
166:
140:
120:
67:
40:
3858:be a filtered probability space. A random variable
1220:The topology associated to a filtration on a group
5015:
4927:
4897:"Stochastic Processes: A very simple introduction"
4824:
4756:
4716:
4626:
4599:
4572:
4526:
4495:
4472:
4440:
4409:
4383:
4352:
4328:
4297:
4276:
4252:
4219:
4198:
4163:
4051:
4028:
4002:
3953:
3894:
3850:
3750:
3722:
3691:
3598:
3574:
3543:
3523:
3489:
3469:
3377:
3355:
3316:
3292:
3272:
3216:
3119:
2997:
2976:
2952:
2852:
2832:
2791:
2741:
2721:
2679:
2620:
2582:
2498:
2474:
2454:
2434:
2414:
2387:
2363:
2343:
2323:
2295:
2273:
2251:
2231:
2211:
2187:
2167:
2147:
2124:
2102:
2081:
2061:
2041:
2019:
1996:
1975:
1955:
1935:
1905:
1885:
1865:
1845:
1821:
1801:
1777:
1754:
1727:
1703:
1680:
1660:
1632:
1589:
1569:
1549:
1519:
1492:
1472:
1452:
1432:
1412:
1382:
1352:
1306:
1286:
1252:
1232:
1209:
1189:
1163:
1133:
1105:
1081:
1054:
1031:
985:
965:
941:
914:
886:
872:In algebra, filtrations are ordinarily indexed by
764:
715:
666:
626:
571:
551:
516:
496:
454:
387:
349:
321:
290:
264:
224:
204:
172:
146:
126:
99:
50:
455:{\displaystyle S_{i}\cdot S_{j}\subseteq S_{i+j}}
5039:
4975:
4973:
4971:
3551:-algebra generated by the infinite union of the
2792:{\displaystyle \{{\mathcal {F}}_{t}\}_{t\geq 0}}
2632:, an ordering on a set corresponds to a maximal
2442:is a field, then an ascending filtration of the
1557:-topology, is continuous if and only if for any
4894:
795:(where they are related in an important way to
4952:
4968:
2510:are one important class of such filtrations.
819:, other terminology is usually used, such as
4467:
4455:
4158:
4138:
4126:
4102:
4086:
3980:
3968:
3053:
3029:
2774:
2756:
2577:
2559:
2553:
2541:
2535:
2529:
1347:
1341:
368:, there is instead the requirement that the
2522:) of the set. For instance, the filtration
361:, because it cannot "see into the future".
2901:
2897:
1765:An important special case is known as the
776:notion of cofiltrations (which consist of
4993:
4705:
3895:{\displaystyle \tau :\Omega \rightarrow }
3839:
3371:
3205:
3061:
1644:Rings and modules: descending filtrations
880:
3524:{\displaystyle {\mathcal {F}}_{\infty }}
2722:{\displaystyle (\Omega ,{\mathcal {F}})}
2395:is an increasing sequence of submodules
2307:Rings and modules: ascending filtrations
1267:The topology associated to a filtration
765:{\displaystyle \bigcup _{i\in I}S_{i}=S}
716:{\displaystyle \bigcap _{i\in I}S_{i}=0}
4979:
4757:{\displaystyle \tau _{1}\leq \tau _{2}}
1735:is a decreasing sequence of submodules
14:
5040:
4527:{\displaystyle {\mathcal {F}}_{\tau }}
4384:{\displaystyle {\mathcal {F}}_{\tau }}
4329:{\displaystyle {\mathcal {F}}_{\tau }}
4253:{\displaystyle {\mathcal {F}}_{\tau }}
4199:{\displaystyle {\mathcal {F}}_{\tau }}
1032:{\displaystyle G_{n+1}\subseteq G_{n}}
399:with respect to some operations (say,
4869:
1633:{\displaystyle G_{m}\subseteq G'_{n}}
1089:, there is a natural way to define a
4870:Björk, Thomas (2005). "Appendix B".
2693:. That is, given a measurable space
2283:, relative to the topology given on
667:{\displaystyle S_{i}\subseteq S_{j}}
627:{\displaystyle S_{i}\supseteq S_{j}}
265:{\displaystyle S_{i}\subseteq S_{j}}
4955:"Filtrations and Adapted Processes"
4953:George Lowther (November 8, 2009).
4872:Arbitrage Theory in Continuous Time
3907:
1353:{\displaystyle \bigcap G_{n}=\{1\}}
24:
4982:Statistics and Probability Letters
4801:
4777:
4674:
4658:
4650:
4559:
4513:
4441:{\displaystyle {\mathcal {F}}_{t}}
4427:
4370:
4345:
4315:
4239:
4185:
4147:
4105:
4097:
4072:
3989:
3924:
3886:
3871:
3808:
3792:
3784:
3681:
3660:
3625:
3619:
3591:
3575:{\displaystyle {\mathcal {F}}_{t}}
3561:
3516:
3510:
3456:
3420:
3403:
3356:{\displaystyle {\mathcal {F}}_{0}}
3342:
3309:
3243:
3174:
3158:
3150:
3108:
2929:
2905:
2825:
2809:
2762:
2711:
2703:
342:
43:
25:
5069:
4175:It is not difficult to show that
2639:
4895:Péter Medvegyev (January 2009).
4360:is finite), the minimal sets of
2729:, a filtration is a sequence of
2662:, a filtration is an increasing
2628:. From the point of view of the
180:, subject to the condition that
100:{\displaystyle (S_{i})_{i\in I}}
4846:Filtration (probability theory)
4417:of the sets of minimal sets of
2963:The exact range of the "times"
2646:Filtration (probability theory)
787:Filtrations are widely used in
594:, which is required to satisfy
4946:
4921:
4888:
4863:
4550:
4544:
4353:{\displaystyle {\mathcal {F}}}
4260:encodes information up to the
3889:
3877:
3874:
3599:{\displaystyle {\mathcal {F}}}
3317:{\displaystyle {\mathcal {F}}}
3111:
3096:
3086:
3074:
2898:
2716:
2700:
2615:
2597:
590:There is also the notion of a
350:{\displaystyle {\mathcal {F}}}
298:is the time parameter of some
82:
68:
51:{\displaystyle {\mathcal {F}}}
13:
1:
4856:
3228:equipped with the filtration
2482:is an increasing sequence of
2259:-adic topology, it becomes a
1527:-topology and the second the
1440:, then the identity map from
462:, where the index set is the
3378:{\displaystyle \mathbb {P} }
2590:corresponds to the ordering
2089:itself, we have defined the
922:, is then a nested sequence
887:{\displaystyle \mathbb {N} }
466:; this is by analogy with a
7:
4930:Probabilities and Potential
4928:Claude Dellacherie (1979).
4834:
4473:{\displaystyle \{\tau =t\}}
4059:-algebra is now defined as
3013:or unbounded. For example,
850:
830:
10:
5074:
3765:
3734:. A typical example is in
3582:'s, which is contained in
3132:filtered probability space
2643:
1480:, where the first copy of
865:
845:
5004:10.1016/j.spl.2012.09.024
4627:{\displaystyle \tau _{2}}
4600:{\displaystyle \tau _{1}}
2513:
861:
587:impose this requirement.
807:for nested sequences of
4410:{\displaystyle t\geq 0}
4220:{\displaystyle \sigma }
4052:{\displaystyle \sigma }
4029:{\displaystyle t\geq 0}
3544:{\displaystyle \sigma }
3293:{\displaystyle \sigma }
2742:{\displaystyle \sigma }
2680:{\displaystyle \sigma }
2621:{\displaystyle (0,1,2)}
1420:are defined on a group
835:
205:{\displaystyle i\leq j}
4826:
4758:
4718:
4628:
4601:
4574:
4528:
4497:
4474:
4442:
4411:
4385:
4354:
4330:
4299:
4278:
4254:
4221:
4200:
4165:
4053:
4030:
4004:
3955:
3896:
3852:
3752:
3724:
3693:
3600:
3576:
3545:
3525:
3491:
3471:
3379:
3357:
3318:
3294:
3274:
3218:
3121:
2999:
2978:
2954:
2854:
2834:
2793:
2743:
2723:
2681:
2630:field with one element
2622:
2584:
2500:
2476:
2456:
2436:
2416:
2389:
2365:
2345:
2325:
2297:
2275:
2253:
2233:
2213:
2189:
2169:
2149:
2126:
2104:
2083:
2063:
2043:
2021:
1998:
1977:
1963:forms a filtration of
1957:
1937:
1936:{\displaystyle I^{n}M}
1907:
1887:
1867:
1847:
1823:
1803:
1779:
1756:
1729:
1705:
1682:
1662:
1634:
1591:
1571:
1551:
1550:{\displaystyle G'_{n}}
1521:
1494:
1474:
1454:
1434:
1414:
1413:{\displaystyle G'_{n}}
1384:
1354:
1308:
1288:
1254:
1234:
1211:
1191:
1190:{\displaystyle a\in G}
1165:
1164:{\displaystyle aG_{n}}
1135:
1107:
1083:
1056:
1033:
987:
967:
943:
916:
898:of natural numbers. A
888:
766:
717:
668:
628:
573:
553:
518:
498:
456:
389:
351:
323:
292:
266:
226:
206:
174:
148:
128:
101:
52:
4827:
4759:
4719:
4629:
4602:
4575:
4529:
4498:
4496:{\displaystyle \tau }
4483:It can be shown that
4475:
4443:
4412:
4386:
4355:
4331:
4300:
4298:{\displaystyle \tau }
4279:
4277:{\displaystyle \tau }
4255:
4222:
4201:
4166:
4054:
4031:
4005:
3956:
3897:
3853:
3753:
3725:
3694:
3601:
3577:
3546:
3526:
3492:
3472:
3380:
3358:
3319:
3295:
3275:
3219:
3122:
3000:
2979:
2955:
2855:
2835:
2794:
2744:
2724:
2682:
2623:
2585:
2501:
2477:
2457:
2437:
2417:
2415:{\displaystyle M_{n}}
2390:
2366:
2346:
2326:
2298:
2276:
2254:
2234:
2214:
2190:
2170:
2150:
2127:
2105:
2084:
2064:
2044:
2022:
1999:
1978:
1958:
1938:
1908:
1888:
1868:
1848:
1824:
1804:
1780:
1757:
1755:{\displaystyle M_{n}}
1730:
1713:descending filtration
1706:
1683:
1663:
1635:
1592:
1572:
1552:
1522:
1520:{\displaystyle G_{n}}
1495:
1475:
1455:
1435:
1415:
1385:
1383:{\displaystyle G_{n}}
1355:
1309:
1289:
1287:{\displaystyle G_{n}}
1255:
1235:
1217:is a natural number.
1212:
1192:
1166:
1136:
1108:
1084:
1082:{\displaystyle G_{n}}
1057:
1034:
988:
968:
944:
942:{\displaystyle G_{n}}
917:
889:
767:
718:
669:
629:
592:descending filtration
574:
554:
552:{\displaystyle S_{i}}
519:
499:
497:{\displaystyle S_{i}}
457:
390:
388:{\displaystyle S_{i}}
352:
324:
322:{\displaystyle S_{i}}
293:
267:
227:
207:
175:
149:
129:
102:
53:
5058:Stochastic processes
5023:. Berlin: Springer.
4851:Filter (mathematics)
4771:
4728:
4642:
4611:
4584:
4538:
4507:
4487:
4452:
4421:
4395:
4364:
4340:
4309:
4289:
4268:
4233:
4211:
4179:
4066:
4043:
4014:
3965:
3913:
3906:with respect to the
3862:
3776:
3742:
3736:mathematical finance
3714:
3613:
3586:
3555:
3535:
3504:
3481:
3397:
3367:
3336:
3304:
3284:
3232:
3142:
3020:
2989:
2968:
2871:
2844:
2803:
2753:
2733:
2697:
2671:
2660:stochastic processes
2594:
2526:
2490:
2466:
2446:
2426:
2422:. In particular, if
2399:
2379:
2373:ascending filtration
2355:
2335:
2315:
2287:
2265:
2243:
2223:
2203:
2179:
2159:
2139:
2116:
2094:
2073:
2053:
2033:
2011:
1988:
1967:
1947:
1917:
1897:
1877:
1857:
1837:
1813:
1793:
1769:
1739:
1719:
1695:
1672:
1652:
1601:
1581:
1561:
1531:
1504:
1484:
1464:
1444:
1424:
1394:
1367:
1322:
1298:
1271:
1244:
1224:
1201:
1175:
1145:
1125:
1097:
1066:
1046:
997:
977:
957:
926:
906:
876:
727:
678:
674:(and, occasionally,
638:
598:
563:
536:
508:
481:
407:
372:
337:
306:
282:
236:
216:
190:
164:
138:
118:
65:
38:
3768:σ-Algebra of τ-past
2654:, in particular in
1629:
1546:
1409:
1363:If two filtrations
813:functional analysis
793:homological algebra
364:Sometimes, as in a
113:algebraic structure
5017:Øksendal, Bernt K.
4841:Natural filtration
4822:
4754:
4714:
4624:
4597:
4570:
4524:
4493:
4470:
4438:
4407:
4381:
4350:
4326:
4295:
4274:
4250:
4217:
4196:
4161:
4049:
4026:
4000:
3951:
3892:
3848:
3748:
3720:
3689:
3656:
3596:
3572:
3541:
3521:
3487:
3467:
3452:
3375:
3353:
3314:
3290:
3270:
3214:
3117:
3094:
2995:
2974:
2950:
2860:is a non-negative
2850:
2830:
2789:
2739:
2719:
2677:
2658:and the theory of
2618:
2580:
2496:
2472:
2452:
2432:
2412:
2385:
2361:
2341:
2321:
2293:
2271:
2249:
2239:is then given the
2229:
2209:
2185:
2165:
2145:
2122:
2100:
2079:
2059:
2039:
2017:
1994:
1973:
1953:
1933:
1903:
1883:
1863:
1843:
1819:
1809:-adic, etc.): Let
1799:
1775:
1752:
1725:
1701:
1678:
1658:
1630:
1617:
1587:
1567:
1547:
1534:
1517:
1490:
1470:
1450:
1430:
1410:
1397:
1380:
1350:
1304:
1284:
1250:
1230:
1207:
1187:
1161:
1131:
1103:
1079:
1052:
1029:
983:
973:(that is, for any
963:
939:
912:
884:
817:numerical analysis
805:probability theory
797:spectral sequences
762:
745:
713:
696:
664:
624:
569:
549:
514:
494:
452:
385:
347:
319:
300:stochastic process
288:
262:
222:
202:
170:
154:running over some
144:
124:
97:
48:
5030:978-3-540-04758-2
4881:978-0-19-927126-9
3751:{\displaystyle t}
3723:{\displaystyle t}
3641:
3490:{\displaystyle t}
3437:
3226:probability space
3134:(also known as a
3093:
2998:{\displaystyle t}
2977:{\displaystyle t}
2853:{\displaystyle t}
2656:martingale theory
2499:{\displaystyle M}
2475:{\displaystyle M}
2455:{\displaystyle R}
2435:{\displaystyle R}
2388:{\displaystyle M}
2364:{\displaystyle M}
2344:{\displaystyle R}
2324:{\displaystyle R}
2296:{\displaystyle R}
2274:{\displaystyle R}
2252:{\displaystyle I}
2232:{\displaystyle M}
2212:{\displaystyle R}
2188:{\displaystyle R}
2168:{\displaystyle I}
2148:{\displaystyle R}
2125:{\displaystyle R}
2103:{\displaystyle I}
2082:{\displaystyle R}
2069:is just the ring
2062:{\displaystyle M}
2042:{\displaystyle M}
2020:{\displaystyle I}
1997:{\displaystyle I}
1976:{\displaystyle M}
1956:{\displaystyle M}
1943:of submodules of
1906:{\displaystyle M}
1886:{\displaystyle R}
1866:{\displaystyle R}
1846:{\displaystyle I}
1822:{\displaystyle R}
1802:{\displaystyle J}
1778:{\displaystyle I}
1728:{\displaystyle M}
1704:{\displaystyle M}
1681:{\displaystyle R}
1661:{\displaystyle R}
1590:{\displaystyle m}
1570:{\displaystyle n}
1493:{\displaystyle G}
1473:{\displaystyle G}
1453:{\displaystyle G}
1433:{\displaystyle G}
1307:{\displaystyle G}
1262:topological group
1253:{\displaystyle G}
1233:{\displaystyle G}
1210:{\displaystyle n}
1134:{\displaystyle G}
1106:{\displaystyle G}
1062:and a filtration
1055:{\displaystyle G}
986:{\displaystyle n}
966:{\displaystyle G}
915:{\displaystyle G}
730:
681:
572:{\displaystyle S}
517:{\displaystyle S}
291:{\displaystyle i}
225:{\displaystyle I}
173:{\displaystyle I}
147:{\displaystyle i}
134:, with the index
127:{\displaystyle S}
16:(Redirected from
5065:
5048:Abstract algebra
5034:
5008:
5007:
4997:
4977:
4966:
4965:
4963:
4961:
4950:
4944:
4943:
4925:
4919:
4918:
4916:
4914:
4909:on April 3, 2015
4908:
4902:. Archived from
4901:
4892:
4886:
4885:
4867:
4831:
4829:
4828:
4823:
4818:
4817:
4816:
4815:
4805:
4804:
4794:
4793:
4792:
4791:
4781:
4780:
4763:
4761:
4760:
4755:
4753:
4752:
4740:
4739:
4723:
4721:
4720:
4715:
4713:
4709:
4708:
4700:
4699:
4688:
4684:
4683:
4678:
4677:
4662:
4661:
4633:
4631:
4630:
4625:
4623:
4622:
4606:
4604:
4603:
4598:
4596:
4595:
4579:
4577:
4576:
4571:
4569:
4568:
4563:
4562:
4533:
4531:
4530:
4525:
4523:
4522:
4517:
4516:
4502:
4500:
4499:
4494:
4479:
4477:
4476:
4471:
4447:
4445:
4444:
4439:
4437:
4436:
4431:
4430:
4416:
4414:
4413:
4408:
4390:
4388:
4387:
4382:
4380:
4379:
4374:
4373:
4359:
4357:
4356:
4351:
4349:
4348:
4335:
4333:
4332:
4327:
4325:
4324:
4319:
4318:
4304:
4302:
4301:
4296:
4283:
4281:
4280:
4275:
4259:
4257:
4256:
4251:
4249:
4248:
4243:
4242:
4226:
4224:
4223:
4218:
4205:
4203:
4202:
4197:
4195:
4194:
4189:
4188:
4170:
4168:
4167:
4162:
4157:
4156:
4151:
4150:
4101:
4100:
4082:
4081:
4076:
4075:
4058:
4056:
4055:
4050:
4035:
4033:
4032:
4027:
4009:
4007:
4006:
4001:
3999:
3998:
3993:
3992:
3960:
3958:
3957:
3952:
3950:
3949:
3938:
3934:
3933:
3928:
3927:
3901:
3899:
3898:
3893:
3857:
3855:
3854:
3849:
3847:
3843:
3842:
3834:
3833:
3822:
3818:
3817:
3812:
3811:
3796:
3795:
3757:
3755:
3754:
3749:
3729:
3727:
3726:
3721:
3698:
3696:
3695:
3690:
3685:
3684:
3675:
3671:
3670:
3669:
3664:
3663:
3655:
3629:
3628:
3623:
3622:
3605:
3603:
3602:
3597:
3595:
3594:
3581:
3579:
3578:
3573:
3571:
3570:
3565:
3564:
3550:
3548:
3547:
3542:
3530:
3528:
3527:
3522:
3520:
3519:
3514:
3513:
3496:
3494:
3493:
3488:
3476:
3474:
3473:
3468:
3466:
3465:
3460:
3459:
3451:
3433:
3432:
3424:
3423:
3413:
3412:
3407:
3406:
3391:right-continuous
3384:
3382:
3381:
3376:
3374:
3362:
3360:
3359:
3354:
3352:
3351:
3346:
3345:
3326:usual conditions
3323:
3321:
3320:
3315:
3313:
3312:
3299:
3297:
3296:
3291:
3279:
3277:
3276:
3271:
3269:
3268:
3257:
3253:
3252:
3247:
3246:
3223:
3221:
3220:
3215:
3213:
3209:
3208:
3200:
3199:
3188:
3184:
3183:
3178:
3177:
3162:
3161:
3136:stochastic basis
3126:
3124:
3123:
3118:
3095:
3091:
3070:
3069:
3064:
3004:
3002:
3001:
2996:
2983:
2981:
2980:
2975:
2959:
2957:
2956:
2951:
2946:
2945:
2944:
2943:
2933:
2932:
2922:
2921:
2920:
2919:
2909:
2908:
2896:
2895:
2883:
2882:
2859:
2857:
2856:
2851:
2839:
2837:
2836:
2831:
2829:
2828:
2819:
2818:
2813:
2812:
2798:
2796:
2795:
2790:
2788:
2787:
2772:
2771:
2766:
2765:
2748:
2746:
2745:
2740:
2728:
2726:
2725:
2720:
2715:
2714:
2691:measurable space
2686:
2684:
2683:
2678:
2627:
2625:
2624:
2619:
2589:
2587:
2586:
2581:
2505:
2503:
2502:
2497:
2484:vector subspaces
2481:
2479:
2478:
2473:
2461:
2459:
2458:
2453:
2441:
2439:
2438:
2433:
2421:
2419:
2418:
2413:
2411:
2410:
2394:
2392:
2391:
2386:
2370:
2368:
2367:
2362:
2350:
2348:
2347:
2342:
2330:
2328:
2327:
2322:
2302:
2300:
2299:
2294:
2280:
2278:
2277:
2272:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2218:
2216:
2215:
2210:
2197:topological ring
2194:
2192:
2191:
2186:
2175:-adic topology,
2174:
2172:
2171:
2166:
2154:
2152:
2151:
2146:
2131:
2129:
2128:
2123:
2109:
2107:
2106:
2101:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2026:
2024:
2023:
2018:
2004:-adic filtration
2003:
2001:
2000:
1995:
1982:
1980:
1979:
1974:
1962:
1960:
1959:
1954:
1942:
1940:
1939:
1934:
1929:
1928:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1831:commutative ring
1828:
1826:
1825:
1820:
1808:
1806:
1805:
1800:
1784:
1782:
1781:
1776:
1761:
1759:
1758:
1753:
1751:
1750:
1734:
1732:
1731:
1726:
1710:
1708:
1707:
1702:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1639:
1637:
1636:
1631:
1625:
1613:
1612:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1556:
1554:
1553:
1548:
1542:
1526:
1524:
1523:
1518:
1516:
1515:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1439:
1437:
1436:
1431:
1419:
1417:
1416:
1411:
1405:
1389:
1387:
1386:
1381:
1379:
1378:
1359:
1357:
1356:
1351:
1337:
1336:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1283:
1282:
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1170:
1168:
1167:
1162:
1160:
1159:
1140:
1138:
1137:
1132:
1112:
1110:
1109:
1104:
1088:
1086:
1085:
1080:
1078:
1077:
1061:
1059:
1058:
1053:
1038:
1036:
1035:
1030:
1028:
1027:
1015:
1014:
992:
990:
989:
984:
972:
970:
969:
964:
951:normal subgroups
948:
946:
945:
940:
938:
937:
921:
919:
918:
913:
893:
891:
890:
885:
883:
857:Filtered algebra
789:abstract algebra
778:quotient objects
771:
769:
768:
763:
755:
754:
744:
722:
720:
719:
714:
706:
705:
695:
673:
671:
670:
665:
663:
662:
650:
649:
633:
631:
630:
625:
623:
622:
610:
609:
578:
576:
575:
570:
558:
556:
555:
550:
548:
547:
523:
521:
520:
515:
503:
501:
500:
495:
493:
492:
461:
459:
458:
453:
451:
450:
432:
431:
419:
418:
394:
392:
391:
386:
384:
383:
366:filtered algebra
359:non-anticipating
356:
354:
353:
348:
346:
345:
333:to a filtration
328:
326:
325:
320:
318:
317:
297:
295:
294:
289:
271:
269:
268:
263:
261:
260:
248:
247:
231:
229:
228:
223:
211:
209:
208:
203:
179:
177:
176:
171:
153:
151:
150:
145:
133:
131:
130:
125:
106:
104:
103:
98:
96:
95:
80:
79:
57:
55:
54:
49:
47:
46:
21:
5073:
5072:
5068:
5067:
5066:
5064:
5063:
5062:
5038:
5037:
5031:
5012:
5011:
4978:
4969:
4959:
4957:
4951:
4947:
4940:
4926:
4922:
4912:
4910:
4906:
4899:
4893:
4889:
4882:
4868:
4864:
4859:
4837:
4811:
4807:
4806:
4800:
4799:
4798:
4787:
4783:
4782:
4776:
4775:
4774:
4772:
4769:
4768:
4748:
4744:
4735:
4731:
4729:
4726:
4725:
4704:
4689:
4679:
4673:
4672:
4671:
4667:
4666:
4657:
4656:
4649:
4645:
4643:
4640:
4639:
4618:
4614:
4612:
4609:
4608:
4591:
4587:
4585:
4582:
4581:
4564:
4558:
4557:
4556:
4539:
4536:
4535:
4518:
4512:
4511:
4510:
4508:
4505:
4504:
4488:
4485:
4484:
4453:
4450:
4449:
4432:
4426:
4425:
4424:
4422:
4419:
4418:
4396:
4393:
4392:
4375:
4369:
4368:
4367:
4365:
4362:
4361:
4344:
4343:
4341:
4338:
4337:
4320:
4314:
4313:
4312:
4310:
4307:
4306:
4290:
4287:
4286:
4269:
4266:
4265:
4244:
4238:
4237:
4236:
4234:
4231:
4230:
4212:
4209:
4208:
4190:
4184:
4183:
4182:
4180:
4177:
4176:
4152:
4146:
4145:
4144:
4096:
4095:
4077:
4071:
4070:
4069:
4067:
4064:
4063:
4044:
4041:
4040:
4015:
4012:
4011:
3994:
3988:
3987:
3986:
3966:
3963:
3962:
3939:
3929:
3923:
3922:
3921:
3917:
3916:
3914:
3911:
3910:
3863:
3860:
3859:
3838:
3823:
3813:
3807:
3806:
3805:
3801:
3800:
3791:
3790:
3783:
3779:
3777:
3774:
3773:
3770:
3764:
3743:
3740:
3739:
3715:
3712:
3711:
3680:
3679:
3665:
3659:
3658:
3657:
3645:
3640:
3636:
3624:
3618:
3617:
3616:
3614:
3611:
3610:
3590:
3589:
3587:
3584:
3583:
3566:
3560:
3559:
3558:
3556:
3553:
3552:
3536:
3533:
3532:
3515:
3509:
3508:
3507:
3505:
3502:
3501:
3482:
3479:
3478:
3461:
3455:
3454:
3453:
3441:
3425:
3419:
3418:
3417:
3408:
3402:
3401:
3400:
3398:
3395:
3394:
3370:
3368:
3365:
3364:
3347:
3341:
3340:
3339:
3337:
3334:
3333:
3308:
3307:
3305:
3302:
3301:
3285:
3282:
3281:
3258:
3248:
3242:
3241:
3240:
3236:
3235:
3233:
3230:
3229:
3204:
3189:
3179:
3173:
3172:
3171:
3167:
3166:
3157:
3156:
3149:
3145:
3143:
3140:
3139:
3089:
3065:
3060:
3059:
3021:
3018:
3017:
3009:or continuous,
2990:
2987:
2986:
2969:
2966:
2965:
2939:
2935:
2934:
2928:
2927:
2926:
2915:
2911:
2910:
2904:
2903:
2902:
2891:
2887:
2878:
2874:
2872:
2869:
2868:
2845:
2842:
2841:
2824:
2823:
2814:
2808:
2807:
2806:
2804:
2801:
2800:
2777:
2773:
2767:
2761:
2760:
2759:
2754:
2751:
2750:
2734:
2731:
2730:
2710:
2709:
2698:
2695:
2694:
2672:
2669:
2668:
2648:
2642:
2595:
2592:
2591:
2527:
2524:
2523:
2516:
2491:
2488:
2487:
2467:
2464:
2463:
2447:
2444:
2443:
2427:
2424:
2423:
2406:
2402:
2400:
2397:
2396:
2380:
2377:
2376:
2356:
2353:
2352:
2336:
2333:
2332:
2316:
2313:
2312:
2309:
2288:
2285:
2284:
2266:
2263:
2262:
2244:
2241:
2240:
2224:
2221:
2220:
2204:
2201:
2200:
2180:
2177:
2176:
2160:
2157:
2156:
2140:
2137:
2136:
2117:
2114:
2113:
2095:
2092:
2091:
2074:
2071:
2070:
2054:
2051:
2050:
2034:
2031:
2030:
2012:
2009:
2008:
1989:
1986:
1985:
1968:
1965:
1964:
1948:
1945:
1944:
1924:
1920:
1918:
1915:
1914:
1913:, the sequence
1898:
1895:
1894:
1878:
1875:
1874:
1858:
1855:
1854:
1838:
1835:
1834:
1814:
1811:
1810:
1794:
1791:
1790:
1770:
1767:
1766:
1746:
1742:
1740:
1737:
1736:
1720:
1717:
1716:
1696:
1693:
1692:
1673:
1670:
1669:
1653:
1650:
1649:
1646:
1621:
1608:
1604:
1602:
1599:
1598:
1582:
1579:
1578:
1562:
1559:
1558:
1538:
1532:
1529:
1528:
1511:
1507:
1505:
1502:
1501:
1485:
1482:
1481:
1465:
1462:
1461:
1445:
1442:
1441:
1425:
1422:
1421:
1401:
1395:
1392:
1391:
1374:
1370:
1368:
1365:
1364:
1332:
1328:
1323:
1320:
1319:
1318:if and only if
1299:
1296:
1295:
1278:
1274:
1272:
1269:
1268:
1245:
1242:
1241:
1225:
1222:
1221:
1202:
1199:
1198:
1176:
1173:
1172:
1155:
1151:
1146:
1143:
1142:
1126:
1123:
1122:
1098:
1095:
1094:
1073:
1069:
1067:
1064:
1063:
1047:
1044:
1043:
1023:
1019:
1004:
1000:
998:
995:
994:
978:
975:
974:
958:
955:
954:
933:
929:
927:
924:
923:
907:
904:
903:
879:
877:
874:
873:
870:
868:Length function
864:
853:
848:
838:
833:
821:scale of spaces
750:
746:
734:
728:
725:
724:
701:
697:
685:
679:
676:
675:
658:
654:
645:
641:
639:
636:
635:
618:
614:
605:
601:
599:
596:
595:
564:
561:
560:
543:
539:
537:
534:
533:
509:
506:
505:
488:
484:
482:
479:
478:
464:natural numbers
440:
436:
427:
423:
414:
410:
408:
405:
404:
401:vector addition
379:
375:
373:
370:
369:
357:is also called
341:
340:
338:
335:
334:
313:
309:
307:
304:
303:
283:
280:
279:
256:
252:
243:
239:
237:
234:
233:
217:
214:
213:
191:
188:
187:
165:
162:
161:
156:totally ordered
139:
136:
135:
119:
116:
115:
85:
81:
75:
71:
66:
63:
62:
42:
41:
39:
36:
35:
23:
22:
15:
12:
11:
5:
5071:
5061:
5060:
5055:
5053:Measure theory
5050:
5036:
5035:
5029:
5010:
5009:
4988:(1): 345–349.
4967:
4945:
4938:
4920:
4887:
4880:
4861:
4860:
4858:
4855:
4854:
4853:
4848:
4843:
4836:
4833:
4821:
4814:
4810:
4803:
4797:
4790:
4786:
4779:
4751:
4747:
4743:
4738:
4734:
4712:
4707:
4703:
4698:
4695:
4692:
4687:
4682:
4676:
4670:
4665:
4660:
4655:
4652:
4648:
4636:stopping times
4621:
4617:
4594:
4590:
4567:
4561:
4555:
4552:
4549:
4546:
4543:
4521:
4515:
4492:
4469:
4466:
4463:
4460:
4457:
4435:
4429:
4406:
4403:
4400:
4378:
4372:
4347:
4323:
4317:
4294:
4273:
4247:
4241:
4216:
4193:
4187:
4173:
4172:
4160:
4155:
4149:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4099:
4094:
4091:
4088:
4085:
4080:
4074:
4048:
4025:
4022:
4019:
3997:
3991:
3985:
3982:
3979:
3976:
3973:
3970:
3948:
3945:
3942:
3937:
3932:
3926:
3920:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3846:
3841:
3837:
3832:
3829:
3826:
3821:
3816:
3810:
3804:
3799:
3794:
3789:
3786:
3782:
3766:Main article:
3763:
3760:
3747:
3719:
3700:
3699:
3688:
3683:
3678:
3674:
3668:
3662:
3654:
3651:
3648:
3644:
3639:
3635:
3632:
3627:
3621:
3593:
3569:
3563:
3540:
3518:
3512:
3486:
3477:for all times
3464:
3458:
3450:
3447:
3444:
3440:
3436:
3431:
3428:
3422:
3416:
3411:
3405:
3373:
3350:
3344:
3311:
3289:
3267:
3264:
3261:
3256:
3251:
3245:
3239:
3212:
3207:
3203:
3198:
3195:
3192:
3187:
3182:
3176:
3170:
3165:
3160:
3155:
3152:
3148:
3128:
3127:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3092: or
3088:
3085:
3082:
3079:
3076:
3073:
3068:
3063:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
2994:
2973:
2961:
2960:
2949:
2942:
2938:
2931:
2925:
2918:
2914:
2907:
2900:
2894:
2890:
2886:
2881:
2877:
2849:
2827:
2822:
2817:
2811:
2786:
2783:
2780:
2776:
2770:
2764:
2758:
2738:
2718:
2713:
2708:
2705:
2702:
2676:
2652:measure theory
2644:Main article:
2641:
2640:Measure theory
2638:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2515:
2512:
2495:
2471:
2462:-vector space
2451:
2431:
2409:
2405:
2384:
2360:
2340:
2320:
2308:
2305:
2292:
2270:
2248:
2228:
2208:
2184:
2164:
2144:
2121:
2110:-adic topology
2099:
2078:
2058:
2038:
2027:-adic topology
2016:
1993:
1972:
1952:
1932:
1927:
1923:
1902:
1882:
1862:
1842:
1818:
1798:
1774:
1749:
1745:
1724:
1700:
1677:
1657:
1645:
1642:
1628:
1624:
1620:
1616:
1611:
1607:
1586:
1566:
1545:
1541:
1537:
1514:
1510:
1489:
1469:
1449:
1429:
1408:
1404:
1400:
1377:
1373:
1349:
1346:
1343:
1340:
1335:
1331:
1327:
1303:
1281:
1277:
1249:
1229:
1206:
1186:
1183:
1180:
1158:
1154:
1150:
1130:
1102:
1076:
1072:
1051:
1042:Given a group
1026:
1022:
1018:
1013:
1010:
1007:
1003:
982:
962:
936:
932:
911:
882:
863:
860:
852:
849:
847:
844:
841:Farey Sequence
837:
834:
832:
829:
801:measure theory
761:
758:
753:
749:
743:
740:
737:
733:
712:
709:
704:
700:
694:
691:
688:
684:
661:
657:
653:
648:
644:
621:
617:
613:
608:
604:
568:
546:
542:
513:
491:
487:
468:graded algebra
449:
446:
443:
439:
435:
430:
426:
422:
417:
413:
382:
378:
344:
316:
312:
287:
276:
275:
274:
273:
259:
255:
251:
246:
242:
221:
201:
198:
195:
169:
143:
123:
94:
91:
88:
84:
78:
74:
70:
60:indexed family
45:
9:
6:
4:
3:
2:
5070:
5059:
5056:
5054:
5051:
5049:
5046:
5045:
5043:
5032:
5026:
5022:
5018:
5014:
5013:
5005:
5001:
4996:
4991:
4987:
4983:
4976:
4974:
4972:
4956:
4949:
4941:
4939:9780720407013
4935:
4931:
4924:
4905:
4898:
4891:
4883:
4877:
4873:
4866:
4862:
4852:
4849:
4847:
4844:
4842:
4839:
4838:
4832:
4819:
4812:
4808:
4795:
4788:
4784:
4766:
4765:almost surely
4749:
4745:
4741:
4736:
4732:
4710:
4701:
4696:
4693:
4690:
4685:
4680:
4668:
4663:
4653:
4646:
4637:
4619:
4615:
4592:
4588:
4565:
4553:
4547:
4541:
4519:
4490:
4481:
4464:
4461:
4458:
4433:
4404:
4401:
4398:
4376:
4321:
4292:
4271:
4263:
4245:
4228:
4214:
4191:
4153:
4141:
4135:
4132:
4129:
4123:
4120:
4117:
4114:
4111:
4108:
4092:
4089:
4083:
4078:
4062:
4061:
4060:
4046:
4039:
4038:stopping time
4023:
4020:
4017:
3995:
3983:
3977:
3974:
3971:
3946:
3943:
3940:
3935:
3930:
3918:
3909:
3905:
3904:stopping time
3883:
3880:
3868:
3865:
3844:
3835:
3830:
3827:
3824:
3819:
3814:
3802:
3797:
3787:
3780:
3769:
3759:
3745:
3737:
3733:
3717:
3709:
3705:
3686:
3676:
3672:
3666:
3652:
3649:
3646:
3642:
3637:
3633:
3630:
3609:
3608:
3607:
3567:
3538:
3498:
3484:
3462:
3448:
3445:
3442:
3438:
3434:
3429:
3426:
3414:
3409:
3392:
3388:
3363:contains all
3348:
3331:
3327:
3287:
3265:
3262:
3259:
3254:
3249:
3237:
3227:
3210:
3201:
3196:
3193:
3190:
3185:
3180:
3168:
3163:
3153:
3146:
3137:
3133:
3130:Similarly, a
3114:
3105:
3102:
3099:
3083:
3080:
3077:
3071:
3066:
3056:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3026:
3023:
3016:
3015:
3014:
3012:
3008:
2992:
2984:
2971:
2947:
2940:
2936:
2923:
2916:
2912:
2892:
2888:
2884:
2879:
2875:
2867:
2866:
2865:
2863:
2847:
2820:
2815:
2784:
2781:
2778:
2768:
2736:
2706:
2692:
2688:
2674:
2665:
2661:
2657:
2653:
2647:
2637:
2635:
2631:
2612:
2609:
2606:
2603:
2600:
2574:
2571:
2568:
2565:
2562:
2556:
2550:
2547:
2544:
2538:
2532:
2521:
2511:
2509:
2493:
2485:
2469:
2449:
2429:
2407:
2403:
2382:
2374:
2358:
2338:
2318:
2311:Given a ring
2304:
2290:
2282:
2268:
2246:
2226:
2206:
2198:
2182:
2162:
2155:is given the
2142:
2133:
2119:
2111:
2097:
2076:
2056:
2036:
2028:
2014:
2005:
1991:
1970:
1950:
1930:
1925:
1921:
1900:
1880:
1860:
1840:
1832:
1816:
1796:
1788:
1787:adic topology
1772:
1763:
1747:
1743:
1722:
1714:
1698:
1691:
1675:
1655:
1648:Given a ring
1641:
1626:
1622:
1618:
1614:
1609:
1605:
1584:
1564:
1543:
1539:
1535:
1512:
1508:
1500:is given the
1487:
1467:
1447:
1427:
1406:
1402:
1398:
1375:
1371:
1361:
1344:
1338:
1333:
1329:
1325:
1317:
1301:
1279:
1275:
1265:
1263:
1247:
1227:
1218:
1204:
1184:
1181:
1178:
1156:
1152:
1148:
1128:
1120:
1116:
1113:, said to be
1100:
1092:
1074:
1070:
1049:
1040:
1024:
1020:
1016:
1011:
1008:
1005:
1001:
980:
960:
952:
934:
930:
909:
901:
897:
869:
859:
858:
843:
842:
828:
826:
825:nested spaces
822:
818:
814:
810:
806:
802:
798:
794:
790:
785:
783:
779:
775:
759:
756:
751:
747:
741:
738:
735:
731:
710:
707:
702:
698:
692:
689:
686:
682:
659:
655:
651:
646:
642:
619:
615:
611:
606:
602:
593:
588:
586:
582:
566:
544:
540:
531:
527:
511:
504:be the whole
489:
485:
476:
471:
469:
465:
447:
444:
441:
437:
433:
428:
424:
420:
415:
411:
402:
398:
380:
376:
367:
362:
360:
332:
314:
310:
301:
285:
278:If the index
257:
253:
249:
244:
240:
219:
199:
196:
193:
185:
184:
183:
182:
181:
167:
160:
157:
141:
121:
114:
110:
92:
89:
86:
76:
72:
61:
34:
30:
19:
18:Filtered ring
5020:
4985:
4981:
4958:. Retrieved
4948:
4932:. Elsevier.
4929:
4923:
4911:. Retrieved
4904:the original
4890:
4871:
4865:
4482:
4448:that lie in
4261:
4206:is indeed a
4174:
4037:
3771:
3703:
3701:
3499:
3325:
3135:
3131:
3129:
2964:
2962:
2649:
2517:
2372:
2310:
2261:topological
2134:
2090:
2007:
1984:
1853:an ideal of
1764:
1712:
1647:
1577:there is an
1362:
1266:
1219:
1114:
1041:
899:
871:
854:
839:
786:
780:rather than
591:
589:
584:
530:direct limit
526:homomorphism
472:
363:
358:
277:
32:
26:
3732:information
3708:probability
2862:real number
2840:where each
2520:permutation
1873:. Given an
1294:on a group
902:of a group
723:instead of
634:in lieu of
581:isomorphism
397:subalgebras
111:of a given
29:mathematics
5042:Categories
4857:References
4229:. The set
3908:filtration
2749:-algebras
2195:becomes a
1597:such that
1115:associated
900:filtration
866:See also:
809:σ-algebras
799:), and in
782:subobjects
109:subobjects
33:filtration
4995:1112.1603
4809:τ
4796:⊆
4785:τ
4746:τ
4742:≤
4733:τ
4694:≥
4651:Ω
4616:τ
4589:τ
4566:τ
4554:≠
4548:τ
4542:σ
4520:τ
4491:τ
4459:τ
4402:≥
4377:τ
4322:τ
4293:τ
4272:τ
4246:τ
4215:σ
4192:τ
4142:∈
4133:≤
4130:τ
4124:∩
4118::
4112:≥
4106:∀
4093:∈
4079:τ
4047:σ
4021:≥
3984:∈
3975:≤
3972:τ
3944:≥
3887:∞
3875:→
3872:Ω
3866:τ
3828:≥
3785:Ω
3677:⊆
3650:≥
3643:⋃
3634:σ
3626:∞
3539:σ
3517:∞
3439:⋂
3387:null sets
3328:if it is
3300:-algebra
3288:σ
3263:≥
3194:≥
3151:Ω
3109:∞
3045:…
3027:∈
3005:might be
2924:⊆
2899:⟹
2885:≤
2821:⊆
2782:≥
2737:σ
2704:Ω
2687:-algebras
2675:σ
2557:⊆
2539:⊆
1615:⊆
1326:⋂
1316:Hausdorff
1182:∈
1017:⊆
739:∈
732:⋃
690:∈
683:⋂
652:⊆
612:⊇
528:from the
434:⊆
421:⋅
250:⊆
197:≤
159:index set
90:∈
5019:(2003).
4960:June 25,
4913:June 25,
4835:See also
4227:-algebra
4010:for all
3330:complete
3007:discrete
2664:sequence
2351:-module
2219:-module
2199:. If an
1893:-module
1627:′
1544:′
1407:′
1171:, where
1091:topology
993:we have
851:Algebras
831:Examples
4767:, then
4036:. The
3531:as the
3332:(i.e.,
3280:of its
3224:, is a
3011:bounded
2331:and an
2281:-module
2006:). The
1668:and an
1260:into a
846:Algebra
532:of the
477:of the
331:adapted
232:, then
5027:
4936:
4878:
4724:, and
4262:random
3393:(i.e.
3389:) and
1833:, and
1690:module
1240:makes
1119:cosets
894:, the
862:Groups
579:is an
58:is an
4990:arXiv
4907:(PDF)
4900:(PDF)
4580:. If
4264:time
3961:, if
3902:is a
2799:with
2689:on a
2508:Flags
2371:, an
2135:When
1983:(the
1829:be a
855:See:
811:. In
475:union
5025:ISBN
4962:2012
4934:ISBN
4915:2012
4876:ISBN
4634:are
4607:and
3772:Let
3446:>
2864:and
2634:flag
2514:Sets
1789:(or
1711:, a
1390:and
1197:and
836:Sets
815:and
803:and
774:dual
31:, a
5000:doi
4638:on
4503:is
4305:is
3497:).
2666:of
2650:In
2486:of
2375:of
2112:on
2029:on
1715:of
1460:to
1314:is
1093:on
953:of
949:of
896:set
823:or
784:).
585:not
559:to
395:be
212:in
186:if
107:of
27:In
5044::
4998:.
4986:83
4984:.
4970:^
4874:.
4480:.
4084::=
3702:A
3606::
3435::=
3138:)
2506:.
2303:.
2132:.
1360:.
1264:.
827:.
791:,
470:.
5033:.
5006:.
5002::
4992::
4964:.
4942:.
4917:.
4884:.
4820:.
4813:2
4802:F
4789:1
4778:F
4750:2
4737:1
4711:)
4706:P
4702:,
4697:0
4691:t
4686:}
4681:t
4675:F
4669:{
4664:,
4659:F
4654:,
4647:(
4620:2
4593:1
4560:F
4551:)
4545:(
4514:F
4468:}
4465:t
4462:=
4456:{
4434:t
4428:F
4405:0
4399:t
4371:F
4346:F
4316:F
4240:F
4186:F
4171:.
4159:}
4154:t
4148:F
4139:}
4136:t
4127:{
4121:A
4115:0
4109:t
4103:|
4098:F
4090:A
4087:{
4073:F
4024:0
4018:t
3996:t
3990:F
3981:}
3978:t
3969:{
3947:0
3941:t
3936:}
3931:t
3925:F
3919:{
3890:]
3884:,
3881:0
3878:[
3869::
3845:)
3840:P
3836:,
3831:0
3825:t
3820:}
3815:t
3809:F
3803:{
3798:,
3793:F
3788:,
3781:(
3746:t
3718:t
3704:σ
3687:.
3682:F
3673:)
3667:t
3661:F
3653:0
3647:t
3638:(
3631:=
3620:F
3592:F
3568:t
3562:F
3511:F
3485:t
3463:s
3457:F
3449:t
3443:s
3430:+
3427:t
3421:F
3415:=
3410:t
3404:F
3385:-
3372:P
3349:0
3343:F
3310:F
3266:0
3260:t
3255:}
3250:t
3244:F
3238:{
3211:)
3206:P
3202:,
3197:0
3191:t
3186:}
3181:t
3175:F
3169:{
3164:,
3159:F
3154:,
3147:(
3115:.
3112:)
3106:+
3103:,
3100:0
3097:[
3087:]
3084:T
3081:,
3078:0
3075:[
3072:,
3067:0
3062:N
3057:,
3054:}
3051:N
3048:,
3042:,
3039:1
3036:,
3033:0
3030:{
3024:t
2993:t
2972:t
2948:.
2941:2
2937:t
2930:F
2917:1
2913:t
2906:F
2893:2
2889:t
2880:1
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2848:t
2826:F
2816:t
2810:F
2785:0
2779:t
2775:}
2769:t
2763:F
2757:{
2717:)
2712:F
2707:,
2701:(
2616:)
2613:2
2610:,
2607:1
2604:,
2601:0
2598:(
2578:}
2575:2
2572:,
2569:1
2566:,
2563:0
2560:{
2554:}
2551:1
2548:,
2545:0
2542:{
2536:}
2533:0
2530:{
2494:M
2470:M
2450:R
2430:R
2408:n
2404:M
2383:M
2359:M
2339:R
2319:R
2291:R
2269:R
2247:I
2227:M
2207:R
2183:R
2163:I
2143:R
2120:R
2098:I
2077:R
2057:M
2037:M
2015:I
1992:I
1971:M
1951:M
1931:M
1926:n
1922:I
1901:M
1881:R
1861:R
1841:I
1817:R
1797:J
1785:-
1773:I
1748:n
1744:M
1723:M
1699:M
1688:-
1676:R
1656:R
1623:n
1619:G
1610:m
1606:G
1585:m
1565:n
1540:n
1536:G
1513:n
1509:G
1488:G
1468:G
1448:G
1428:G
1403:n
1399:G
1376:n
1372:G
1348:}
1345:1
1342:{
1339:=
1334:n
1330:G
1302:G
1280:n
1276:G
1248:G
1228:G
1205:n
1185:G
1179:a
1157:n
1153:G
1149:a
1129:G
1101:G
1075:n
1071:G
1050:G
1025:n
1021:G
1012:1
1009:+
1006:n
1002:G
981:n
961:G
935:n
931:G
910:G
881:N
760:S
757:=
752:i
748:S
742:I
736:i
711:0
708:=
703:i
699:S
693:I
687:i
660:j
656:S
647:i
643:S
620:j
616:S
607:i
603:S
567:S
545:i
541:S
512:S
490:i
486:S
448:j
445:+
442:i
438:S
429:j
425:S
416:i
412:S
381:i
377:S
343:F
315:i
311:S
286:i
272:.
258:j
254:S
245:i
241:S
220:I
200:j
194:i
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122:S
93:I
87:i
83:)
77:i
73:S
69:(
44:F
20:)
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