45:
645:
2031:. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete
249:
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers.
1898:
524:
519:
454:
1803:
1690:
1733:
1799:
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792:
1225:
1133:
383:
337:
179:
1568:
1536:
1350:
1283:
1652:
1480:
919:
750:
79:
2928:
2574:
2671:
1448:
1413:
1248:
1081:
3111:
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2622:
2477:
1623:
1380:
1159:
1049:
1027:
308:
205:
240:
2853:
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2325:
1601:
3194:
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2289:
946:
2355:
2254:
2184:
2018:
2720:
2539:
2506:
2420:
877:
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816:
2980:
3068:
3045:
2067:
3131:
2873:
2814:
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2642:
2219:
2150:
2127:
2107:
2087:
1990:
1970:
1950:
1930:
1504:
1303:
1179:
1005:
700:
665:
283:
131:
zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has
Lebesgue measure zero and therefore is null.
4320:
4398:
4415:
3347:
640:{\displaystyle A\subseteq \bigcup _{n=1}^{\infty }U_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (U_{n})<\varepsilon \,,}
3439:
3723:
3582:
3339:
459:
392:
4238:
4069:
3551:
3532:
3513:
3277:
3140:
The term refers to the null invariance of the measures of translates, associating it with the complete invariance found with
3609:
4230:
4016:
4410:
829:
4367:
4357:
1660:
4167:
4076:
3840:
1703:
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4405:
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4152:
3295:
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1738:
4271:
4251:
4215:
4139:
3859:
3575:
2729:
17:
856:
762:
4393:
4172:
4134:
4086:
1184:
1092:
342:
795:
313:
140:
4298:
4266:
4256:
4177:
4144:
3775:
3684:
1541:
1509:
1323:
1256:
1083:
1893:{\displaystyle \pi (A)=0\iff \lambda \left(\left\{x:\lambda \left(A_{x}\right)>0\right\}\right)=0.}
1628:
1456:
886:
717:
55:
4315:
4220:
3996:
3924:
2892:
2544:
4305:
3398:
4388:
3834:
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2647:
1428:
1393:
1230:
1063:
3701:
3072:
2933:
2579:
2429:
1606:
1363:
1142:
1032:
1010:
291:
184:
4476:
4157:
3915:
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1353:
94:
210:
4440:
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4162:
3884:
3730:
3393:
2819:
2132:
First, we have to know that every set of positive measure contains a nonmeasurable subset. Let
1136:
386:
113:
3151:
have been related to the size of subsets and Haar null sets. Haar null sets have been used in
2360:
2294:
1586:
49:
4001:
3954:
3949:
3944:
3786:
3669:
3627:
3229:
3166:
2985:
2259:
931:
671:
86:
2330:
2224:
2159:
1450:
however other constructions are possible which assign the Cantor set any measure whatsoever.
4310:
4276:
4184:
3894:
3849:
3691:
3614:
3478:
3458:
3415:
3324:
1996:
1909:
2696:
2515:
2482:
2396:
862:
835:
801:
8:
4481:
4293:
4283:
4129:
4093:
3919:
3648:
3605:
3012:
2959:
2723:
2069:
which is closed hence Borel measurable, and which has measure zero, and to find a subset
2043:
The Borel measure is not complete. One simple construction is to start with the standard
1693:
703:
3971:
3462:
3050:
3027:
2049:
4445:
4205:
4190:
3889:
3770:
3748:
3482:
3448:
3419:
3366:
3312:
3116:
2858:
2799:
2676:
2627:
2423:
2189:
2135:
2112:
2092:
2072:
1975:
1955:
1935:
1915:
1489:
1288:
1164:
990:
825:
685:
650:
268:
4362:
4098:
4059:
4054:
3961:
3879:
3664:
3637:
3547:
3528:
3509:
3486:
3273:
3205:
3148:
953:
3423:
4379:
4288:
4064:
4049:
4039:
4024:
3991:
3986:
3976:
3854:
3829:
3644:
3466:
3403:
3356:
3308:
3304:
3265:
3197:
2027:
1387:
965:
128:
105:
99:
3361:
4455:
4435:
4210:
4108:
4103:
4081:
3939:
3904:
3824:
3718:
3474:
3411:
3320:
3217:
2884:
2153:
1655:
1572:
1422:
1383:
1306:
1058:
981:
4345:
4200:
4195:
4006:
3981:
3934:
3864:
3844:
3804:
3794:
3591:
3201:
949:
3407:
4470:
4450:
4113:
4034:
4029:
3929:
3899:
3869:
3819:
3814:
3809:
3799:
3713:
3632:
2509:
2109:
which is not Borel measurable. (Since the
Lebesgue measure is complete, this
2032:
1357:
926:
753:
135:
2022:
as sets of equivalence classes of functions which differ only on null sets.
4044:
3966:
3706:
3152:
3141:
3021:
2887:
2038:
1992:
is, and their integrals are equal. This motivates the formal definition of
3743:
3470:
3909:
1309:
instead of intervals. In fact, the idea can be made to make sense on any
922:
3753:
3370:
3316:
3269:
3160:
2044:
1418:
679:
254:
124:
44:
3735:
3679:
3674:
3437:
Dodos, Pandelis (2009). "The
Steinhaus property and Haar-null sets".
3384:
Solecki, S. (2005). "Sizes of subsets of groups and Haar null sets".
3223:
1483:
1055:
822:
286:
120:
109:
31:
3619:
1993:
1692:
In terms of null sets, the following equivalence has been styled a
1310:
1087:
675:
38:
3453:
3560:
3235:
3238: â Complete absence of anything; the opposite of everything
1029:
has null
Lebesgue measure and is considered to be a null set in
977:
969:
855:
Any (measurable) subset of a null set is itself a null set (by
3293:
van Douwen, Eric K. (1989). "Fubini's theorem for null sets".
2722:
would also be Borel measurable (here we use the fact that the
2025:
A measure in which all subsets of null sets are measurable is
3220: â Continuous function that is not absolutely continuous
2327:
is countable, since it contains one point per component of
973:
2693:
is a null set. However, if it were Borel measurable, then
514:{\displaystyle \operatorname {length} (U_{n})=b_{n}-a_{n}}
449:{\displaystyle U_{n}=(a_{n},b_{n})\subseteq \mathbb {R} }
2039:
A subset of the Cantor set which is not Borel measurable
1538:
For instance straight lines or circles are null sets in
3232: â Generalization of mass, length, area and volume
2726:
of a Borel set by a continuous function is measurable;
119:
The notion of null set should not be confused with the
3169:
3119:
3075:
3053:
3030:
2988:
2962:
2936:
2895:
2861:
2822:
2802:
2732:
2699:
2679:
2650:
2630:
2582:
2547:
2518:
2485:
2432:
2399:
2363:
2333:
2297:
2262:
2227:
2192:
2162:
2156:, a continuous function which is locally constant on
2138:
2115:
2095:
2075:
2052:
1999:
1978:
1958:
1938:
1918:
1806:
1741:
1706:
1663:
1631:
1609:
1589:
1544:
1512:
1492:
1459:
1431:
1396:
1366:
1326:
1291:
1259:
1233:
1187:
1167:
1145:
1095:
1066:
1035:
1013:
993:
934:
889:
865:
838:
804:
765:
720:
688:
653:
527:
462:
395:
345:
316:
294:
271:
213:
187:
143:
58:
667:
is a null set, also known as a set of zero-content.
1908:Null sets play a key role in the definition of the
3260:. The Student Mathematical Library. Vol. 48.
3188:
3125:
3105:
3062:
3039:
3003:
2974:
2948:
2922:
2867:
2847:
2808:
2788:
2714:
2685:
2665:
2636:
2616:
2568:
2533:
2500:
2471:
2414:
2385:
2349:
2319:
2283:
2248:
2213:
2178:
2144:
2121:
2101:
2081:
2061:
2012:
1984:
1964:
1944:
1924:
1892:
1793:
1727:
1684:
1646:
1617:
1595:
1562:
1530:
1498:
1474:
1442:
1407:
1374:
1344:
1297:
1277:
1242:
1219:
1173:
1153:
1127:
1075:
1043:
1021:
999:
940:
913:
871:
844:
810:
786:
744:
694:
659:
639:
513:
448:
377:
331:
302:
277:
234:
199:
173:
104:. This can be characterized as a set that can be
73:
883:Together, these facts show that the null sets of
4468:
3348:Proceedings of the American Mathematical Society
3226: â Mathematical set containing no elements
1227:and the total length of the union is less than
948:. Accordingly, null sets may be interpreted as
3258:A (Terse) Introduction to Lebesgue Integration
2508:is strictly monotonic and continuous, it is a
1313:, even if there is no Lebesgue measure there.
3576:
1685:{\displaystyle \lambda \times \lambda =\pi .}
4321:RieszâMarkovâKakutani representation theorem
3503:
2911:
2905:
1785:
1755:
3440:Bulletin of the London Mathematical Society
674:, this definition requires that there be a
4416:Vitale's random BrunnâMinkowski inequality
3583:
3569:
3337:
3292:
1829:
1825:
952:, yielding a measure-theoretic notion of "
257:is an example of an uncountable null set.
3452:
3397:
3360:
2875:is a null, but non-Borel measurable set.
1728:{\displaystyle A\subset \mathbb {R} ^{2}}
1715:
1634:
1611:
1547:
1515:
1462:
1433:
1398:
1368:
1329:
1262:
1147:
1037:
1015:
633:
442:
296:
61:
52:is an example of a null set of points in
3525:Lebesgue Integration on Euclidean Spaces
3504:Capinski, Marek; Kopp, Ekkehard (2005).
43:
37:For the set of zeros of a function, see
3383:
1794:{\displaystyle A_{x}=\{y:(x,y)\in A\},}
14:
4469:
3541:
3255:
1579:of a smooth function has measure zero.
828:of null sets is itself a null set (by
706:of the lengths of the covers is zero.
3564:
3527:. Jones & Bartlett. p. 107.
3522:
3436:
3196:contains an open neighborhood of the
2789:{\displaystyle g(F)=(g^{-1})^{-1}(F)}
1952:are equal except on a null set, then
1253:This condition can be generalised to
4429:Applications & related
2422:has measure one. We need a strictly
787:{\displaystyle \mu (\varnothing )=0}
116:of arbitrarily small total length.
27:Measurable set whose measure is zero
2129:is of course Lebesgue measurable.)
1220:{\displaystyle I_{1},I_{2},\ldots }
1128:{\displaystyle I_{1},I_{2},\ldots }
968:is the standard way of assigning a
959:
378:{\displaystyle U_{1},U_{2},\ldots }
24:
3590:
3497:
3204:since it is the conclusion of the
935:
899:
730:
600:
550:
332:{\displaystyle \varepsilon >0,}
194:
174:{\displaystyle M=(X,\Sigma ,\mu )}
159:
30:For the set with no elements, see
25:
4493:
3506:Measure, Integral and Probability
3386:Geometric and Functional Analysis
1563:{\displaystyle \mathbb {R} ^{2}.}
1531:{\displaystyle \mathbb {R} ^{n}.}
1417:The standard construction of the
1360:are null. In particular, the set
1345:{\displaystyle \mathbb {R} ^{n},}
1278:{\displaystyle \mathbb {R} ^{n},}
1181:is contained in the union of the
772:
4358:Lebesgue differentiation theorem
4239:Carathéodory's extension theorem
2816:through the continuous function
2186:and monotonically increasing on
1647:{\displaystyle \mathbb {R} ^{2}}
1475:{\displaystyle \mathbb {R} ^{n}}
914:{\displaystyle (X,\Sigma ,\mu )}
745:{\displaystyle (X,\Sigma ,\mu )}
74:{\displaystyle \mathbb {R} ^{2}}
2923:{\displaystyle (X,\|\cdot \|).}
2569:{\displaystyle E\subseteq g(K)}
3546:. Springer-Verlag. p. 3.
3430:
3377:
3340:"Convexity and Haar Null Sets"
3331:
3309:10.1080/00029890.1989.11972270
3286:
3249:
3091:
3079:
2914:
2896:
2783:
2777:
2765:
2748:
2742:
2736:
2709:
2703:
2608:
2602:
2563:
2557:
2528:
2522:
2495:
2489:
2457:
2451:
2442:
2436:
2409:
2403:
2380:
2367:
2314:
2301:
2272:
2266:
2237:
2231:
2205:
2193:
1826:
1816:
1810:
1776:
1764:
1625:and Ï is Lebesgue measure for
1506:have null Lebesgue measure in
908:
890:
775:
769:
739:
721:
624:
611:
482:
469:
435:
409:
223:
217:
168:
150:
13:
1:
3362:10.1090/S0002-9939-97-03776-3
3296:American Mathematical Monthly
3262:American Mathematical Society
3242:
3200:. This property is named for
3147:Some algebraic properties of
2666:{\displaystyle F\subseteq K,}
1972:is integrable if and only if
1443:{\displaystyle \mathbb {R} ;}
1408:{\displaystyle \mathbb {R} .}
1386:is a null set, despite being
1243:{\displaystyle \varepsilon .}
1076:{\displaystyle \varepsilon ,}
709:
260:
127:. Although the empty set has
3106:{\displaystyle \mu (A+x)=0,}
2949:{\displaystyle A\subseteq X}
2878:
2617:{\displaystyle F=g^{-1}(E).}
2472:{\displaystyle g(x)=f(x)+x.}
1618:{\displaystyle \mathbb {R} }
1375:{\displaystyle \mathbb {Q} }
1356:are null, and therefore all
1154:{\displaystyle \mathbb {R} }
1044:{\displaystyle \mathbb {R} }
1022:{\displaystyle \mathbb {R} }
303:{\displaystyle \mathbb {R} }
200:{\displaystyle S\in \Sigma }
7:
4411:PrĂ©kopaâLeindler inequality
3211:
2644:is injective, we have that
2576:be non-measurable, and let
244:
134:More generally, on a given
10:
4498:
4353:Lebesgue's density theorem
2930:addition moves any subset
235:{\displaystyle \mu (S)=0.}
36:
29:
4428:
4406:MinkowskiâSteiner formula
4376:
4336:
4329:
4229:
4221:Projection-valued measure
4122:
4015:
3784:
3657:
3598:
3408:10.1007/s00039-005-0505-z
2848:{\displaystyle h=g^{-1}.}
97:of real numbers that has
4389:Isoperimetric inequality
4368:VitaliâHahnâSaks theorem
3697:Carathéodory's criterion
3542:Oxtoby, John C. (1971).
3508:. Springer. p. 16.
3338:Matouskova, Eva (1997).
2386:{\displaystyle f(K^{c})}
2320:{\displaystyle f(K^{c})}
1603:is Lebesgue measure for
1596:{\displaystyle \lambda }
1421:is an example of a null
339:there exists a sequence
4394:BrunnâMinkowski theorem
4263:Decomposition theorems
3189:{\displaystyle A^{-1}A}
3004:{\displaystyle x\in X.}
2284:{\displaystyle f(1)=1.}
1903:
941:{\displaystyle \Sigma }
830:countable subadditivity
95:Lebesgue measurable set
4441:Descriptive set theory
4341:Disintegration theorem
3776:Universally measurable
3190:
3127:
3107:
3064:
3041:
3005:
2976:
2950:
2924:
2869:
2849:
2810:
2790:
2716:
2687:
2667:
2638:
2618:
2570:
2535:
2502:
2473:
2416:
2387:
2351:
2350:{\displaystyle K^{c}.}
2321:
2285:
2250:
2249:{\displaystyle f(0)=0}
2215:
2180:
2179:{\displaystyle K^{c},}
2146:
2123:
2103:
2083:
2063:
2014:
1986:
1966:
1946:
1926:
1894:
1795:
1729:
1686:
1648:
1619:
1597:
1564:
1532:
1500:
1476:
1444:
1409:
1376:
1346:
1299:
1279:
1244:
1221:
1175:
1155:
1129:
1077:
1045:
1023:
1001:
942:
915:
873:
846:
812:
788:
746:
696:
661:
641:
604:
554:
515:
450:
379:
333:
304:
279:
236:
201:
175:
82:
75:
4243:Convergence theorems
3702:Cylindrical Ï-algebra
3523:Jones, Frank (1993).
3256:Franks, John (2009).
3230:Measure (mathematics)
3191:
3128:
3108:
3065:
3042:
3006:
2977:
2951:
2925:
2870:
2850:
2811:
2791:
2717:
2688:
2668:
2639:
2619:
2571:
2541:has measure one. Let
2536:
2503:
2474:
2417:
2393:has measure zero, so
2388:
2352:
2322:
2286:
2251:
2216:
2181:
2147:
2124:
2104:
2084:
2064:
2015:
2013:{\displaystyle L^{p}}
1987:
1967:
1947:
1927:
1895:
1796:
1730:
1687:
1649:
1620:
1598:
1565:
1533:
1501:
1477:
1445:
1410:
1377:
1347:
1300:
1280:
1245:
1222:
1176:
1156:
1130:
1078:
1046:
1024:
1002:
943:
916:
874:
847:
813:
789:
747:
697:
672:mathematical analysis
662:
642:
584:
534:
516:
451:
380:
334:
305:
280:
237:
202:
176:
87:mathematical analysis
76:
47:
4311:Minkowski inequality
4185:Cylinder set measure
4070:Infinite-dimensional
3685:equivalence relation
3615:Lebesgue integration
3544:Measure and Category
3167:
3117:
3073:
3051:
3028:
3020:on the Ï-algebra of
2986:
2960:
2934:
2893:
2859:
2820:
2800:
2730:
2715:{\displaystyle f(F)}
2697:
2677:
2648:
2628:
2580:
2545:
2534:{\displaystyle g(K)}
2516:
2501:{\displaystyle f(x)}
2483:
2430:
2415:{\displaystyle f(K)}
2397:
2361:
2331:
2295:
2260:
2225:
2190:
2160:
2136:
2113:
2093:
2073:
2050:
1997:
1976:
1956:
1936:
1916:
1804:
1739:
1704:
1661:
1629:
1607:
1587:
1542:
1510:
1490:
1457:
1429:
1394:
1364:
1324:
1289:
1257:
1231:
1185:
1165:
1143:
1093:
1064:
1033:
1011:
991:
932:
887:
872:{\displaystyle \mu }
863:
845:{\displaystyle \mu }
836:
811:{\displaystyle \mu }
802:
763:
718:
686:
651:
525:
460:
393:
343:
314:
310:such that for every
292:
269:
211:
185:
181:a null set is a set
141:
56:
4306:Hölder's inequality
4168:of random variables
4130:Measurable function
4017:Particular measures
3606:Absolute continuity
3471:10.1112/blms/bdp014
3463:2010arXiv1006.2675D
3013:probability measure
2975:{\displaystyle A+x}
2796:is the preimage of
1453:All the subsets of
285:is a subset of the
50:SierpiĆski triangle
4446:Probability theory
3771:Transverse measure
3749:Non-measurable set
3731:Locally measurable
3186:
3155:to show that when
3149:topological groups
3123:
3103:
3063:{\displaystyle x,}
3060:
3047:such that for all
3040:{\displaystyle X,}
3037:
3001:
2972:
2956:to the translates
2946:
2920:
2865:
2845:
2806:
2786:
2712:
2683:
2663:
2634:
2614:
2566:
2531:
2498:
2469:
2424:monotonic function
2412:
2383:
2347:
2317:
2281:
2246:
2211:
2176:
2142:
2119:
2099:
2079:
2062:{\displaystyle K,}
2059:
2010:
1982:
1962:
1942:
1922:
1890:
1791:
1725:
1682:
1644:
1615:
1593:
1560:
1528:
1496:
1472:
1440:
1405:
1372:
1342:
1295:
1275:
1240:
1217:
1171:
1151:
1125:
1073:
1041:
1019:
997:
938:
911:
869:
842:
808:
784:
742:
692:
670:In terminology of
657:
637:
511:
446:
375:
329:
300:
275:
232:
197:
171:
83:
71:
4464:
4463:
4424:
4423:
4153:almost everywhere
4099:Spherical measure
3997:Strictly positive
3925:Projection-valued
3665:Almost everywhere
3638:Probability space
3553:978-0-387-05349-3
3534:978-0-86720-203-8
3515:978-1-85233-781-0
3279:978-0-8218-4862-3
3206:Steinhaus theorem
3126:{\displaystyle A}
2868:{\displaystyle F}
2809:{\displaystyle F}
2686:{\displaystyle F}
2637:{\displaystyle g}
2214:{\displaystyle ,}
2145:{\displaystyle f}
2122:{\displaystyle F}
2102:{\displaystyle K}
2082:{\displaystyle F}
1985:{\displaystyle g}
1965:{\displaystyle f}
1945:{\displaystyle g}
1925:{\displaystyle f}
1910:Lebesgue integral
1499:{\displaystyle n}
1298:{\displaystyle n}
1174:{\displaystyle N}
1000:{\displaystyle N}
954:almost everywhere
695:{\displaystyle A}
660:{\displaystyle A}
583:
580:
575:
570:
567:
278:{\displaystyle A}
16:(Redirected from
4489:
4399:Milman's reverse
4382:
4380:Lebesgue measure
4334:
4333:
3738:
3724:infimum/supremum
3645:Measurable space
3585:
3578:
3571:
3562:
3561:
3557:
3538:
3519:
3491:
3490:
3456:
3434:
3428:
3427:
3401:
3381:
3375:
3374:
3364:
3355:(6): 1793â1799.
3344:
3335:
3329:
3328:
3290:
3284:
3283:
3270:10.1090/stml/048
3253:
3198:identity element
3195:
3193:
3192:
3187:
3182:
3181:
3158:
3132:
3130:
3129:
3124:
3112:
3110:
3109:
3104:
3069:
3067:
3066:
3061:
3046:
3044:
3043:
3038:
3019:
3011:When there is a
3010:
3008:
3007:
3002:
2981:
2979:
2978:
2973:
2955:
2953:
2952:
2947:
2929:
2927:
2926:
2921:
2874:
2872:
2871:
2866:
2854:
2852:
2851:
2846:
2841:
2840:
2815:
2813:
2812:
2807:
2795:
2793:
2792:
2787:
2776:
2775:
2763:
2762:
2721:
2719:
2718:
2713:
2692:
2690:
2689:
2684:
2672:
2670:
2669:
2664:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2601:
2600:
2575:
2573:
2572:
2567:
2540:
2538:
2537:
2532:
2507:
2505:
2504:
2499:
2478:
2476:
2475:
2470:
2421:
2419:
2418:
2413:
2392:
2390:
2389:
2384:
2379:
2378:
2356:
2354:
2353:
2348:
2343:
2342:
2326:
2324:
2323:
2318:
2313:
2312:
2290:
2288:
2287:
2282:
2255:
2253:
2252:
2247:
2220:
2218:
2217:
2212:
2185:
2183:
2182:
2177:
2172:
2171:
2151:
2149:
2148:
2143:
2128:
2126:
2125:
2120:
2108:
2106:
2105:
2100:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2019:
2017:
2016:
2011:
2009:
2008:
1991:
1989:
1988:
1983:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1931:
1929:
1928:
1923:
1899:
1897:
1896:
1891:
1883:
1879:
1875:
1868:
1864:
1863:
1800:
1798:
1797:
1792:
1751:
1750:
1734:
1732:
1731:
1726:
1724:
1723:
1718:
1694:Fubini's theorem
1691:
1689:
1688:
1683:
1653:
1651:
1650:
1645:
1643:
1642:
1637:
1624:
1622:
1621:
1616:
1614:
1602:
1600:
1599:
1594:
1569:
1567:
1566:
1561:
1556:
1555:
1550:
1537:
1535:
1534:
1529:
1524:
1523:
1518:
1505:
1503:
1502:
1497:
1486:is smaller than
1481:
1479:
1478:
1473:
1471:
1470:
1465:
1449:
1447:
1446:
1441:
1436:
1414:
1412:
1411:
1406:
1401:
1384:rational numbers
1381:
1379:
1378:
1373:
1371:
1351:
1349:
1348:
1343:
1338:
1337:
1332:
1320:With respect to
1304:
1302:
1301:
1296:
1284:
1282:
1281:
1276:
1271:
1270:
1265:
1249:
1247:
1246:
1241:
1226:
1224:
1223:
1218:
1210:
1209:
1197:
1196:
1180:
1178:
1177:
1172:
1160:
1158:
1157:
1152:
1150:
1134:
1132:
1131:
1126:
1118:
1117:
1105:
1104:
1082:
1080:
1079:
1074:
1051:if and only if:
1050:
1048:
1047:
1042:
1040:
1028:
1026:
1025:
1020:
1018:
1006:
1004:
1003:
998:
966:Lebesgue measure
960:Lebesgue measure
947:
945:
944:
939:
920:
918:
917:
912:
878:
876:
875:
870:
851:
849:
848:
843:
817:
815:
814:
809:
793:
791:
790:
785:
751:
749:
748:
743:
701:
699:
698:
693:
666:
664:
663:
658:
646:
644:
643:
638:
623:
622:
603:
598:
581:
578:
577:
576:
573:
568:
565:
564:
563:
553:
548:
520:
518:
517:
512:
510:
509:
497:
496:
481:
480:
455:
453:
452:
447:
445:
434:
433:
421:
420:
405:
404:
389:(where interval
384:
382:
381:
376:
368:
367:
355:
354:
338:
336:
335:
330:
309:
307:
306:
301:
299:
284:
282:
281:
276:
241:
239:
238:
233:
206:
204:
203:
198:
180:
178:
177:
172:
129:Lebesgue measure
80:
78:
77:
72:
70:
69:
64:
21:
4497:
4496:
4492:
4491:
4490:
4488:
4487:
4486:
4467:
4466:
4465:
4460:
4456:Spectral theory
4436:Convex analysis
4420:
4377:
4372:
4325:
4225:
4173:in distribution
4118:
4011:
3841:Logarithmically
3780:
3736:
3719:Essential range
3653:
3594:
3589:
3554:
3535:
3516:
3500:
3498:Further reading
3495:
3494:
3435:
3431:
3399:10.1.1.133.7074
3382:
3378:
3342:
3336:
3332:
3291:
3287:
3280:
3254:
3250:
3245:
3218:Cantor function
3214:
3174:
3170:
3168:
3165:
3164:
3156:
3118:
3115:
3114:
3074:
3071:
3070:
3052:
3049:
3048:
3029:
3026:
3025:
3015:
2987:
2984:
2983:
2961:
2958:
2957:
2935:
2932:
2931:
2894:
2891:
2890:
2881:
2860:
2857:
2856:
2833:
2829:
2821:
2818:
2817:
2801:
2798:
2797:
2768:
2764:
2755:
2751:
2731:
2728:
2727:
2698:
2695:
2694:
2678:
2675:
2674:
2649:
2646:
2645:
2629:
2626:
2625:
2593:
2589:
2581:
2578:
2577:
2546:
2543:
2542:
2517:
2514:
2513:
2512:. Furthermore,
2484:
2481:
2480:
2431:
2428:
2427:
2398:
2395:
2394:
2374:
2370:
2362:
2359:
2358:
2338:
2334:
2332:
2329:
2328:
2308:
2304:
2296:
2293:
2292:
2261:
2258:
2257:
2226:
2223:
2222:
2191:
2188:
2187:
2167:
2163:
2161:
2158:
2157:
2154:Cantor function
2137:
2134:
2133:
2114:
2111:
2110:
2094:
2091:
2090:
2074:
2071:
2070:
2051:
2048:
2047:
2041:
2004:
2000:
1998:
1995:
1994:
1977:
1974:
1973:
1957:
1954:
1953:
1937:
1934:
1933:
1917:
1914:
1913:
1912:: if functions
1906:
1859:
1855:
1851:
1841:
1837:
1833:
1805:
1802:
1801:
1746:
1742:
1740:
1737:
1736:
1719:
1714:
1713:
1705:
1702:
1701:
1662:
1659:
1658:
1656:product measure
1638:
1633:
1632:
1630:
1627:
1626:
1610:
1608:
1605:
1604:
1588:
1585:
1584:
1577:critical values
1551:
1546:
1545:
1543:
1540:
1539:
1519:
1514:
1513:
1511:
1508:
1507:
1491:
1488:
1487:
1466:
1461:
1460:
1458:
1455:
1454:
1432:
1430:
1427:
1426:
1423:uncountable set
1397:
1395:
1392:
1391:
1367:
1365:
1362:
1361:
1333:
1328:
1327:
1325:
1322:
1321:
1290:
1287:
1286:
1266:
1261:
1260:
1258:
1255:
1254:
1232:
1229:
1228:
1205:
1201:
1192:
1188:
1186:
1183:
1182:
1166:
1163:
1162:
1146:
1144:
1141:
1140:
1113:
1109:
1100:
1096:
1094:
1091:
1090:
1065:
1062:
1061:
1059:positive number
1036:
1034:
1031:
1030:
1014:
1012:
1009:
1008:
992:
989:
988:
982:Euclidean space
962:
950:negligible sets
933:
930:
929:
888:
885:
884:
864:
861:
860:
837:
834:
833:
803:
800:
799:
764:
761:
760:
719:
716:
715:
712:
687:
684:
683:
652:
649:
648:
618:
614:
599:
588:
572:
571:
559:
555:
549:
538:
526:
523:
522:
505:
501:
492:
488:
476:
472:
461:
458:
457:
441:
429:
425:
416:
412:
400:
396:
394:
391:
390:
363:
359:
350:
346:
344:
341:
340:
315:
312:
311:
295:
293:
290:
289:
270:
267:
266:
263:
247:
212:
209:
208:
186:
183:
182:
142:
139:
138:
65:
60:
59:
57:
54:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4495:
4485:
4484:
4479:
4477:Measure theory
4462:
4461:
4459:
4458:
4453:
4448:
4443:
4438:
4432:
4430:
4426:
4425:
4422:
4421:
4419:
4418:
4413:
4408:
4403:
4402:
4401:
4391:
4385:
4383:
4374:
4373:
4371:
4370:
4365:
4363:Sard's theorem
4360:
4355:
4350:
4349:
4348:
4346:Lifting theory
4337:
4331:
4327:
4326:
4324:
4323:
4318:
4313:
4308:
4303:
4302:
4301:
4299:FubiniâTonelli
4291:
4286:
4281:
4280:
4279:
4274:
4269:
4261:
4260:
4259:
4254:
4249:
4241:
4235:
4233:
4227:
4226:
4224:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4187:
4182:
4181:
4180:
4178:in probability
4175:
4165:
4160:
4155:
4149:
4148:
4147:
4142:
4137:
4126:
4124:
4120:
4119:
4117:
4116:
4111:
4106:
4101:
4096:
4091:
4090:
4089:
4079:
4074:
4073:
4072:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4027:
4021:
4019:
4013:
4012:
4010:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3958:
3957:
3952:
3947:
3937:
3932:
3927:
3922:
3912:
3907:
3902:
3897:
3892:
3887:
3885:Locally finite
3882:
3872:
3867:
3862:
3857:
3852:
3847:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3791:
3789:
3782:
3781:
3779:
3778:
3773:
3768:
3763:
3758:
3757:
3756:
3746:
3741:
3733:
3728:
3727:
3726:
3716:
3711:
3710:
3709:
3699:
3694:
3689:
3688:
3687:
3677:
3672:
3667:
3661:
3659:
3655:
3654:
3652:
3651:
3642:
3641:
3640:
3630:
3625:
3617:
3612:
3602:
3600:
3599:Basic concepts
3596:
3595:
3592:Measure theory
3588:
3587:
3580:
3573:
3565:
3559:
3558:
3552:
3539:
3533:
3520:
3514:
3499:
3496:
3493:
3492:
3429:
3376:
3330:
3285:
3278:
3264:. p. 28.
3247:
3246:
3244:
3241:
3240:
3239:
3233:
3227:
3221:
3213:
3210:
3202:Hugo Steinhaus
3185:
3180:
3177:
3173:
3122:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3059:
3056:
3036:
3033:
3000:
2997:
2994:
2991:
2971:
2968:
2965:
2945:
2942:
2939:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2880:
2877:
2864:
2844:
2839:
2836:
2832:
2828:
2825:
2805:
2785:
2782:
2779:
2774:
2771:
2767:
2761:
2758:
2754:
2750:
2747:
2744:
2741:
2738:
2735:
2711:
2708:
2705:
2702:
2682:
2662:
2659:
2656:
2653:
2633:
2613:
2610:
2607:
2604:
2599:
2596:
2592:
2588:
2585:
2565:
2562:
2559:
2556:
2553:
2550:
2530:
2527:
2524:
2521:
2497:
2494:
2491:
2488:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2426:, so consider
2411:
2408:
2405:
2402:
2382:
2377:
2373:
2369:
2366:
2346:
2341:
2337:
2316:
2311:
2307:
2303:
2300:
2280:
2277:
2274:
2271:
2268:
2265:
2245:
2242:
2239:
2236:
2233:
2230:
2210:
2207:
2204:
2201:
2198:
2195:
2175:
2170:
2166:
2141:
2118:
2098:
2078:
2058:
2055:
2040:
2037:
2007:
2003:
1981:
1961:
1941:
1921:
1905:
1902:
1901:
1900:
1889:
1886:
1882:
1878:
1874:
1871:
1867:
1862:
1858:
1854:
1850:
1847:
1844:
1840:
1836:
1832:
1828:
1824:
1821:
1818:
1815:
1812:
1809:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1749:
1745:
1722:
1717:
1712:
1709:
1681:
1678:
1675:
1672:
1669:
1666:
1641:
1636:
1613:
1592:
1581:
1580:
1570:
1559:
1554:
1549:
1527:
1522:
1517:
1495:
1469:
1464:
1451:
1439:
1435:
1415:
1404:
1400:
1370:
1358:countable sets
1354:singleton sets
1341:
1336:
1331:
1316:For instance:
1294:
1274:
1269:
1264:
1251:
1250:
1239:
1236:
1216:
1213:
1208:
1204:
1200:
1195:
1191:
1170:
1149:
1124:
1121:
1116:
1112:
1108:
1103:
1099:
1072:
1069:
1039:
1017:
996:
980:to subsets of
961:
958:
937:
910:
907:
904:
901:
898:
895:
892:
881:
880:
868:
853:
841:
819:
807:
783:
780:
777:
774:
771:
768:
741:
738:
735:
732:
729:
726:
723:
711:
708:
702:for which the
691:
656:
636:
632:
629:
626:
621:
617:
613:
610:
607:
602:
597:
594:
591:
587:
562:
558:
552:
547:
544:
541:
537:
533:
530:
508:
504:
500:
495:
491:
487:
484:
479:
475:
471:
468:
465:
444:
440:
437:
432:
428:
424:
419:
415:
411:
408:
403:
399:
374:
371:
366:
362:
358:
353:
349:
328:
325:
322:
319:
298:
274:
262:
259:
246:
243:
231:
228:
225:
222:
219:
216:
196:
193:
190:
170:
167:
164:
161:
158:
155:
152:
149:
146:
123:as defined in
68:
63:
26:
9:
6:
4:
3:
2:
4494:
4483:
4480:
4478:
4475:
4474:
4472:
4457:
4454:
4452:
4451:Real analysis
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4433:
4431:
4427:
4417:
4414:
4412:
4409:
4407:
4404:
4400:
4397:
4396:
4395:
4392:
4390:
4387:
4386:
4384:
4381:
4375:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4347:
4344:
4343:
4342:
4339:
4338:
4335:
4332:
4330:Other results
4328:
4322:
4319:
4317:
4316:RadonâNikodym
4314:
4312:
4309:
4307:
4304:
4300:
4297:
4296:
4295:
4292:
4290:
4289:Fatou's lemma
4287:
4285:
4282:
4278:
4275:
4273:
4270:
4268:
4265:
4264:
4262:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4242:
4240:
4237:
4236:
4234:
4232:
4228:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4188:
4186:
4183:
4179:
4176:
4174:
4171:
4170:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:Convergence:
4150:
4146:
4143:
4141:
4138:
4136:
4133:
4132:
4131:
4128:
4127:
4125:
4121:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4088:
4085:
4084:
4083:
4080:
4078:
4075:
4071:
4068:
4067:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4022:
4020:
4018:
4014:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3956:
3955:Outer regular
3953:
3951:
3950:Inner regular
3948:
3946:
3945:Borel regular
3943:
3942:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3917:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3877:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3842:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3792:
3790:
3788:
3783:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3755:
3752:
3751:
3750:
3747:
3745:
3742:
3740:
3734:
3732:
3729:
3725:
3722:
3721:
3720:
3717:
3715:
3712:
3708:
3705:
3704:
3703:
3700:
3698:
3695:
3693:
3690:
3686:
3683:
3682:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3660:
3656:
3650:
3646:
3643:
3639:
3636:
3635:
3634:
3633:Measure space
3631:
3629:
3626:
3624:
3622:
3618:
3616:
3613:
3611:
3607:
3604:
3603:
3601:
3597:
3593:
3586:
3581:
3579:
3574:
3572:
3567:
3566:
3563:
3555:
3549:
3545:
3540:
3536:
3530:
3526:
3521:
3517:
3511:
3507:
3502:
3501:
3488:
3484:
3480:
3476:
3472:
3468:
3464:
3460:
3455:
3450:
3447:(2): 377â44.
3446:
3442:
3441:
3433:
3425:
3421:
3417:
3413:
3409:
3405:
3400:
3395:
3391:
3387:
3380:
3372:
3368:
3363:
3358:
3354:
3350:
3349:
3341:
3334:
3326:
3322:
3318:
3314:
3310:
3306:
3303:(8): 718â21.
3302:
3298:
3297:
3289:
3281:
3275:
3271:
3267:
3263:
3259:
3252:
3248:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3215:
3209:
3207:
3203:
3199:
3183:
3178:
3175:
3171:
3162:
3154:
3153:Polish groups
3150:
3145:
3143:
3138:
3136:
3135:Haar null set
3120:
3100:
3097:
3094:
3088:
3085:
3082:
3076:
3057:
3054:
3034:
3031:
3023:
3022:Borel subsets
3018:
3014:
2998:
2995:
2992:
2989:
2969:
2966:
2963:
2943:
2940:
2937:
2917:
2908:
2902:
2899:
2889:
2886:
2876:
2862:
2855:) Therefore,
2842:
2837:
2834:
2830:
2826:
2823:
2803:
2780:
2772:
2769:
2759:
2756:
2752:
2745:
2739:
2733:
2725:
2706:
2700:
2680:
2660:
2657:
2654:
2651:
2631:
2611:
2605:
2597:
2594:
2590:
2586:
2583:
2560:
2554:
2551:
2548:
2525:
2519:
2511:
2510:homeomorphism
2492:
2486:
2466:
2463:
2460:
2454:
2448:
2445:
2439:
2433:
2425:
2406:
2400:
2375:
2371:
2364:
2344:
2339:
2335:
2309:
2305:
2298:
2278:
2275:
2269:
2263:
2243:
2240:
2234:
2228:
2208:
2202:
2199:
2196:
2173:
2168:
2164:
2155:
2139:
2130:
2116:
2096:
2076:
2056:
2053:
2046:
2036:
2034:
2033:Borel measure
2030:
2029:
2023:
2021:
2005:
2001:
1979:
1959:
1939:
1919:
1911:
1887:
1884:
1880:
1876:
1872:
1869:
1865:
1860:
1856:
1852:
1848:
1845:
1842:
1838:
1834:
1830:
1822:
1819:
1813:
1807:
1788:
1782:
1779:
1773:
1770:
1767:
1761:
1758:
1752:
1747:
1743:
1720:
1710:
1707:
1699:
1698:
1697:
1695:
1679:
1676:
1673:
1670:
1667:
1664:
1657:
1639:
1590:
1578:
1575:: the set of
1574:
1571:
1557:
1552:
1525:
1520:
1493:
1485:
1467:
1452:
1437:
1424:
1420:
1416:
1402:
1389:
1385:
1359:
1355:
1339:
1334:
1319:
1318:
1317:
1314:
1312:
1308:
1292:
1272:
1267:
1237:
1234:
1214:
1211:
1206:
1202:
1198:
1193:
1189:
1168:
1138:
1122:
1119:
1114:
1110:
1106:
1101:
1097:
1089:
1085:
1070:
1067:
1060:
1057:
1054:
1053:
1052:
994:
985:
983:
979:
975:
971:
967:
957:
955:
951:
928:
924:
905:
902:
896:
893:
866:
858:
854:
839:
831:
827:
824:
820:
805:
797:
781:
778:
766:
759:
758:
757:
755:
754:measure space
736:
733:
727:
724:
707:
705:
689:
681:
677:
673:
668:
654:
634:
630:
627:
619:
615:
608:
605:
595:
592:
589:
585:
560:
556:
545:
542:
539:
535:
531:
528:
506:
502:
498:
493:
489:
485:
477:
473:
466:
463:
438:
430:
426:
422:
417:
413:
406:
401:
397:
388:
372:
369:
364:
360:
356:
351:
347:
326:
323:
320:
317:
288:
272:
258:
256:
251:
242:
229:
226:
220:
214:
191:
188:
165:
162:
156:
153:
147:
144:
137:
136:measure space
132:
130:
126:
122:
117:
115:
111:
107:
103:
101:
96:
92:
88:
66:
51:
46:
40:
33:
19:
4231:Main results
3967:Set function
3895:Metric outer
3850:Decomposable
3760:
3707:Cylinder set
3620:
3543:
3524:
3505:
3444:
3438:
3432:
3389:
3385:
3379:
3352:
3346:
3333:
3300:
3294:
3288:
3257:
3251:
3146:
3142:Haar measure
3139:
3134:
3016:
2888:Banach space
2882:
2131:
2042:
2026:
2024:
1907:
1582:
1576:
1573:Sard's lemma
1315:
1252:
986:
963:
882:
857:monotonicity
713:
669:
264:
252:
248:
133:
118:
98:
90:
84:
18:Measure zero
4191:compact set
4158:of measures
4094:Pushforward
4087:Projections
4077:Logarithmic
3920:Probability
3910:Pre-measure
3692:Borel space
3610:of measures
2291:Obviously,
1654:, then the
756:. We have:
680:open covers
456:has length
4482:Set theory
4471:Categories
4163:in measure
3890:Maximising
3860:Equivalent
3754:Vitali set
3392:: 246â73.
3243:References
3161:meagre set
2045:Cantor set
1419:Cantor set
1161:such that
927:đ-algebra
796:definition
710:Properties
521:such that
261:Definition
255:Cantor set
207:such that
125:set theory
4277:Maharam's
4247:Dominated
4060:Intensity
4055:Hausdorff
3962:Saturated
3880:Invariant
3785:Types of
3744:Ï-algebra
3714:đ-system
3680:Borel set
3675:Baire set
3487:119174196
3454:1006.2675
3394:CiteSeerX
3224:Empty set
3176:−
3159:is not a
3077:μ
2993:∈
2941:⊆
2912:‖
2909:⋅
2906:‖
2885:separable
2879:Haar null
2835:−
2770:−
2757:−
2655:⊆
2595:−
2552:⊆
1849:λ
1831:λ
1827:⟺
1808:π
1780:∈
1711:⊂
1677:π
1671:λ
1668:×
1665:λ
1591:λ
1484:dimension
1235:ε
1215:…
1137:intervals
1123:…
1068:ε
1056:Given any
987:A subset
936:Σ
906:μ
900:Σ
867:μ
840:μ
823:countable
806:μ
773:∅
767:μ
737:μ
731:Σ
631:ε
609:
601:∞
586:∑
551:∞
536:⋃
532:⊆
499:−
467:
439:⊆
387:intervals
373:…
318:ε
287:real line
215:μ
195:Σ
192:∈
166:μ
160:Σ
121:empty set
114:intervals
112:union of
110:countable
32:Empty set
4294:Fubini's
4284:Egorov's
4252:Monotone
4211:variable
4189:Random:
4140:Strongly
4065:Lebesgue
4050:Harmonic
4040:Gaussian
4025:Counting
3992:Spectral
3987:Singular
3977:s-finite
3972:Ï-finite
3855:Discrete
3830:Complete
3787:Measures
3761:Null set
3649:function
3424:11511821
3212:See also
2982:for any
2724:preimage
2624:Because
2028:complete
1311:manifold
1088:sequence
1084:there is
923:đ-ideal
676:sequence
385:of open
265:Suppose
245:Examples
91:null set
39:Zero set
4206:process
4201:measure
4196:element
4135:Bochner
4109:Trivial
4104:Tangent
4082:Product
3940:Regular
3918:)
3905:Perfect
3878:)
3843:)
3835:Content
3825:Complex
3766:Support
3739:-system
3628:Measure
3479:4296513
3459:Bibcode
3416:2140632
3371:2162223
3325:1019152
3317:2324722
3236:Nothing
2673:and so
2152:be the
925:of the
921:form a
106:covered
100:measure
4272:Jordan
4257:Vitali
4216:vector
4145:Weakly
4007:Vector
3982:Signed
3935:Random
3876:Quasi-
3865:Finite
3845:Convex
3805:Banach
3795:Atomic
3623:spaces
3608:
3550:
3531:
3512:
3485:
3477:
3422:
3414:
3396:
3369:
3323:
3315:
3276:
2479:Since
2357:Hence
2020:spaces
1482:whose
1285:using
978:volume
970:length
606:length
582:
579:
569:
566:
464:length
4114:Young
4035:Euler
4030:Dirac
4002:Tight
3930:Radon
3900:Outer
3870:Inner
3820:Brown
3815:Borel
3810:Besov
3800:Baire
3483:S2CID
3449:arXiv
3420:S2CID
3367:JSTOR
3343:(PDF)
3313:JSTOR
3163:then
3133:is a
3113:then
2883:In a
2221:with
1388:dense
1307:cubes
826:union
752:be a
704:limit
647:then
108:by a
93:is a
4378:For
4267:Hahn
4123:Maps
4045:Haar
3916:Sub-
3670:Atom
3658:Sets
3548:ISBN
3529:ISBN
3510:ISBN
3274:ISBN
2256:and
1932:and
1904:Uses
1870:>
1735:and
1700:For
1352:all
974:area
964:The
821:Any
794:(by
714:Let
628:<
321:>
253:The
102:zero
89:, a
48:The
3467:doi
3404:doi
3357:doi
3353:125
3305:doi
3266:doi
3024:of
2089:of
1696::
1583:If
1425:in
1390:in
1382:of
1139:in
1135:of
1007:of
976:or
956:".
859:of
832:of
798:of
682:of
678:of
574:and
85:In
4473::
3481:.
3475:MR
3473:.
3465:.
3457:.
3445:41
3443:.
3418:.
3412:MR
3410:.
3402:.
3390:15
3388:.
3365:.
3351:.
3345:.
3321:MR
3319:.
3311:.
3301:96
3299:.
3272:.
3208:.
3144:.
3137:.
2279:1.
2035:.
1888:0.
1086:a
984:.
972:,
879:).
852:).
818:).
230:0.
3914:(
3874:(
3839:(
3737:Ï
3647:/
3621:L
3584:e
3577:t
3570:v
3556:.
3537:.
3518:.
3489:.
3469::
3461::
3451::
3426:.
3406::
3373:.
3359::
3327:.
3307::
3282:.
3268::
3184:A
3179:1
3172:A
3157:A
3121:A
3101:,
3098:0
3095:=
3092:)
3089:x
3086:+
3083:A
3080:(
3058:,
3055:x
3035:,
3032:X
3017:Ό
2999:.
2996:X
2990:x
2970:x
2967:+
2964:A
2944:X
2938:A
2918:.
2915:)
2903:,
2900:X
2897:(
2863:F
2843:.
2838:1
2831:g
2827:=
2824:h
2804:F
2784:)
2781:F
2778:(
2773:1
2766:)
2760:1
2753:g
2749:(
2746:=
2743:)
2740:F
2737:(
2734:g
2710:)
2707:F
2704:(
2701:f
2681:F
2661:,
2658:K
2652:F
2632:g
2612:.
2609:)
2606:E
2603:(
2598:1
2591:g
2587:=
2584:F
2564:)
2561:K
2558:(
2555:g
2549:E
2529:)
2526:K
2523:(
2520:g
2496:)
2493:x
2490:(
2487:f
2467:.
2464:x
2461:+
2458:)
2455:x
2452:(
2449:f
2446:=
2443:)
2440:x
2437:(
2434:g
2410:)
2407:K
2404:(
2401:f
2381:)
2376:c
2372:K
2368:(
2365:f
2345:.
2340:c
2336:K
2315:)
2310:c
2306:K
2302:(
2299:f
2276:=
2273:)
2270:1
2267:(
2264:f
2244:0
2241:=
2238:)
2235:0
2232:(
2229:f
2209:,
2206:]
2203:1
2200:,
2197:0
2194:[
2174:,
2169:c
2165:K
2140:f
2117:F
2097:K
2077:F
2057:,
2054:K
2006:p
2002:L
1980:g
1960:f
1940:g
1920:f
1885:=
1881:)
1877:}
1873:0
1866:)
1861:x
1857:A
1853:(
1846::
1843:x
1839:{
1835:(
1823:0
1820:=
1817:)
1814:A
1811:(
1789:,
1786:}
1783:A
1777:)
1774:y
1771:,
1768:x
1765:(
1762::
1759:y
1756:{
1753:=
1748:x
1744:A
1721:2
1716:R
1708:A
1680:.
1674:=
1640:2
1635:R
1612:R
1558:.
1553:2
1548:R
1526:.
1521:n
1516:R
1494:n
1468:n
1463:R
1438:;
1434:R
1403:.
1399:R
1369:Q
1340:,
1335:n
1330:R
1305:-
1293:n
1273:,
1268:n
1263:R
1238:.
1212:,
1207:2
1203:I
1199:,
1194:1
1190:I
1169:N
1148:R
1120:,
1115:2
1111:I
1107:,
1102:1
1098:I
1071:,
1038:R
1016:R
995:N
909:)
903:,
897:,
894:X
891:(
782:0
779:=
776:)
770:(
740:)
734:,
728:,
725:X
722:(
690:A
655:A
635:,
625:)
620:n
616:U
612:(
596:1
593:=
590:n
561:n
557:U
546:1
543:=
540:n
529:A
507:n
503:a
494:n
490:b
486:=
483:)
478:n
474:U
470:(
443:R
436:)
431:n
427:b
423:,
418:n
414:a
410:(
407:=
402:n
398:U
370:,
365:2
361:U
357:,
352:1
348:U
327:,
324:0
297:R
273:A
227:=
224:)
221:S
218:(
189:S
169:)
163:,
157:,
154:X
151:(
148:=
145:M
81:.
67:2
62:R
41:.
34:.
20:)
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