2029:
915:
536:
3715:
742:
366:
3612:
1081:
910:{\displaystyle \lambda ^{\!*\!}(E)=\inf \left\{\sum _{k=1}^{\infty }\operatorname {vol} (C_{k}):{(C_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of products of open intervals with }}E\subset \bigcup _{k=1}^{\infty }C_{k}\right\}.}
699:
3829:
2330:
of
Lebesgue-measurable sets are Lebesgue-measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions does not need to be closed under countable unions:
2457:
3918:
agrees with the
Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not
3578:
2227:
531:{\displaystyle \lambda ^{\!*\!}(E)=\inf \left\{\sum _{k=1}^{\infty }\ell (I_{k}):{(I_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of open intervals with }}E\subset \bigcup _{k=1}^{\infty }I_{k}\right\}.}
2500:
A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set. This property has been used as an alternative definition of
Lebesgue measurability. More precisely,
3228:
2666:
1747:
is used as a "mask" to "clip" that set, hinting at the existence of sets for which the
Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.)
1407:
That characterizes the
Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets
617:
3110:
1161:
2990:
734:
3033:
3710:{\displaystyle \lambda ^{*}(A)=\inf \left\{\sum _{B\in {\mathcal {C}}}\operatorname {vol} (B):{\mathcal {C}}{\text{ is a countable collection of boxes whose union covers }}A\right\}.}
970:
314:
3467:
2527:
2625:
355:
2553:
1972:
2897:
1878:
286:
245:
5428:
2939:
1998:
223:
2076:
978:
1338:
185:
2919:
2593:
2573:
2050:
1745:
1725:
1705:
1685:
1665:
1645:
1625:
1605:
1585:
1565:
1545:
1525:
1505:
1485:
1465:
1445:
1425:
1402:
1378:
1358:
1315:
1295:
1275:
1255:
1211:
1181:
1104:
938:
562:
5067:
622:
3738:
5635:
5145:
5652:
5162:
3973:
2334:
3494:
2131:
3116:
All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following):
5452:
4470:
4329:
3144:
4985:
3365:
4816:
2630:
1257:
of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals
4356:
2497:
is a
Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure.
5584:
5235:
4977:
17:
5740:
5510:
4763:
1384:
of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit
570:
5647:
5157:
134:
Henri
Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the
5566:
4258:
4107:
3314:
3064:
1113:
5114:
5104:
2945:
4914:
4823:
4587:
4000:
3899:
704:
5546:
5526:
2995:
4443:
1214:
1107:
941:
5642:
5576:
5480:
5363:
5152:
5099:
4993:
4899:
3995:
947:
291:
5018:
4998:
4962:
4886:
4606:
4322:
3898:
showed that the existence of sets that are not
Lebesgue-measurable is not provable within the framework of
3395:
2504:
1110:
are said to be
Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure:
5630:
5500:
5140:
4919:
4881:
4833:
2598:
322:
5594:
5536:
5393:
5045:
5013:
5003:
4924:
4891:
4522:
4431:
4274:
Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is
Lebesgue-measurable".
2532:
3488:, and the product symbol here represents a Cartesian product. The volume of this box is defined to be
1948:
5599:
5531:
5062:
4967:
4743:
4671:
3888:
2774:
2327:
1902:
holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the
5551:
5052:
1727:
must not have some curious properties which causes a discrepancy in the measure of another set when
1360:
is a subset of the union of the intervals, and so the intervals may include points which are not in
5625:
5604:
5541:
5135:
4581:
4512:
4017:
3135:
2778:
4448:
2876:
1861:
250:
228:
5406:
5228:
4904:
4662:
4622:
4315:
2267:
4123:
5556:
5485:
5378:
5358:
5187:
5087:
4909:
4631:
4477:
3867:
The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical
2759:
2099:
1757:
147:
1850:
set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of
5383:
5275:
4748:
4701:
4696:
4691:
4533:
4416:
4374:
4276:
3931:
3302:
3138:
3042:
2924:
2763:
1977:
1843:
is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.
190:
51:
31:
4250:
4244:
1076:{\displaystyle \lambda ^{\!*\!}(A)=\lambda ^{\!*\!}(A\cap E)+\lambda ^{\!*\!}(A\cap E^{c}).}
5561:
5447:
5332:
5057:
5023:
4931:
4641:
4596:
4438:
4361:
3333:
3234:
1899:
1381:
112:
8:
5717:
5505:
5388:
5327:
5296:
5040:
5030:
4876:
4840:
4666:
4395:
4352:
4191:
3966:
3278:
2055:
2012:
4718:
3945:
is a generalization of the Lebesgue measure that is useful for measuring the subsets of
1320:
152:
5687:
5589:
5495:
5442:
5368:
5301:
5221:
5192:
4952:
4937:
4636:
4517:
4495:
4293:
4225:
4077:
4005:
3903:
3884:
2904:
2578:
2558:
2323:
2035:
1913:
1730:
1710:
1690:
1670:
1650:
1630:
1610:
1590:
1570:
1550:
1530:
1510:
1490:
1470:
1450:
1430:
1410:
1387:
1363:
1343:
1300:
1280:
1260:
1240:
1222:
1196:
1166:
1089:
923:
547:
542:
5475:
5109:
4845:
4806:
4801:
4708:
4626:
4411:
4384:
4254:
4217:
4103:
4081:
3942:
3895:
2095:
1851:
1809:
694:{\displaystyle \operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})}
565:
135:
3356:
can be generated using countable unions and intersections from open or closed sets.
5677:
5437:
5337:
5292:
5035:
4796:
4786:
4771:
4738:
4733:
4723:
4601:
4576:
4391:
4285:
4207:
4069:
3920:
3824:{\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap A)+\lambda ^{*}(S\setminus A)\,.}
3317:
which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.
3298:
3290:
3132:
1888:
59:
5202:
5182:
4957:
4855:
4850:
4828:
4686:
4651:
4571:
4465:
4170:"Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras?"
3868:
2256:
1931:
1917:
1903:
1892:
541:
The above definition can be generalised to higher dimensions as follows. For any
62:
1317:. The total length of any covering interval set may overestimate the measure of
5490:
5306:
5092:
4947:
4942:
4753:
4728:
4681:
4611:
4591:
4551:
4541:
4338:
2236:
1790:
47:
5734:
5702:
5697:
5682:
5672:
5373:
5287:
5262:
5197:
4860:
4781:
4776:
4676:
4646:
4616:
4566:
4561:
4556:
4546:
4460:
4379:
4221:
4095:
3915:
3887:
with many surprising properties have been demonstrated, such as those of the
3596:
3121:
2767:
2005:
1775:
1184:
317:
108:
3289:-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the
5459:
5258:
4791:
4713:
4453:
3990:
3927:
3871:, which is independent from many of the conventional systems of axioms for
2483:
1939:
1923:
1802:
1707:
gives its Lebesgue measure. Intuitively, this condition means that the set
4490:
4169:
4146:
3835:
3262:
if, for every Δ > 0, it can be covered with countably many products of
2452:{\displaystyle \{\emptyset ,\{1,2,3,4\},\{1,2\},\{3,4\},\{1,3\},\{2,4\}\}}
4656:
3954:
1935:
1761:
35:
3573:{\displaystyle \operatorname {vol} (B)=\prod _{i=1}^{n}(b_{i}-a_{i})\,.}
5712:
5403:
4500:
4297:
4229:
4073:
4010:
3876:
3872:
2807:
2471:
2028:
2001:
1909:
1884:
1812:
of intervals and is Lebesgue-measurable, and its Lebesgue measure is
1226:
5707:
5692:
4482:
4426:
4421:
3965:
sets. The Hausdorff measure is not to be confused with the notion of
3364:
The modern construction of the Lebesgue measure is an application of
3267:
2479:
2240:
1916:
with respect to the Lebesgue measure. Their existence relies on the
1855:
1847:
1840:
1833:
4289:
4212:
4195:
4057:
2708:
2695:
2682:
2672:
5347:
5316:
5267:
5244:
4507:
4366:
3985:
3879:, which follows from the axiom, states that there exist subsets of
3249:
2799:
2467:
1607:
are subject to the outer measure. If for all possible such subsets
4307:
4043:
3962:
3693: is a countable collection of boxes whose union covers
2671:
A Lebesgue-measurable set can be "squeezed" between a containing
358:
3883:
that are not Lebesgue-measurable. Assuming the axiom of choice,
3324:
is Lebesgue-measurable, one usually tries to find a "nicer" set
4038:
2222:{\displaystyle \lambda (A)=|I_{1}|\cdot |I_{2}|\cdots |I_{n}|.}
1934:
Lebesgue measure (it can be obtained by small variation of the
1764:
is Lebesgue-measurable, and its Lebesgue measure is the length
82:
74:
55:
43:
3974:
there is no infinite-dimensional analogue of Lebesgue measure
1927:
5213:
859: is a sequence of products of open intervals with
2862:}, is also Lebesgue-measurable and has the same measure as
138:. Both were published as part of his dissertation in 1902.
78:
3313:-dimensional Lebesgue measure. An example of this is the
701:(a real number product) denote its volume. For any subset
115:. Sets that can be assigned a Lebesgue measure are called
3223:{\displaystyle \lambda (\times \times \cdots \times )=1.}
1218:
2000:, has a zero Lebesgue measure. In general, every proper
1297:
in a sense, since the union of these intervals contains
1237:
The first part of the definition states that the subset
73:= 1, 2, or 3, it coincides with the standard measure of
3359:
2661:{\displaystyle \lambda (G\setminus F)<\varepsilon }
1667:
have outer measures whose sum is the outer measure of
5409:
3741:
3615:
3497:
3398:
3301:
equivalent to it). On the other hand, a set may have
3147:
3067:
2998:
2948:
2927:
2907:
2879:
2633:
2601:
2581:
2561:
2535:
2507:
2337:
2134:
2058:
2038:
2019:
can be calculated in terms of Euler's gamma function.
1980:
1951:
1864:
1793:
between the two sets consists only of the end points
1733:
1713:
1693:
1673:
1653:
1633:
1613:
1593:
1573:
1553:
1533:
1513:
1493:
1473:
1453:
1433:
1413:
1390:
1366:
1346:
1323:
1303:
1283:
1263:
1243:
1199:
1169:
1116:
1092:
981:
950:
926:
745:
707:
625:
573:
550:
369:
325:
294:
253:
231:
193:
155:
3233:
The Lebesgue measure also has the property of being
5422:
4249:. Cambridge: Cambridge University Press. pp.
3823:
3709:
3572:
3461:
3222:
3104:
3061:) is also Lebesgue-measurable and has the measure
3027:
2984:
2933:
2913:
2891:
2660:
2619:
2587:
2567:
2547:
2521:
2451:
2259:) of the measures of the involved measurable sets.
2221:
2070:
2044:
1992:
1966:
1872:
1739:
1719:
1699:
1679:
1659:
1639:
1619:
1599:
1579:
1559:
1539:
1519:
1499:
1479:
1459:
1439:
1419:
1396:
1372:
1352:
1332:
1309:
1289:
1269:
1249:
1205:
1175:
1155:
1098:
1075:
964:
932:
909:
728:
693:
611:
556:
530:
349:
308:
280:
239:
217:
179:
4200:Transactions of the American Mathematical Society
4102:(3rd ed.). New York: Macmillan. p. 56.
3934:and is a generalization of the Lebesgue measure (
1141:
1137:
1045:
1041:
1015:
1011:
991:
987:
755:
751:
612:{\displaystyle C=I_{1}\times \cdots \times I_{n}}
379:
375:
335:
331:
5732:
4001:Lebesgue measure of the set of Liouville numbers
3638:
3073:
2529:is Lebesgue-measurable if and only if for every
2032:Translation invariance: The Lebesgue measure of
770:
480: is a sequence of open intervals with
394:
3266:intervals whose total volume is at most Δ. All
3105:{\displaystyle \left|\det(T)\right|\lambda (A)}
1156:{\displaystyle \lambda (E)=\lambda ^{\!*\!}(E)}
3724:to be Lebesgue-measurable if for every subset
5229:
4323:
4206:(1). American Mathematical Society: 107â112.
3909:
119:; the measure of the Lebesgue-measurable set
5068:RieszâMarkovâKakutani representation theorem
2992:is also Lebesgue-measurable and has measure
2985:{\displaystyle \delta A=\{\delta x:x\in A\}}
2979:
2958:
2446:
2443:
2431:
2425:
2413:
2407:
2395:
2389:
2377:
2371:
2347:
2338:
1380:. The Lebesgue outer measure emerges as the
3938:with addition is a locally compact group).
3902:in the absence of the axiom of choice (see
729:{\displaystyle E\subseteq \mathbb {R^{n}} }
5653:Vitale's random BrunnâMinkowski inequality
5236:
5222:
5163:Vitale's random BrunnâMinkowski inequality
4330:
4316:
3332:only by a null set (in the sense that the
3127:containing all products of intervals, and
2694:is Lebesgue-measurable then there exist a
4242:
4211:
4167:
4144:
3841:, and the Lebesgue measure is defined by
3817:
3566:
3455:
3028:{\displaystyle \delta ^{n}\lambda \,(A).}
3012:
2515:
1954:
1866:
958:
850:
720:
716:
471:
302:
233:
4055:
2280:) â„ 0 for every Lebesgue-measurable set
2243:disjoint Lebesgue-measurable sets, then
2027:
4273:
2266:is Lebesgue-measurable, then so is its
1627:of the real numbers, the partitions of
965:{\displaystyle A\subseteq \mathbb {R} }
309:{\displaystyle E\subseteq \mathbb {R} }
14:
5733:
4190:
4094:
4062:Annali di Matematica Pura ed Applicata
3834:These Lebesgue-measurable sets form a
3285:then it is a null set with respect to
5217:
4311:
3957:, for example, surfaces or curves in
3462:{\displaystyle B=\prod _{i=1}^{n}\,,}
2522:{\displaystyle E\subset \mathbb {R} }
1687:, then the outer Lebesgue measure of
50:, is the standard way of assigning a
5666:Applications & related
5176:Applications & related
4147:"What sets are Lebesgue-measurable?"
3360:Construction of the Lebesgue measure
3352:) is a null set) and then show that
3120:The Lebesgue-measurable sets form a
5585:Marcinkiewicz interpolation theorem
2777:on non-empty open sets, and so its
2620:{\displaystyle F\subset E\subset G}
2004:has a zero Lebesgue measure in its
350:{\displaystyle \lambda ^{\!*\!}(E)}
24:
5511:Symmetric decreasing rearrangement
5415:
4337:
4042:is also used, more strictly, as a
3686:
3659:
3320:In order to show that a given set
2792:is a Lebesgue-measurable set with
2341:
2247:is itself Lebesgue-measurable and
884:
794:
505:
418:
288:denote its length. For any subset
25:
5752:
4196:"A Jordan Curve of Positive Area"
3808:
2643:
2548:{\displaystyle \varepsilon >0}
1467:into two partitions: the part of
1404:most tightly and do not overlap.
5105:Lebesgue differentiation theorem
4986:Carathéodory's extension theorem
3860:for any Lebesgue-measurable set
3366:Carathéodory's extension theorem
1967:{\displaystyle \mathbb {R} ^{n}}
1832:, the area of the corresponding
1789:has the same measure, since the
944:, which requires that for every
85:. In general, it is also called
27:Concept of area in any dimension
4267:
4236:
4184:
4161:
4138:
4116:
4088:
4049:
4030:
3814:
3802:
3786:
3774:
3758:
3752:
3678:
3672:
3632:
3626:
3563:
3537:
3510:
3504:
3452:
3426:
3211:
3208:
3196:
3184:
3172:
3166:
3154:
3151:
3099:
3093:
3082:
3076:
3019:
3013:
2649:
2637:
2212:
2197:
2189:
2174:
2166:
2151:
2144:
2138:
2086:has the following properties:
1912:are examples of sets that are
1382:greatest lower bound (infimum)
1150:
1144:
1126:
1120:
1067:
1048:
1030:
1018:
1000:
994:
839:
825:
818:
805:
764:
758:
688:
675:
660:
647:
638:
632:
460:
446:
439:
426:
388:
382:
344:
338:
263:
257:
212:
200:
174:
162:
13:
1:
5481:Convergence almost everywhere
5243:
4023:
2023:
1895:that have Lebesgue measure 0.
1854:is 0, even though the set is
141:
3243:
3141:on that σ-algebra with
2892:{\displaystyle \delta >0}
2295:are Lebesgue-measurable and
1873:{\displaystyle \mathbb {R} }
1232:
1225:do exist; an example is the
1217:is not Lebesgue-measurable.
281:{\displaystyle \ell (I)=b-a}
240:{\displaystyle \mathbb {R} }
7:
5648:PrĂ©kopaâLeindler inequality
5501:Locally integrable function
5423:{\displaystyle L^{\infty }}
5158:PrĂ©kopaâLeindler inequality
4058:"Intégrale, Longueur, Aire"
3979:
3900:ZermeloâFraenkel set theory
2873:is Lebesgue-measurable and
2822:is Lebesgue-measurable and
2128:is Lebesgue-measurable and
1942:is another unusual example.
1750:
10:
5757:
5394:Square-integrable function
5100:Lebesgue's density theorem
3996:Lebesgue's density theorem
3910:Relation to other measures
3368:. It proceeds as follows.
3247:
3049:is a measurable subset of
2255:) is equal to the sum (or
1507:and the remaining part of
1447:as an instrument to split
1427:of the real numbers using
1213:that does not satisfy the
111:, in particular to define
5741:Measures (measure theory)
5665:
5643:MinkowskiâSteiner formula
5613:
5575:
5519:
5468:
5402:
5346:
5315:
5251:
5175:
5153:MinkowskiâSteiner formula
5123:
5083:
5076:
4976:
4968:Projection-valued measure
4869:
4762:
4531:
4404:
4345:
4243:Carothers, N. L. (2000).
3949:of lower dimensions than
3315:SmithâVolterraâCantor set
2758:Lebesgue measure is both
5626:Isoperimetric inequality
5136:Isoperimetric inequality
5115:VitaliâHahnâSaks theorem
4444:Carathéodory's criterion
2802:), then every subset of
2555:there exist an open set
2319:). (A consequence of 2.)
2082:The Lebesgue measure on
1547:: the set difference of
107:. It is used throughout
5631:BrunnâMinkowski theorem
5141:BrunnâMinkowski theorem
5010:Decomposition theorems
4046:of 3-dimensional volume
3720:We then define the set
2934:{\displaystyle \delta }
2490:is Lebesgue-measurable.
1993:{\displaystyle n\geq 2}
1587:. These partitions of
619:of open intervals, let
218:{\displaystyle I=(a,b)}
69:. For lower dimensions
5486:Convergence in measure
5424:
5188:Descriptive set theory
5088:Disintegration theorem
4523:Universally measurable
4134:– via Knowledge.
3930:can be defined on any
3825:
3711:
3574:
3536:
3463:
3425:
3224:
3106:
3029:
2986:
2935:
2915:
2893:
2662:
2621:
2589:
2569:
2549:
2523:
2453:
2223:
2079:
2072:
2046:
1994:
1968:
1874:
1741:
1721:
1701:
1681:
1661:
1641:
1621:
1601:
1581:
1561:
1541:
1521:
1501:
1487:which intersects with
1481:
1461:
1441:
1421:
1398:
1374:
1354:
1334:
1311:
1291:
1271:
1251:
1215:Carathéodory criterion
1207:
1177:
1163:. The set of all such
1157:
1108:Carathéodory criterion
1100:
1077:
966:
942:Carathéodory criterion
934:
911:
888:
798:
730:
695:
613:
558:
532:
509:
422:
351:
310:
282:
241:
219:
181:
5600:RieszâFischer theorem
5425:
5384:Polarization identity
4990:Convergence theorems
4449:Cylindrical Ï-algebra
4277:Annals of Mathematics
4172:. math stack exchange
4149:. math stack exchange
4056:Lebesgue, H. (1902).
3972:It can be shown that
3932:locally compact group
3889:BanachâTarski paradox
3826:
3712:
3575:
3516:
3464:
3405:
3389:is a set of the form
3303:topological dimension
3225:
3136:translation-invariant
3107:
3043:linear transformation
3030:
2987:
2936:
2916:
2894:
2663:
2622:
2590:
2570:
2550:
2524:
2454:
2224:
2073:
2047:
2031:
1995:
1969:
1875:
1742:
1722:
1702:
1682:
1662:
1642:
1622:
1602:
1582:
1562:
1542:
1522:
1502:
1482:
1462:
1442:
1422:
1399:
1375:
1355:
1335:
1312:
1292:
1272:
1252:
1208:
1178:
1158:
1101:
1078:
967:
935:
912:
868:
778:
731:
696:
614:
559:
533:
489:
402:
352:
311:
283:
247:of real numbers, let
242:
220:
182:
5605:RieszâThorin theorem
5448:Infimum and supremum
5407:
5333:Lebesgue integration
5058:Minkowski inequality
4932:Cylinder set measure
4817:Infinite-dimensional
4432:equivalence relation
4362:Lebesgue integration
4018:PeanoâJordan measure
3739:
3613:
3595:, we can define its
3495:
3396:
3334:symmetric difference
3270:sets are null sets.
3145:
3065:
2996:
2946:
2925:
2905:
2877:
2806:is also a null set.
2773:Lebesgue measure is
2631:
2599:
2579:
2559:
2533:
2505:
2335:
2132:
2056:
2036:
1978:
1949:
1900:axiom of determinacy
1862:
1731:
1711:
1691:
1671:
1651:
1631:
1611:
1591:
1571:
1551:
1531:
1511:
1491:
1471:
1451:
1431:
1411:
1388:
1364:
1344:
1321:
1301:
1281:
1261:
1241:
1197:
1167:
1114:
1090:
979:
948:
924:
743:
705:
623:
571:
548:
367:
323:
292:
251:
229:
191:
153:
113:Lebesgue integration
5567:Young's convolution
5506:Measurable function
5389:Pythagorean theorem
5379:Parseval's identity
5328:Integrable function
5053:Hölder's inequality
4915:of random variables
4877:Measurable function
4764:Particular measures
4353:Absolute continuity
3967:Hausdorff dimension
3885:non-measurable sets
3328:which differs from
3279:Hausdorff dimension
3037:More generally, if
2071:{\displaystyle A+t}
1938:construction). The
1223:non-measurable sets
123:is here denoted by
117:Lebesgue-measurable
90:-dimensional volume
18:Lebesgue measurable
5688:Probability theory
5590:Plancherel theorem
5496:Integral transform
5443:Chebyshev distance
5420:
5369:Euclidean distance
5302:Minkowski distance
5193:Probability theory
4518:Transverse measure
4496:Non-measurable set
4478:Locally measurable
4192:Osgood, William F.
4074:10.1007/BF02420592
4006:Non-measurable set
3821:
3707:
3665:
3570:
3459:
3309:and have positive
3220:
3102:
3025:
2982:
2931:
2911:
2889:
2810:, every subset of
2658:
2617:
2585:
2565:
2545:
2519:
2449:
2219:
2080:
2068:
2042:
1990:
1964:
1870:
1801:, which each have
1737:
1717:
1697:
1677:
1657:
1637:
1617:
1597:
1577:
1557:
1537:
1517:
1497:
1477:
1457:
1437:
1417:
1394:
1370:
1350:
1333:{\displaystyle E,}
1330:
1307:
1287:
1267:
1247:
1203:
1173:
1153:
1096:
1073:
962:
930:
907:
726:
691:
609:
554:
543:rectangular cuboid
528:
347:
306:
278:
237:
215:
180:{\displaystyle I=}
177:
60:higher dimensional
5728:
5727:
5661:
5660:
5476:Almost everywhere
5261: &
5211:
5210:
5171:
5170:
4900:almost everywhere
4846:Spherical measure
4744:Strictly positive
4672:Projection-valued
4412:Almost everywhere
4385:Probability space
4280:. Second Series.
3943:Hausdorff measure
3896:Robert M. Solovay
3694:
3646:
2914:{\displaystyle A}
2826:is an element of
2775:strictly positive
2766:, and so it is a
2588:{\displaystyle F}
2575:and a closed set
2568:{\displaystyle G}
2096:cartesian product
2045:{\displaystyle A}
1926:are simple plane
1889:Liouville numbers
1852:algebraic numbers
1810:Cartesian product
1740:{\displaystyle E}
1720:{\displaystyle E}
1700:{\displaystyle E}
1680:{\displaystyle A}
1660:{\displaystyle E}
1640:{\displaystyle A}
1620:{\displaystyle A}
1600:{\displaystyle A}
1580:{\displaystyle E}
1560:{\displaystyle A}
1540:{\displaystyle E}
1520:{\displaystyle A}
1500:{\displaystyle E}
1480:{\displaystyle A}
1460:{\displaystyle A}
1440:{\displaystyle E}
1420:{\displaystyle A}
1397:{\displaystyle E}
1373:{\displaystyle E}
1353:{\displaystyle E}
1310:{\displaystyle E}
1290:{\displaystyle E}
1270:{\displaystyle I}
1250:{\displaystyle E}
1206:{\displaystyle E}
1176:{\displaystyle E}
1106:that satisfy the
1099:{\displaystyle E}
933:{\displaystyle E}
860:
566:Cartesian product
557:{\displaystyle C}
481:
357:is defined as an
136:Lebesgue integral
16:(Redirected from
5748:
5678:Fourier analysis
5636:Milman's reverse
5619:
5617:Lebesgue measure
5611:
5610:
5595:RiemannâLebesgue
5438:Bounded function
5429:
5427:
5426:
5421:
5419:
5418:
5338:Taxicab geometry
5293:Measurable space
5238:
5231:
5224:
5215:
5214:
5146:Milman's reverse
5129:
5127:Lebesgue measure
5081:
5080:
4485:
4471:infimum/supremum
4392:Measurable space
4332:
4325:
4318:
4309:
4308:
4302:
4301:
4271:
4265:
4264:
4240:
4234:
4233:
4215:
4194:(January 1903).
4188:
4182:
4181:
4179:
4177:
4165:
4159:
4158:
4156:
4154:
4142:
4136:
4135:
4133:
4131:
4126:. 29 August 2022
4120:
4114:
4113:
4092:
4086:
4085:
4053:
4047:
4034:
3859:
3830:
3828:
3827:
3822:
3801:
3800:
3773:
3772:
3751:
3750:
3716:
3714:
3713:
3708:
3703:
3699:
3695:
3692:
3690:
3689:
3664:
3663:
3662:
3625:
3624:
3579:
3577:
3576:
3571:
3562:
3561:
3549:
3548:
3535:
3530:
3487:
3468:
3466:
3465:
3460:
3451:
3450:
3438:
3437:
3424:
3419:
3380:
3291:Euclidean metric
3229:
3227:
3226:
3221:
3111:
3109:
3108:
3103:
3089:
3085:
3034:
3032:
3031:
3026:
3008:
3007:
2991:
2989:
2988:
2983:
2940:
2938:
2937:
2932:
2920:
2918:
2917:
2912:
2898:
2896:
2895:
2890:
2781:is the whole of
2755:) = 0.
2681:and a contained
2667:
2665:
2664:
2659:
2626:
2624:
2623:
2618:
2594:
2592:
2591:
2586:
2574:
2572:
2571:
2566:
2554:
2552:
2551:
2546:
2528:
2526:
2525:
2520:
2518:
2458:
2456:
2455:
2450:
2228:
2226:
2225:
2220:
2215:
2210:
2209:
2200:
2192:
2187:
2186:
2177:
2169:
2164:
2163:
2154:
2077:
2075:
2074:
2069:
2051:
2049:
2048:
2043:
1999:
1997:
1996:
1991:
1973:
1971:
1970:
1965:
1963:
1962:
1957:
1893:uncountable sets
1891:are examples of
1879:
1877:
1876:
1871:
1869:
1839:Moreover, every
1831:
1788:
1773:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1686:
1684:
1683:
1678:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1626:
1624:
1623:
1618:
1606:
1604:
1603:
1598:
1586:
1584:
1583:
1578:
1566:
1564:
1563:
1558:
1546:
1544:
1543:
1538:
1527:which is not in
1526:
1524:
1523:
1518:
1506:
1504:
1503:
1498:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1426:
1424:
1423:
1418:
1403:
1401:
1400:
1395:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1276:
1274:
1273:
1268:
1256:
1254:
1253:
1248:
1212:
1210:
1209:
1204:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1143:
1142:
1105:
1103:
1102:
1097:
1082:
1080:
1079:
1074:
1066:
1065:
1047:
1046:
1017:
1016:
993:
992:
971:
969:
968:
963:
961:
939:
937:
936:
931:
916:
914:
913:
908:
903:
899:
898:
897:
887:
882:
861:
858:
856:
855:
854:
853:
837:
836:
817:
816:
797:
792:
757:
756:
735:
733:
732:
727:
725:
724:
723:
700:
698:
697:
692:
687:
686:
659:
658:
618:
616:
615:
610:
608:
607:
589:
588:
563:
561:
560:
555:
537:
535:
534:
529:
524:
520:
519:
518:
508:
503:
482:
479:
477:
476:
475:
474:
458:
457:
438:
437:
421:
416:
381:
380:
356:
354:
353:
348:
337:
336:
315:
313:
312:
307:
305:
287:
285:
284:
279:
246:
244:
243:
238:
236:
224:
222:
221:
216:
186:
184:
183:
178:
40:Lebesgue measure
21:
5756:
5755:
5751:
5750:
5749:
5747:
5746:
5745:
5731:
5730:
5729:
5724:
5657:
5614:
5609:
5571:
5547:HausdorffâYoung
5527:BabenkoâBeckner
5515:
5464:
5414:
5410:
5408:
5405:
5404:
5398:
5342:
5311:
5307:Sequence spaces
5247:
5242:
5212:
5207:
5203:Spectral theory
5183:Convex analysis
5167:
5124:
5119:
5072:
4972:
4920:in distribution
4865:
4758:
4588:Logarithmically
4527:
4483:
4466:Essential range
4400:
4341:
4336:
4306:
4305:
4290:10.2307/1970696
4272:
4268:
4261:
4241:
4237:
4213:10.2307/1986455
4189:
4185:
4175:
4173:
4168:Asaf Karagila.
4166:
4162:
4152:
4150:
4145:Asaf Karagila.
4143:
4139:
4129:
4127:
4122:
4121:
4117:
4110:
4093:
4089:
4054:
4050:
4035:
4031:
4026:
3982:
3912:
3904:Solovay's model
3869:axiom of choice
3842:
3796:
3792:
3768:
3764:
3746:
3742:
3740:
3737:
3736:
3691:
3685:
3684:
3658:
3657:
3650:
3645:
3641:
3620:
3616:
3614:
3611:
3610:
3557:
3553:
3544:
3540:
3531:
3520:
3496:
3493:
3492:
3485:
3478:
3473:
3446:
3442:
3433:
3429:
3420:
3409:
3397:
3394:
3393:
3372:
3362:
3297:(or any metric
3273:If a subset of
3252:
3246:
3146:
3143:
3142:
3072:
3068:
3066:
3063:
3062:
3003:
2999:
2997:
2994:
2993:
2947:
2944:
2943:
2926:
2923:
2922:
2906:
2903:
2902:
2878:
2875:
2874:
2832:translation of
2714:
2701:
2688:
2678:
2632:
2629:
2628:
2600:
2597:
2596:
2580:
2577:
2576:
2560:
2557:
2556:
2534:
2531:
2530:
2514:
2506:
2503:
2502:
2336:
2333:
2332:
2299:is a subset of
2257:infinite series
2211:
2205:
2201:
2196:
2188:
2182:
2178:
2173:
2165:
2159:
2155:
2150:
2133:
2130:
2129:
2123:
2114:
2107:
2057:
2054:
2053:
2037:
2034:
2033:
2026:
1979:
1976:
1975:
1958:
1953:
1952:
1950:
1947:
1946:
1918:axiom of choice
1904:axiom of choice
1887:and the set of
1865:
1863:
1860:
1859:
1813:
1778:
1765:
1753:
1732:
1729:
1728:
1712:
1709:
1708:
1692:
1689:
1688:
1672:
1669:
1668:
1652:
1649:
1648:
1632:
1629:
1628:
1612:
1609:
1608:
1592:
1589:
1588:
1572:
1569:
1568:
1552:
1549:
1548:
1532:
1529:
1528:
1512:
1509:
1508:
1492:
1489:
1488:
1472:
1469:
1468:
1452:
1449:
1448:
1432:
1429:
1428:
1412:
1409:
1408:
1389:
1386:
1385:
1365:
1362:
1361:
1345:
1342:
1341:
1322:
1319:
1318:
1302:
1299:
1298:
1282:
1279:
1278:
1262:
1259:
1258:
1242:
1239:
1238:
1235:
1198:
1195:
1194:
1168:
1165:
1164:
1136:
1132:
1115:
1112:
1111:
1091:
1088:
1087:
1061:
1057:
1040:
1036:
1010:
1006:
986:
982:
980:
977:
976:
957:
949:
946:
945:
925:
922:
921:
893:
889:
883:
872:
857:
849:
842:
838:
832:
828:
824:
812:
808:
793:
782:
777:
773:
750:
746:
744:
741:
740:
719:
715:
714:
706:
703:
702:
682:
678:
654:
650:
624:
621:
620:
603:
599:
584:
580:
572:
569:
568:
549:
546:
545:
514:
510:
504:
493:
478:
470:
463:
459:
453:
449:
445:
433:
429:
417:
406:
401:
397:
374:
370:
368:
365:
364:
330:
326:
324:
321:
320:
316:, the Lebesgue
301:
293:
290:
289:
252:
249:
248:
232:
230:
227:
226:
192:
189:
188:
154:
151:
150:
144:
28:
23:
22:
15:
12:
11:
5:
5754:
5744:
5743:
5726:
5725:
5723:
5722:
5721:
5720:
5715:
5705:
5700:
5695:
5690:
5685:
5680:
5675:
5669:
5667:
5663:
5662:
5659:
5658:
5656:
5655:
5650:
5645:
5640:
5639:
5638:
5628:
5622:
5620:
5608:
5607:
5602:
5597:
5592:
5587:
5581:
5579:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5523:
5521:
5517:
5516:
5514:
5513:
5508:
5503:
5498:
5493:
5491:Function space
5488:
5483:
5478:
5472:
5470:
5466:
5465:
5463:
5462:
5457:
5456:
5455:
5445:
5440:
5434:
5432:
5417:
5413:
5400:
5399:
5397:
5396:
5391:
5386:
5381:
5376:
5371:
5366:
5364:CauchyâSchwarz
5361:
5355:
5353:
5344:
5343:
5341:
5340:
5335:
5330:
5324:
5322:
5313:
5312:
5310:
5309:
5304:
5299:
5290:
5285:
5284:
5283:
5273:
5265:
5263:Hilbert spaces
5255:
5253:
5252:Basic concepts
5249:
5248:
5241:
5240:
5233:
5226:
5218:
5209:
5208:
5206:
5205:
5200:
5195:
5190:
5185:
5179:
5177:
5173:
5172:
5169:
5168:
5166:
5165:
5160:
5155:
5150:
5149:
5148:
5138:
5132:
5130:
5121:
5120:
5118:
5117:
5112:
5110:Sard's theorem
5107:
5102:
5097:
5096:
5095:
5093:Lifting theory
5084:
5078:
5074:
5073:
5071:
5070:
5065:
5060:
5055:
5050:
5049:
5048:
5046:FubiniâTonelli
5038:
5033:
5028:
5027:
5026:
5021:
5016:
5008:
5007:
5006:
5001:
4996:
4988:
4982:
4980:
4974:
4973:
4971:
4970:
4965:
4960:
4955:
4950:
4945:
4940:
4934:
4929:
4928:
4927:
4925:in probability
4922:
4912:
4907:
4902:
4896:
4895:
4894:
4889:
4884:
4873:
4871:
4867:
4866:
4864:
4863:
4858:
4853:
4848:
4843:
4838:
4837:
4836:
4826:
4821:
4820:
4819:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4768:
4766:
4760:
4759:
4757:
4756:
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4705:
4704:
4699:
4694:
4684:
4679:
4674:
4669:
4659:
4654:
4649:
4644:
4639:
4634:
4632:Locally finite
4629:
4619:
4614:
4609:
4604:
4599:
4594:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4538:
4536:
4529:
4528:
4526:
4525:
4520:
4515:
4510:
4505:
4504:
4503:
4493:
4488:
4480:
4475:
4474:
4473:
4463:
4458:
4457:
4456:
4446:
4441:
4436:
4435:
4434:
4424:
4419:
4414:
4408:
4406:
4402:
4401:
4399:
4398:
4389:
4388:
4387:
4377:
4372:
4364:
4359:
4349:
4347:
4346:Basic concepts
4343:
4342:
4339:Measure theory
4335:
4334:
4327:
4320:
4312:
4304:
4303:
4266:
4259:
4235:
4183:
4160:
4137:
4124:"Lebesgue-MaĂ"
4115:
4108:
4087:
4048:
4028:
4027:
4025:
4022:
4021:
4020:
4015:
4014:
4013:
4003:
3998:
3993:
3988:
3981:
3978:
3911:
3908:
3877:Vitali theorem
3832:
3831:
3820:
3816:
3813:
3810:
3807:
3804:
3799:
3795:
3791:
3788:
3785:
3782:
3779:
3776:
3771:
3767:
3763:
3760:
3757:
3754:
3749:
3745:
3718:
3717:
3706:
3702:
3698:
3688:
3683:
3680:
3677:
3674:
3671:
3668:
3661:
3656:
3653:
3649:
3644:
3640:
3637:
3634:
3631:
3628:
3623:
3619:
3581:
3580:
3569:
3565:
3560:
3556:
3552:
3547:
3543:
3539:
3534:
3529:
3526:
3523:
3519:
3515:
3512:
3509:
3506:
3503:
3500:
3483:
3476:
3470:
3469:
3458:
3454:
3449:
3445:
3441:
3436:
3432:
3428:
3423:
3418:
3415:
3412:
3408:
3404:
3401:
3361:
3358:
3248:Main article:
3245:
3242:
3231:
3230:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3131:is the unique
3114:
3113:
3101:
3098:
3095:
3092:
3088:
3084:
3081:
3078:
3075:
3071:
3035:
3024:
3021:
3018:
3015:
3011:
3006:
3002:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2930:
2910:
2888:
2885:
2882:
2867:
2816:
2814:is measurable.
2786:
2771:
2760:locally finite
2756:
2743:) =
2710:
2697:
2684:
2674:
2669:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2616:
2613:
2610:
2607:
2604:
2584:
2564:
2544:
2541:
2538:
2517:
2513:
2510:
2498:
2491:
2460:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2320:
2285:
2271:
2260:
2241:countably many
2237:disjoint union
2229:
2218:
2214:
2208:
2204:
2199:
2195:
2191:
2185:
2181:
2176:
2172:
2168:
2162:
2158:
2153:
2149:
2146:
2143:
2140:
2137:
2119:
2112:
2105:
2067:
2064:
2061:
2041:
2025:
2022:
2021:
2020:
2009:
1989:
1986:
1983:
1961:
1956:
1943:
1921:
1914:not measurable
1907:
1896:
1881:
1868:
1844:
1837:
1806:
1752:
1749:
1736:
1716:
1696:
1676:
1656:
1636:
1616:
1596:
1576:
1556:
1536:
1516:
1496:
1476:
1456:
1436:
1416:
1393:
1369:
1349:
1329:
1326:
1306:
1286:
1266:
1246:
1234:
1231:
1202:
1172:
1152:
1149:
1146:
1140:
1135:
1131:
1128:
1125:
1122:
1119:
1095:
1084:
1083:
1072:
1069:
1064:
1060:
1056:
1053:
1050:
1044:
1039:
1035:
1032:
1029:
1026:
1023:
1020:
1014:
1009:
1005:
1002:
999:
996:
990:
985:
960:
956:
953:
929:
918:
917:
906:
902:
896:
892:
886:
881:
878:
875:
871:
867:
864:
852:
848:
845:
841:
835:
831:
827:
823:
820:
815:
811:
807:
804:
801:
796:
791:
788:
785:
781:
776:
772:
769:
766:
763:
760:
754:
749:
722:
718:
713:
710:
690:
685:
681:
677:
674:
671:
668:
665:
662:
657:
653:
649:
646:
643:
640:
637:
634:
631:
628:
606:
602:
598:
595:
592:
587:
583:
579:
576:
553:
539:
538:
527:
523:
517:
513:
507:
502:
499:
496:
492:
488:
485:
473:
469:
466:
462:
456:
452:
448:
444:
441:
436:
432:
428:
425:
420:
415:
412:
409:
405:
400:
396:
393:
390:
387:
384:
378:
373:
346:
343:
340:
334:
329:
304:
300:
297:
277:
274:
271:
268:
265:
262:
259:
256:
235:
214:
211:
208:
205:
202:
199:
196:
176:
173:
170:
167:
164:
161:
158:
143:
140:
48:Henri Lebesgue
46:mathematician
42:, named after
34:, a branch of
32:measure theory
26:
9:
6:
4:
3:
2:
5753:
5742:
5739:
5738:
5736:
5719:
5716:
5714:
5711:
5710:
5709:
5706:
5704:
5703:Sobolev space
5701:
5699:
5698:Real analysis
5696:
5694:
5691:
5689:
5686:
5684:
5683:Lorentz space
5681:
5679:
5676:
5674:
5673:Bochner space
5671:
5670:
5668:
5664:
5654:
5651:
5649:
5646:
5644:
5641:
5637:
5634:
5633:
5632:
5629:
5627:
5624:
5623:
5621:
5618:
5612:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5582:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5524:
5522:
5518:
5512:
5509:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5473:
5471:
5467:
5461:
5458:
5454:
5451:
5450:
5449:
5446:
5444:
5441:
5439:
5436:
5435:
5433:
5431:
5411:
5401:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5374:Hilbert space
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5356:
5354:
5352:
5350:
5345:
5339:
5336:
5334:
5331:
5329:
5326:
5325:
5323:
5321:
5319:
5314:
5308:
5305:
5303:
5300:
5298:
5294:
5291:
5289:
5288:Measure space
5286:
5282:
5279:
5278:
5277:
5274:
5272:
5270:
5266:
5264:
5260:
5257:
5256:
5254:
5250:
5246:
5239:
5234:
5232:
5227:
5225:
5220:
5219:
5216:
5204:
5201:
5199:
5198:Real analysis
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5180:
5178:
5174:
5164:
5161:
5159:
5156:
5154:
5151:
5147:
5144:
5143:
5142:
5139:
5137:
5134:
5133:
5131:
5128:
5122:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5094:
5091:
5090:
5089:
5086:
5085:
5082:
5079:
5077:Other results
5075:
5069:
5066:
5064:
5063:RadonâNikodym
5061:
5059:
5056:
5054:
5051:
5047:
5044:
5043:
5042:
5039:
5037:
5036:Fatou's lemma
5034:
5032:
5029:
5025:
5022:
5020:
5017:
5015:
5012:
5011:
5009:
5005:
5002:
5000:
4997:
4995:
4992:
4991:
4989:
4987:
4984:
4983:
4981:
4979:
4975:
4969:
4966:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4935:
4933:
4930:
4926:
4923:
4921:
4918:
4917:
4916:
4913:
4911:
4908:
4906:
4903:
4901:
4898:Convergence:
4897:
4893:
4890:
4888:
4885:
4883:
4880:
4879:
4878:
4875:
4874:
4872:
4868:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4835:
4832:
4831:
4830:
4827:
4825:
4822:
4818:
4815:
4814:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4769:
4767:
4765:
4761:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4703:
4702:Outer regular
4700:
4698:
4697:Inner regular
4695:
4693:
4692:Borel regular
4690:
4689:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4664:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4624:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4593:
4589:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4539:
4537:
4535:
4530:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4502:
4499:
4498:
4497:
4494:
4492:
4489:
4487:
4481:
4479:
4476:
4472:
4469:
4468:
4467:
4464:
4462:
4459:
4455:
4452:
4451:
4450:
4447:
4445:
4442:
4440:
4437:
4433:
4430:
4429:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4409:
4407:
4403:
4397:
4393:
4390:
4386:
4383:
4382:
4381:
4380:Measure space
4378:
4376:
4373:
4371:
4369:
4365:
4363:
4360:
4358:
4354:
4351:
4350:
4348:
4344:
4340:
4333:
4328:
4326:
4321:
4319:
4314:
4313:
4310:
4299:
4295:
4291:
4287:
4283:
4279:
4278:
4270:
4262:
4260:9780521497565
4256:
4252:
4248:
4247:
4246:Real Analysis
4239:
4231:
4227:
4223:
4219:
4214:
4209:
4205:
4201:
4197:
4193:
4187:
4171:
4164:
4148:
4141:
4125:
4119:
4111:
4109:0-02-404151-3
4105:
4101:
4100:Real Analysis
4097:
4096:Royden, H. L.
4091:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4052:
4045:
4041:
4040:
4033:
4029:
4019:
4016:
4012:
4009:
4008:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3983:
3977:
3975:
3970:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3939:
3937:
3933:
3929:
3924:
3922:
3917:
3916:Borel measure
3907:
3905:
3901:
3897:
3892:
3890:
3886:
3882:
3878:
3874:
3870:
3865:
3863:
3857:
3853:
3849:
3845:
3840:
3838:
3818:
3811:
3805:
3797:
3793:
3789:
3783:
3780:
3777:
3769:
3765:
3761:
3755:
3747:
3743:
3735:
3734:
3733:
3731:
3727:
3723:
3704:
3700:
3696:
3681:
3675:
3669:
3666:
3654:
3651:
3647:
3642:
3635:
3629:
3621:
3617:
3609:
3608:
3607:
3605:
3601:
3598:
3597:outer measure
3594:
3590:
3586:
3567:
3558:
3554:
3550:
3545:
3541:
3532:
3527:
3524:
3521:
3517:
3513:
3507:
3501:
3498:
3491:
3490:
3489:
3486:
3479:
3456:
3447:
3443:
3439:
3434:
3430:
3421:
3416:
3413:
3410:
3406:
3402:
3399:
3392:
3391:
3390:
3388:
3384:
3379:
3375:
3369:
3367:
3357:
3355:
3351:
3347:
3343:
3339:
3335:
3331:
3327:
3323:
3318:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3271:
3269:
3265:
3261:
3257:
3251:
3241:
3239:
3237:
3217:
3214:
3205:
3202:
3199:
3193:
3190:
3187:
3181:
3178:
3175:
3169:
3163:
3160:
3157:
3148:
3140:
3137:
3134:
3130:
3126:
3124:
3119:
3118:
3117:
3096:
3090:
3086:
3079:
3069:
3060:
3056:
3052:
3048:
3044:
3040:
3036:
3022:
3016:
3009:
3004:
3000:
2976:
2973:
2970:
2967:
2964:
2961:
2955:
2952:
2949:
2941:
2928:
2908:
2886:
2883:
2880:
2872:
2868:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2838:, defined by
2837:
2833:
2829:
2825:
2821:
2817:
2815:
2811:
2809:
2803:
2801:
2795:
2791:
2787:
2784:
2780:
2776:
2772:
2769:
2768:Radon measure
2765:
2764:inner regular
2761:
2757:
2754:
2751: \
2750:
2746:
2742:
2739: \
2738:
2734:
2730:
2727: â
2726:
2723: â
2722:
2718:
2715:
2713:
2706:
2703:
2700:
2693:
2689:
2687:
2680:
2677:
2670:
2655:
2652:
2646:
2640:
2634:
2614:
2611:
2608:
2605:
2602:
2582:
2562:
2542:
2539:
2536:
2511:
2508:
2499:
2496:
2492:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2440:
2437:
2434:
2428:
2422:
2419:
2416:
2410:
2404:
2401:
2398:
2392:
2386:
2383:
2380:
2374:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2344:
2329:
2328:intersections
2325:
2321:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2283:
2279:
2275:
2272:
2269:
2265:
2261:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2216:
2206:
2202:
2193:
2183:
2179:
2170:
2160:
2156:
2147:
2141:
2135:
2127:
2122:
2118:
2111:
2104:
2101:
2097:
2093:
2089:
2088:
2087:
2085:
2078:are the same.
2065:
2062:
2059:
2039:
2030:
2018:
2016:
2013:volume of an
2010:
2007:
2006:ambient space
2003:
1987:
1984:
1981:
1959:
1944:
1941:
1937:
1933:
1929:
1925:
1924:Osgood curves
1922:
1919:
1915:
1911:
1908:
1905:
1901:
1897:
1894:
1890:
1886:
1882:
1857:
1853:
1849:
1845:
1842:
1838:
1835:
1829:
1825:
1821:
1817:
1811:
1807:
1804:
1800:
1796:
1792:
1786:
1782:
1777:
1776:open interval
1772:
1768:
1763:
1759:
1755:
1754:
1748:
1734:
1714:
1694:
1674:
1654:
1647:cut apart by
1634:
1614:
1594:
1574:
1554:
1534:
1514:
1494:
1474:
1454:
1434:
1414:
1405:
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1383:
1367:
1347:
1327:
1324:
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1284:
1264:
1244:
1230:
1228:
1224:
1220:
1216:
1200:
1191:
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1187:
1170:
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1109:
1093:
1070:
1062:
1058:
1054:
1051:
1042:
1037:
1033:
1027:
1024:
1021:
1012:
1007:
1003:
997:
988:
983:
975:
974:
973:
954:
951:
943:
927:
904:
900:
894:
890:
879:
876:
873:
869:
865:
862:
846:
843:
833:
829:
821:
813:
809:
802:
799:
789:
786:
783:
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774:
767:
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752:
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738:
737:
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708:
683:
679:
672:
669:
666:
663:
655:
651:
644:
641:
635:
629:
626:
604:
600:
596:
593:
590:
585:
581:
577:
574:
567:
551:
544:
525:
521:
515:
511:
500:
497:
494:
490:
486:
483:
467:
464:
454:
450:
442:
434:
430:
423:
413:
410:
407:
403:
398:
391:
385:
376:
371:
363:
362:
361:
360:
341:
332:
327:
319:
318:outer measure
298:
295:
275:
272:
269:
266:
260:
254:
225:, in the set
209:
206:
203:
197:
194:
171:
168:
165:
159:
156:
149:
139:
137:
132:
130:
126:
122:
118:
114:
110:
109:real analysis
106:
102:
98:
96:
91:
89:
84:
80:
76:
72:
68:
66:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5616:
5520:Inequalities
5460:Uniform norm
5348:
5317:
5280:
5268:
5126:
4978:Main results
4811:
4714:Set function
4642:Metric outer
4597:Decomposable
4454:Cylinder set
4367:
4281:
4275:
4269:
4245:
4238:
4203:
4199:
4186:
4176:26 September
4174:. Retrieved
4163:
4153:26 September
4151:. Retrieved
4140:
4128:. Retrieved
4118:
4099:
4090:
4065:
4061:
4051:
4037:
4032:
3991:Edison Farah
3971:
3958:
3955:submanifolds
3950:
3946:
3940:
3935:
3928:Haar measure
3925:
3913:
3893:
3880:
3866:
3861:
3855:
3851:
3847:
3843:
3836:
3833:
3729:
3725:
3721:
3719:
3603:
3599:
3592:
3588:
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3329:
3325:
3321:
3319:
3310:
3306:
3294:
3286:
3282:
3274:
3272:
3263:
3259:
3255:
3254:A subset of
3253:
3235:
3232:
3128:
3122:
3115:
3058:
3054:
3050:
3046:
3038:
2901:dilation of
2900:
2870:
2863:
2859:
2855:
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2839:
2835:
2831:
2827:
2823:
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2813:
2805:
2797:
2793:
2789:
2782:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2711:
2704:
2698:
2691:
2685:
2675:
2494:
2487:
2484:metric space
2475:
2463:
2316:
2312:
2308:
2304:
2300:
2296:
2292:
2288:
2281:
2277:
2273:
2263:
2252:
2248:
2244:
2232:
2125:
2120:
2116:
2109:
2102:
2091:
2083:
2081:
2014:
1945:Any line in
1940:dragon curve
1827:
1823:
1819:
1815:
1803:measure zero
1798:
1794:
1784:
1780:
1770:
1766:
1762:real numbers
1406:
1236:
1221:proves that
1192:
1185:
1085:
940:satisfy the
919:
540:
145:
133:
128:
124:
120:
116:
104:
103:, or simply
100:
94:
93:
87:
86:
70:
64:
39:
29:
5718:Von Neumann
5532:Chebyshev's
4938:compact set
4905:of measures
4841:Pushforward
4834:Projections
4824:Logarithmic
4667:Probability
4657:Pre-measure
4439:Borel space
4357:of measures
4284:(1): 1â56.
4068:: 231â359.
2942:defined by
2899:, then the
2830:, then the
1936:Peano curve
1910:Vitali sets
1756:Any closed
1227:Vitali sets
564:which is a
101:hypervolume
36:mathematics
5713:C*-algebra
5537:Clarkson's
4910:in measure
4637:Maximising
4607:Equivalent
4501:Vitali set
4024:References
4011:Vitali set
3873:set theory
3305:less than
3281:less than
2808:A fortiori
2719:such that
2690:. I.e, if
2595:such that
2474:subset of
2322:Countable
2268:complement
2024:Properties
2002:hyperplane
1885:Cantor set
1791:difference
920:Some sets
142:Definition
63:Euclidean
5708:*-algebra
5693:Quasinorm
5562:Minkowski
5453:Essential
5416:∞
5245:Lp spaces
5024:Maharam's
4994:Dominated
4807:Intensity
4802:Hausdorff
4709:Saturated
4627:Invariant
4532:Types of
4491:Ï-algebra
4461:đ-system
4427:Borel set
4422:Baire set
4222:0002-9947
4082:121256884
4036:The term
3894:In 1970,
3809:∖
3798:∗
3794:λ
3781:∩
3770:∗
3766:λ
3748:∗
3744:λ
3670:
3655:∈
3648:∑
3622:∗
3618:λ
3551:−
3518:∏
3502:
3407:∏
3299:Lipschitz
3268:countable
3244:Null sets
3194:×
3191:⋯
3188:×
3170:×
3149:λ
3091:λ
3010:λ
3001:δ
2974:∈
2962:δ
2950:δ
2929:δ
2881:δ
2798:) = 0 (a
2656:ε
2644:∖
2635:λ
2612:⊂
2606:⊂
2537:ε
2512:⊂
2480:Borel set
2478:(or even
2342:∅
2194:⋯
2171:⋅
2136:λ
2100:intervals
1985:≥
1848:countable
1841:Borel set
1834:rectangle
1233:Intuition
1139:∗
1134:λ
1118:λ
1086:The sets
1055:∩
1043:∗
1038:λ
1025:∩
1013:∗
1008:λ
989:∗
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955:⊆
885:∞
870:⋃
866:⊂
847:∈
803:
795:∞
780:∑
753:∗
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712:⊆
673:ℓ
670:×
667:⋯
664:×
645:ℓ
630:
597:×
594:⋯
591:×
506:∞
491:⋃
487:⊂
468:∈
424:ℓ
419:∞
404:∑
377:∗
372:λ
333:∗
328:λ
299:⊆
273:−
255:ℓ
5735:Category
5557:Markov's
5552:Hölder's
5542:Hanner's
5359:Bessel's
5297:function
5281:Lebesgue
5041:Fubini's
5031:Egorov's
4999:Monotone
4958:variable
4936:Random:
4887:Strongly
4812:Lebesgue
4797:Harmonic
4787:Gaussian
4772:Counting
4739:Spectral
4734:Singular
4724:s-finite
4719:Ï-finite
4602:Discrete
4577:Complete
4534:Measures
4508:Null set
4396:function
4098:(1988).
3986:4-volume
3980:See also
3921:complete
3839:-algebra
3480:≥
3376:∈
3260:null set
3250:Null set
3133:complete
3125:-algebra
2854: :
2800:null set
2486:), then
1932:positive
1826:−
1818:−
1769:−
1758:interval
1751:Examples
1340:because
1188:-algebra
1183:forms a
148:interval
146:For any
5577:Results
5276:Measure
4953:process
4948:measure
4943:element
4882:Bochner
4856:Trivial
4851:Tangent
4829:Product
4687:Regular
4665:)
4652:Perfect
4625:)
4590:)
4582:Content
4572:Complex
4513:Support
4486:-system
4375:Measure
4298:1970696
4230:1986455
4130:9 March
4044:synonym
3963:fractal
3953:, like
3587:subset
3238:-finite
3139:measure
3053:, then
2779:support
2707:and an
2303:, then
2124:, then
1898:If the
1277:covers
359:infimum
97:-volume
67:-spaces
56:subsets
52:measure
5430:spaces
5351:spaces
5320:spaces
5271:spaces
5259:Banach
5019:Jordan
5004:Vitali
4963:vector
4892:Weakly
4754:Vector
4729:Signed
4682:Random
4623:Quasi-
4612:Finite
4592:Convex
4552:Banach
4542:Atomic
4370:spaces
4355:
4296:
4257:
4228:
4220:
4106:
4080:
4039:volume
3875:. The
3606:) by:
3472:where
3129:λ
2482:, see
2472:closed
2466:is an
2324:unions
2115:à ⯠Ă
1974:, for
1928:curves
1774:. The
1193:A set
105:volume
83:volume
75:length
44:French
38:, the
4861:Young
4782:Euler
4777:Dirac
4749:Tight
4677:Radon
4647:Outer
4617:Inner
4567:Brown
4562:Borel
4557:Besov
4547:Baire
4294:JSTOR
4226:JSTOR
4078:S2CID
3344:) âȘ (
3258:is a
3041:is a
2235:is a
2094:is a
2017:-ball
1930:with
1856:dense
187:, or
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5615:For
5469:Maps
5125:For
5014:Hahn
4870:Maps
4792:Haar
4663:Sub-
4417:Atom
4405:Sets
4255:ISBN
4218:ISSN
4178:2015
4155:2015
4132:2023
4104:ISBN
3961:and
3941:The
3926:The
3914:The
3850:) =
3583:For
3381:. A
3371:Fix
3277:has
3045:and
2884:>
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2762:and
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2468:open
2326:and
2311:) â€
2291:and
2052:and
2011:The
1883:The
1846:Any
1808:Any
1797:and
1567:and
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4286:doi
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4070:doi
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3728:of
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