6796:
3029:
3175:
123:
courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that the measure of some sets could be infinite, or that the underlying space could
1328:
In a measure theoretical or probabilistic context setwise convergence is often referred to as strong convergence (as opposed to weak convergence). This can lead to some ambiguity because in functional analysis, strong convergence usually refers to convergence with respect to a norm.
634:
2916:
827:
3040:
292:
is required. Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence. As a matter of fact, when considering sequences of measures with uniformly bounded variation on a
4618:
The weak limit of a sequence of probability measures, provided it exists, is a probability measure. In general, if tightness is not assumed, a sequence of probability (or sub-probability) measures may not necessarily converge
5254:
2046:
1960:
4463:
4273:
4927:
4778:
in the context of functional analysis, weak convergence of measures is actually an example of weak-* convergence. The definitions of weak and weak-* convergences used in functional analysis are as follows:
1032:
5032:
1875:
1789:
4331:
4138:
4026:
1163:
5175:
5537:
3573:
930:
445:
3926:
3515:
191:
2611:
4744:
348:
4704:
4593:
4521:
1242:
1282:
5471:
3024:{\displaystyle \left\{\ U_{\varphi ,x,\delta }\ \left|\quad \varphi :S\to \mathbf {R} {\text{ is bounded and continuous, }}x\in \mathbf {R} {\text{ and }}\delta >0\ \right.\right\},}
2225:
2126:
527:
3473:
108:. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
3433:
1608:
284:
5101:
1493:
2399:
1196:
3400:
2324:
1682:
1581:
1076:
503:
5657:
4986:
3352:
3296:
3235:
2876:
2639:
461:. As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant.
5415:
720:
5579:
4656:
4396:
4206:
62:
on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure
5280:
5058:
4953:
3716:
2813:
2743:
2908:
1525:
5373:
5124:
3963:
3845:
3787:
3170:{\displaystyle U_{\varphi ,x,\delta }:=\left\{\ \mu \in {\mathcal {P}}(S)\ \left|\quad \left|\int _{S}\varphi \,\mathrm {d} \mu -x\right|<\delta \ \right.\right\}.}
1435:
1414:
1360:
of weak convergence of a sequence of measures, some of which are (apparently) more general than others. The equivalence of these conditions is sometimes known as the
5201:
124:
exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if
4831:
4080:
4053:
2689:
2524:
2477:
2426:
1709:
1662:
1635:
5337:
4764:
4613:
4545:
4354:
4164:
2771:
2721:
5599:
5304:
4874:
4851:
4800:
3865:
3807:
3694:
3319:
3259:
3198:
2836:
2791:
2659:
2544:
2497:
2446:
2367:
2347:
2268:
2248:
2169:
2149:
2070:
1984:
1898:
1816:
1729:
1548:
1388:
5581:. Applying the definition of weak-* convergence in terms of linear functionals, the characterization of vague convergence of measures is obtained. For compact
3748:
833:
6643:
5307:
839:
To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that we are given two probability measures
6721:
5286:
To illustrate how weak convergence of measures is an example of weak-* convergence, we give an example in terms of vague convergence (see above). Let
6738:
5206:
5834:
1991:
1905:
17:
195:
To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.
140:
The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge:
5703:
Madras, Neil; Sezer, Deniz (25 Feb 2011). "Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances".
836:. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.
4401:
4211:
143:
4879:
982:
6046:
5905:
4991:
863:
but we do not know which one of the two. Assume that these two measures have prior probabilities 0.5 each of being the true law of
1823:
1737:
4286:
4093:
119:
This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in
6561:
3972:
6392:
5789:
5757:
1101:
5129:
238:
5932:
5476:
3534:
881:
383:
3870:
6553:
3478:
66:
that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking
6866:
6339:
2549:
6733:
4709:
6839:
6817:
5874:
5851:
5818:
6810:
4661:
4550:
4478:
1204:
300:
6690:
6680:
2449:
1247:
5420:
6490:
6399:
6163:
5668:
4471:
In a probability setting, vague convergence and weak convergence of probability measures are equivalent assuming
3526:
629:{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=\sup _{f}\left\{\int _{X}f\,d\mu -\int _{X}f\,d\nu \right\}.}
2176:
2077:
6019:
6728:
6675:
6569:
6475:
3438:
3405:
1586:
6594:
6574:
6538:
6462:
6182:
5898:
5066:
1443:
5673:
3750:). The following spaces of test functions are commonly used in the convergence of probability measures.
3238:
2376:
6716:
6495:
6457:
6409:
2745:
with the usual topology), but it does not converge setwise. This is intuitively clear: we only know that
1171:
822:{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=2\cdot \sup _{A\in {\mathcal {F}}}|\mu (A)-\nu (A)|.}
3372:
2275:
1667:
1553:
1045:
875:
and that we are then asked to guess which one of the two distributions describes that law. The quantity
472:
6621:
6589:
6579:
6500:
6467:
6098:
6007:
5604:
4958:
3324:
3268:
3207:
2848:
2616:
1288:
5378:
1353:. It depends on a topology on the underlying space and thus is not a purely measure-theoretic notion.
6638:
6543:
6319:
6247:
6628:
6804:
6711:
6157:
6088:
5542:
4626:
4359:
4169:
6024:
5259:
5037:
4932:
3699:
2796:
2726:
6861:
6238:
6198:
5891:
2881:
1498:
6821:
6763:
6663:
6485:
6207:
6053:
5678:
5339:
of Radon measures is isomorphic to a subspace of the space of continuous linear functionals on
31:
5342:
5109:
3932:
3814:
3756:
469:
This is the strongest notion of convergence shown on this page and is defined as follows. Let
6324:
6277:
6272:
6267:
6109:
5992:
5950:
5683:
4472:
2049:
1963:
1420:
1399:
5180:
4623:
to a true probability measure, but rather to a sub-probability measure (a measure such that
3369:
There are many "arrow notations" for this kind of convergence: the most frequently used are
6633:
6599:
6507:
6217:
6172:
6014:
5937:
4809:
4058:
4031:
2667:
2502:
2455:
2404:
1687:
1640:
1613:
935:
then provides a sharp upper bound on the prior probability that our guess will be correct.
5313:
4766:
is not specified to be a probability measure is not guaranteed to imply weak convergence.
4749:
4598:
4530:
4339:
4149:
8:
6616:
6606:
6452:
6416:
6242:
5971:
5928:
3635:
2748:
2698:
1796:
1438:
1395:
711:
644:
67:
6294:
5659:, so in this case weak convergence of measures is a special case of weak-* convergence.
6768:
6528:
6513:
6212:
6093:
6071:
5863:
5828:
5730:
5712:
5584:
5289:
4859:
4836:
4785:
3850:
3792:
3679:
3304:
3244:
3183:
2821:
2776:
2644:
2529:
2482:
2431:
2352:
2332:
2253:
2233:
2154:
2134:
2055:
1969:
1883:
1878:
1801:
1714:
1533:
1373:
672:
656:
5779:
3721:
832:
The equivalence between these two definitions can be seen as a particular case of the
6685:
6421:
6382:
6377:
6284:
6202:
5987:
5960:
5870:
5847:
5814:
5785:
5753:
3576:
2842:
5734:
2723:
converges weakly to the Dirac measure located at 0 (if we view these as measures on
1325:
converges setwise to
Lebesgue measure, but it does not converge in total variation.
6702:
6611:
6387:
6372:
6362:
6347:
6314:
6309:
6299:
6177:
6152:
5967:
5722:
4769:
1792:
1079:
506:
235:
formalizes the assertion that the measure of each measurable set should converge:
6778:
6758:
6533:
6431:
6426:
6404:
6262:
6227:
6147:
6041:
3597:
3201:
510:
365:
formalizes the assertion that the measure of all measurable sets should converge
5203:. That is, convergence occurs in the point-wise sense. In this case, one writes
6668:
6523:
6518:
6329:
6304:
6257:
6187:
6167:
6127:
6117:
5914:
5249:{\displaystyle \varphi _{n}\mathrel {\stackrel {w^{*}}{\rightarrow }} \varphi }
4524:
4142:
2327:
1350:
42:
202:
requires this convergence to take place for every continuous bounded function
6855:
6773:
6436:
6357:
6352:
6252:
6222:
6192:
6142:
6137:
6132:
6122:
6036:
5955:
4775:
3355:
2692:
6367:
6289:
6029:
3262:
1391:
696:
676:
659:, where the definition is of the same form, but the supremum is taken over
294:
134:
6066:
6232:
3359:
1338:
38:
5811:
Gradient Flows in Metric Spaces and in the Space of
Probability Measures
5808:
6076:
2910:. The weak topology is generated by the following basis of open sets:
2839:
2129:
1357:
1342:
5726:
6058:
6002:
5997:
3363:
2041:{\displaystyle \liminf \operatorname {E} _{n}\geq \operatorname {E} }
1955:{\displaystyle \limsup \operatorname {E} _{n}\leq \operatorname {E} }
224:
to be approximated equally well (thus, convergence is non-uniform in
111:
Three of the most common notions of convergence are described below.
3496:
3456:
1265:
1222:
6083:
5942:
2228:
1349:
is one of many types of convergence relating to the convergence of
938:
Given the above definition of total variation distance, a sequence
120:
5717:
4458:{\displaystyle \int _{X}f\,d\mu _{n}\rightarrow \int _{X}f\,d\mu }
4268:{\displaystyle \int _{X}f\,d\mu _{n}\rightarrow \int _{X}f\,d\mu }
5883:
4922:{\displaystyle \varphi \left(x_{n}\right)\rightarrow \varphi (x)}
4770:
Weak convergence of measures as an example of weak-* convergence
4468:
In general, these two convergence notions are not equivalent.
5784:. Internet Archive. New York, Academic Press. pp. 84â99.
1583:) if any of the following equivalent conditions is true (here
1027:{\displaystyle \|\mu _{n}-\mu \|_{\text{TV}}<\varepsilon .}
699:, the total variation metric coincides with the Radon metric.
5027:{\displaystyle x_{n}\mathrel {\stackrel {w}{\rightarrow }} x}
1870:{\displaystyle \operatorname {E} _{n}\to \operatorname {E} }
1784:{\displaystyle \operatorname {E} _{n}\to \operatorname {E} }
3156:
3010:
464:
206:. This notion treats convergence for different functions
4326:{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}
4133:{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}
947:
of measures defined on the same measure space is said to
4021:{\displaystyle C_{c}\subset C_{0}\subset C_{B}\subset C}
2818:
This definition of weak convergence can be extended for
210:
independently of one another, i.e., different functions
3520:
1158:{\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}
5170:{\displaystyle \varphi _{n}(x)\rightarrow \varphi (x)}
303:
5607:
5587:
5545:
5532:{\displaystyle \varphi _{n}(f)=\int _{X}f\,d\mu _{n}}
5479:
5423:
5381:
5345:
5316:
5292:
5262:
5209:
5183:
5132:
5112:
5069:
5040:
4994:
4961:
4935:
4882:
4862:
4839:
4812:
4788:
4752:
4712:
4664:
4629:
4601:
4553:
4533:
4481:
4404:
4362:
4342:
4289:
4214:
4172:
4152:
4096:
4061:
4034:
3975:
3935:
3873:
3853:
3817:
3795:
3759:
3724:
3702:
3682:
3568:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
3537:
3481:
3441:
3408:
3375:
3327:
3307:
3271:
3247:
3210:
3186:
3043:
2919:
2884:
2851:
2824:
2799:
2779:
2751:
2729:
2701:
2670:
2647:
2619:
2552:
2532:
2505:
2485:
2458:
2434:
2407:
2379:
2355:
2335:
2278:
2256:
2236:
2179:
2157:
2137:
2080:
2058:
1994:
1972:
1908:
1886:
1826:
1804:
1740:
1717:
1690:
1670:
1643:
1616:
1589:
1556:
1536:
1501:
1446:
1423:
1402:
1376:
1250:
1207:
1174:
1104:
1048:
985:
925:{\displaystyle {2+\|\mu -\nu \|_{\text{TV}} \over 4}}
884:
723:
714:, then the total variation distance is also given by
530:
475:
440:{\displaystyle |\mu _{n}(A)-\mu (A)|<\varepsilon }
386:
241:
146:
49:. For an intuitive general sense of what is meant by
4475:. That is, a tight sequence of probability measures
3921:{\displaystyle \lim _{|x|\rightarrow \infty }f(x)=0}
3671:
5862:
5702:
5651:
5593:
5573:
5531:
5465:
5409:
5367:
5331:
5298:
5274:
5248:
5195:
5169:
5118:
5095:
5052:
5026:
4980:
4947:
4921:
4868:
4845:
4825:
4794:
4758:
4738:
4698:
4650:
4607:
4587:
4539:
4515:
4457:
4390:
4348:
4325:
4267:
4200:
4158:
4132:
4074:
4047:
4020:
3957:
3920:
3859:
3839:
3801:
3781:
3742:
3710:
3688:
3664:) in the sense of weak convergence of measures on
3567:
3510:{\displaystyle P_{n}\xrightarrow {\mathcal {D}} P}
3509:
3467:
3427:
3394:
3346:
3313:
3290:
3253:
3229:
3192:
3169:
3023:
2902:
2870:
2830:
2807:
2785:
2765:
2737:
2715:
2683:
2653:
2633:
2605:
2538:
2518:
2491:
2471:
2440:
2420:
2393:
2361:
2341:
2318:
2262:
2242:
2219:
2163:
2143:
2120:
2064:
2040:
1978:
1954:
1892:
1869:
1810:
1783:
1723:
1703:
1676:
1656:
1629:
1602:
1575:
1542:
1519:
1487:
1429:
1408:
1382:
1276:
1236:
1190:
1157:
1070:
1037:
1026:
924:
871:single sample distributed according to the law of
821:
683:ranging over the set of continuous functions from
663:ranging over the set of measurable functions from
628:
497:
439:
342:
278:
186:{\displaystyle \int f\,d\mu _{n}\to \int f\,d\mu }
185:
5809:Ambrosio, L., Gigli, N. & Savaré, G. (2005).
2878:, the set of all probability measures defined on
2606:{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}
6853:
4802:be a topological vector space or Banach space.
4739:{\displaystyle \mu _{n}{\overset {v}{\to }}\mu }
3875:
3237:is metrizable and separable, for example by the
2554:
2279:
2180:
2081:
1995:
1909:
1106:
760:
561:
354:. As before, this convergence is non-uniform in
1332:
343:{\textstyle \int f\,d\mu _{n}\to \int f\,d\mu }
297:, setwise convergence implies the convergence
4699:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
4588:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
4516:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
1237:{\displaystyle \mu _{n}\xrightarrow {sw} \mu }
5899:
5306:be a locally compact Hausdorff space. By the
1277:{\displaystyle \mu _{n}\xrightarrow {s} \mu }
6644:RieszâMarkovâKakutani representation theorem
5869:. New York, NY: John Wiley & Sons, Inc.
5846:. New York, NY: John Wiley & Sons, Inc.
5833:: CS1 maint: multiple names: authors list (
5466:{\displaystyle \varphi _{n}\in C_{0}(X)^{*}}
4658:). Thus, a sequence of probability measures
1006:
986:
907:
894:
675:at most 1; and also in contrast to the
5860:
5841:
655:. This is in contrast, for example, to the
133:is a sequence of probability measures on a
6739:Vitale's random BrunnâMinkowski inequality
5906:
5892:
6840:Learn how and when to remove this message
5716:
5515:
4690:
4579:
4507:
4448:
4418:
4317:
4258:
4228:
4124:
3965:the class of continuous bounded functions
3704:
3558:
3126:
2220:{\displaystyle \liminf P_{n}(U)\geq P(U)}
2121:{\displaystyle \limsup P_{n}(C)\leq P(C)}
1457:
955:in total variation distance if for every
611:
588:
513:distance between two (positive) measures
333:
310:
176:
153:
6803:This article includes a list of general
5813:. Basel: ETH ZĂŒrich, BirkhĂ€user Verlag.
1529:converge weakly to a probability measure
114:
5689:
3468:{\displaystyle P_{n}\xrightarrow {w} P}
3321:is separable, it naturally embeds into
1091:is said to converge setwise to a limit
465:Total variation convergence of measures
14:
6854:
5747:
3428:{\displaystyle P_{n}\rightharpoonup P}
2978: is bounded and continuous,
1603:{\displaystyle \operatorname {E} _{n}}
279:{\displaystyle \mu _{n}(A)\to \mu (A)}
5887:
5777:
5096:{\displaystyle \varphi _{n}\in V^{*}}
4082:with respect to uniform convergence.
3809:each vanishing outside a compact set.
2845:. It also defines a weak topology on
1488:{\displaystyle P_{n}\,(n=1,2,\dots )}
1287:For example, as a consequence of the
6789:
6752:Applications & related
5773:
5771:
5769:
5375:. Therefore, for each Radon measure
4085:
3521:Weak convergence of random variables
2394:{\displaystyle S\equiv \mathbf {R} }
350:for any bounded measurable function
5865:Convergence of Probability Measures
4278:
1191:{\displaystyle A\in {\mathcal {F}}}
679:, where the supremum is taken over
91:to ensure the 'difference' between
45:, there are various notions of the
24:
6809:it lacks sufficient corresponding
5913:
5802:
5269:
5047:
4942:
3895:
3847:the class of continuous functions
3789:the class of continuous functions
3549:
3541:
3498:
3395:{\displaystyle P_{n}\Rightarrow P}
3330:
3274:
3213:
3128:
3085:
2894:
2854:
2564:
2319:{\displaystyle \lim P_{n}(A)=P(A)}
2023:
1999:
1937:
1913:
1852:
1828:
1766:
1742:
1677:{\displaystyle \operatorname {E} }
1671:
1591:
1576:{\displaystyle P_{n}\Rightarrow P}
1511:
1424:
1183:
1116:
1071:{\displaystyle (X,{\mathcal {F}})}
1060:
772:
498:{\displaystyle (X,{\mathcal {F}})}
487:
288:Again, no uniformity over the set
53:, consider a sequence of measures
25:
6878:
5766:
5652:{\displaystyle C_{0}(X)=C_{B}(X)}
4981:{\displaystyle \varphi \in V^{*}}
3672:Comparison with vague convergence
3347:{\displaystyle {\mathcal {P}}(S)}
3291:{\displaystyle {\mathcal {P}}(S)}
3230:{\displaystyle {\mathcal {P}}(S)}
2871:{\displaystyle {\mathcal {P}}(S)}
2634:{\displaystyle x\in \mathbf {R} }
2450:cumulative distribution functions
1437:. A bounded sequence of positive
6794:
6681:Lebesgue differentiation theorem
6562:Carathéodory's extension theorem
5410:{\displaystyle \mu _{n}\in M(X)}
5104:converges in the weak-* topology
4774:Despite having the same name as
2988:
2972:
2801:
2731:
2664:For example, the sequence where
2627:
2387:
639:Here the supremum is taken over
214:may require different values of
5669:Convergence of random variables
5417:, there is a linear functional
3696:be a metric space (for example
3527:Convergence of random variables
3107:
2958:
1038:Setwise convergence of measures
867:. Assume now that we are given
847:, as well as a random variable
5781:A course in probability theory
5741:
5696:
5646:
5640:
5624:
5618:
5568:
5562:
5496:
5490:
5454:
5447:
5404:
5398:
5362:
5356:
5326:
5320:
5266:
5224:
5164:
5158:
5152:
5149:
5143:
5044:
5009:
4939:
4916:
4910:
4904:
4725:
4679:
4665:
4639:
4633:
4568:
4554:
4496:
4482:
4432:
4385:
4379:
4242:
4195:
4189:
3952:
3946:
3909:
3903:
3892:
3888:
3880:
3834:
3828:
3776:
3770:
3737:
3725:
3562:
3538:
3419:
3386:
3341:
3335:
3285:
3279:
3224:
3218:
3096:
3090:
2968:
2897:
2885:
2865:
2859:
2600:
2594:
2585:
2579:
2561:
2313:
2307:
2298:
2292:
2214:
2208:
2199:
2193:
2115:
2109:
2100:
2094:
2035:
2029:
2017:
2011:
1949:
1943:
1931:
1925:
1864:
1858:
1849:
1846:
1840:
1778:
1772:
1763:
1760:
1754:
1567:
1514:
1502:
1482:
1458:
1152:
1146:
1137:
1131:
1113:
1065:
1049:
812:
808:
802:
793:
787:
780:
740:
726:
547:
533:
492:
476:
427:
423:
417:
408:
402:
388:
324:
273:
267:
261:
258:
252:
167:
13:
1:
5861:Billingsley, Patrick (1999).
5842:Billingsley, Patrick (1995).
5574:{\displaystyle f\in C_{0}(X)}
4651:{\displaystyle \mu (X)\leq 1}
4391:{\displaystyle f\in C_{B}(X)}
4201:{\displaystyle f\in C_{c}(X)}
1356:There are several equivalent
457:and for every measurable set
5308:Riesz-Representation theorem
5275:{\displaystyle n\to \infty }
5053:{\displaystyle n\to \infty }
4948:{\displaystyle n\to \infty }
3711:{\displaystyle \mathbb {R} }
2808:{\displaystyle \mathbf {R} }
2738:{\displaystyle \mathbf {R} }
2401:with its usual topology, if
1333:Weak convergence of measures
1300:of measures on the interval
1201:Typical arrow notations are
643:ranging over the set of all
18:Weak convergence of measures
7:
6734:PrĂ©kopaâLeindler inequality
5662:
2903:{\displaystyle (S,\Sigma )}
2793:because of the topology of
1684:denotes expectation or the
1610:denotes expectation or the
1520:{\displaystyle (S,\Sigma )}
363:total variation convergence
10:
6883:
6676:Lebesgue's density theorem
3524:
70:; for any error tolerance
29:
6867:Convergence (mathematics)
6751:
6729:MinkowskiâSteiner formula
6699:
6659:
6652:
6552:
6544:Projection-valued measure
6445:
6338:
6107:
5980:
5921:
4527:to a probability measure
3619:) to the random variable
834:MongeâKantorovich duality
6712:Isoperimetric inequality
6691:VitaliâHahnâSaks theorem
6020:Carathéodory's criterion
5368:{\displaystyle C_{0}(X)}
5119:{\displaystyle \varphi }
3958:{\displaystyle C_{B}(X)}
3840:{\displaystyle C_{0}(X)}
3782:{\displaystyle C_{c}(X)}
30:Not to be confused with
6824:more precise citations.
6717:BrunnâMinkowski theorem
6586:Decomposition theorems
5844:Probability and Measure
5778:Chung, Kai Lai (1974).
4283:A sequence of measures
4090:A sequence of measures
3354:as the (closed) set of
1430:{\displaystyle \Sigma }
1409:{\displaystyle \sigma }
81:sufficiently large for
51:convergence of measures
47:convergence of measures
6764:Descriptive set theory
6664:Disintegration theorem
6099:Universally measurable
5748:Klenke, Achim (2006).
5653:
5595:
5575:
5533:
5467:
5411:
5369:
5333:
5300:
5276:
5250:
5197:
5196:{\displaystyle x\in V}
5171:
5120:
5097:
5054:
5028:
4982:
4949:
4923:
4870:
4847:
4827:
4796:
4760:
4740:
4700:
4652:
4609:
4589:
4541:
4517:
4459:
4392:
4350:
4327:
4269:
4202:
4160:
4134:
4076:
4049:
4022:
3959:
3922:
3861:
3841:
3803:
3783:
3744:
3712:
3690:
3653:) converges weakly to
3583:be a metric space. If
3569:
3511:
3469:
3429:
3396:
3348:
3315:
3292:
3255:
3231:
3194:
3171:
3025:
2904:
2872:
2832:
2809:
2787:
2767:
2739:
2717:
2685:
2655:
2635:
2607:
2540:
2520:
2493:
2473:
2442:
2422:
2395:
2363:
2343:
2320:
2264:
2244:
2221:
2165:
2145:
2122:
2066:
2042:
1980:
1956:
1894:
1871:
1812:
1785:
1725:
1705:
1678:
1658:
1631:
1604:
1577:
1544:
1521:
1489:
1431:
1410:
1384:
1289:RiemannâLebesgue lemma
1278:
1238:
1192:
1159:
1072:
1028:
926:
823:
630:
499:
441:
344:
280:
187:
32:Convergence in measure
6566:Convergence theorems
6025:Cylindrical Ï-algebra
5684:Tightness of measures
5674:LĂ©vyâProkhorov metric
5654:
5596:
5576:
5534:
5468:
5412:
5370:
5334:
5301:
5277:
5251:
5198:
5172:
5121:
5098:
5055:
5029:
4983:
4950:
4924:
4871:
4848:
4828:
4826:{\displaystyle x_{n}}
4797:
4761:
4741:
4701:
4653:
4610:
4590:
4542:
4518:
4460:
4393:
4351:
4328:
4270:
4203:
4161:
4135:
4077:
4075:{\displaystyle C_{c}}
4050:
4048:{\displaystyle C_{0}}
4023:
3960:
3923:
3862:
3842:
3804:
3784:
3745:
3713:
3691:
3570:
3512:
3470:
3430:
3397:
3349:
3316:
3293:
3256:
3239:LĂ©vyâProkhorov metric
3232:
3195:
3172:
3026:
2905:
2873:
2833:
2810:
2788:
2768:
2740:
2718:
2686:
2684:{\displaystyle P_{n}}
2656:
2636:
2608:
2541:
2521:
2519:{\displaystyle P_{n}}
2499:, respectively, then
2494:
2474:
2472:{\displaystyle P_{n}}
2443:
2423:
2421:{\displaystyle F_{n}}
2396:
2364:
2344:
2321:
2265:
2245:
2222:
2166:
2146:
2123:
2067:
2050:lower semi-continuous
2043:
1981:
1964:upper semi-continuous
1957:
1895:
1872:
1813:
1786:
1726:
1711:norm with respect to
1706:
1704:{\displaystyle L^{1}}
1679:
1659:
1657:{\displaystyle P_{n}}
1637:norm with respect to
1632:
1630:{\displaystyle L^{1}}
1605:
1578:
1545:
1522:
1490:
1432:
1411:
1385:
1279:
1239:
1193:
1160:
1073:
1029:
927:
824:
631:
500:
442:
345:
281:
188:
115:Informal descriptions
6634:Minkowski inequality
6508:Cylinder set measure
6393:Infinite-dimensional
6008:equivalence relation
5938:Lebesgue integration
5690:Notes and references
5605:
5585:
5543:
5477:
5421:
5379:
5343:
5332:{\displaystyle M(X)}
5314:
5290:
5260:
5207:
5181:
5130:
5110:
5067:
5038:
4992:
4959:
4933:
4880:
4860:
4837:
4810:
4786:
4759:{\displaystyle \mu }
4750:
4710:
4662:
4627:
4608:{\displaystyle \mu }
4599:
4595:converges weakly to
4551:
4540:{\displaystyle \mu }
4531:
4479:
4402:
4360:
4349:{\displaystyle \mu }
4340:
4287:
4212:
4170:
4159:{\displaystyle \mu }
4150:
4094:
4059:
4032:
3973:
3933:
3871:
3851:
3815:
3793:
3757:
3722:
3700:
3680:
3668:, as defined above.
3636:pushforward measures
3535:
3479:
3439:
3406:
3373:
3325:
3305:
3269:
3245:
3208:
3184:
3041:
2917:
2882:
2849:
2822:
2797:
2777:
2749:
2727:
2699:
2668:
2645:
2617:
2550:
2530:
2526:converges weakly to
2503:
2483:
2456:
2432:
2405:
2377:
2353:
2333:
2276:
2254:
2234:
2177:
2155:
2135:
2078:
2056:
1992:
1970:
1906:
1884:
1877:for all bounded and
1824:
1802:
1797:continuous functions
1738:
1715:
1688:
1668:
1641:
1614:
1587:
1554:
1534:
1499:
1444:
1439:probability measures
1421:
1400:
1374:
1248:
1205:
1172:
1102:
1046:
983:
882:
721:
712:probability measures
691:. In the case where
645:measurable functions
528:
473:
384:
301:
239:
144:
77:we require there be
41:, more specifically
27:Mathematical concept
6629:Hölder's inequality
6491:of random variables
6453:Measurable function
6340:Particular measures
5929:Absolute continuity
5752:. Springer-Verlag.
5679:Prokhorov's theorem
3634:if the sequence of
3502:
3460:
3261:is also compact or
2766:{\displaystyle 1/n}
2716:{\displaystyle 1/n}
2072:bounded from below;
1986:bounded from above;
1879:Lipschitz functions
1362:Portmanteau theorem
1269:
1229:
233:setwise convergence
6769:Probability theory
6094:Transverse measure
6072:Non-measurable set
6054:Locally measurable
5750:Probability Theory
5649:
5591:
5571:
5529:
5463:
5407:
5365:
5329:
5296:
5272:
5246:
5193:
5167:
5116:
5093:
5050:
5024:
4978:
4945:
4919:
4866:
4843:
4823:
4792:
4756:
4736:
4696:
4648:
4605:
4585:
4537:
4513:
4455:
4388:
4346:
4323:
4265:
4198:
4156:
4130:
4072:
4055:is the closure of
4045:
4018:
3955:
3918:
3899:
3857:
3837:
3799:
3779:
3740:
3708:
3686:
3565:
3507:
3465:
3425:
3392:
3344:
3311:
3288:
3251:
3227:
3190:
3167:
3021:
2900:
2868:
2828:
2805:
2783:
2763:
2735:
2713:
2681:
2651:
2631:
2603:
2568:
2536:
2516:
2489:
2469:
2438:
2418:
2391:
2359:
2339:
2316:
2260:
2240:
2217:
2161:
2141:
2118:
2062:
2038:
1976:
1952:
1890:
1867:
1808:
1781:
1721:
1701:
1674:
1654:
1627:
1600:
1573:
1540:
1517:
1485:
1427:
1406:
1380:
1274:
1234:
1188:
1155:
1120:
1068:
1024:
966:such that for all
962:, there exists an
922:
819:
778:
673:Lipschitz constant
657:Wasserstein metric
626:
569:
495:
437:
340:
276:
183:
6850:
6849:
6842:
6787:
6786:
6747:
6746:
6476:almost everywhere
6422:Spherical measure
6320:Strictly positive
6248:Projection-valued
5988:Almost everywhere
5961:Probability space
5791:978-0-12-174151-8
5759:978-1-84800-047-6
5727:10.3150/09-BEJ238
5594:{\displaystyle X}
5299:{\displaystyle X}
5240:
5018:
4869:{\displaystyle x}
4846:{\displaystyle V}
4795:{\displaystyle V}
4731:
4086:Vague Convergence
3874:
3860:{\displaystyle f}
3802:{\displaystyle f}
3689:{\displaystyle X}
3596:is a sequence of
3577:probability space
3503:
3461:
3314:{\displaystyle S}
3254:{\displaystyle S}
3193:{\displaystyle S}
3154:
3101:
3076:
3008:
2995:
2979:
2952:
2927:
2843:topological space
2831:{\displaystyle S}
2786:{\displaystyle 0}
2654:{\displaystyle F}
2553:
2539:{\displaystyle P}
2492:{\displaystyle P}
2441:{\displaystyle F}
2362:{\displaystyle P}
2342:{\displaystyle A}
2263:{\displaystyle S}
2243:{\displaystyle U}
2164:{\displaystyle S}
2144:{\displaystyle C}
2065:{\displaystyle f}
1979:{\displaystyle f}
1893:{\displaystyle f}
1811:{\displaystyle f}
1724:{\displaystyle P}
1543:{\displaystyle P}
1383:{\displaystyle S}
1270:
1230:
1105:
1012:
920:
913:
759:
747:
560:
554:
521:is then given by
369:, i.e. for every
16:(Redirected from
6874:
6845:
6838:
6834:
6831:
6825:
6820:this article by
6811:inline citations
6798:
6797:
6790:
6722:Milman's reverse
6705:
6703:Lebesgue measure
6657:
6656:
6061:
6047:infimum/supremum
5968:Measurable space
5908:
5901:
5894:
5885:
5884:
5880:
5868:
5857:
5838:
5832:
5824:
5796:
5795:
5775:
5764:
5763:
5745:
5739:
5738:
5720:
5700:
5658:
5656:
5655:
5650:
5639:
5638:
5617:
5616:
5600:
5598:
5597:
5592:
5580:
5578:
5577:
5572:
5561:
5560:
5538:
5536:
5535:
5530:
5528:
5527:
5511:
5510:
5489:
5488:
5472:
5470:
5469:
5464:
5462:
5461:
5446:
5445:
5433:
5432:
5416:
5414:
5413:
5408:
5391:
5390:
5374:
5372:
5371:
5366:
5355:
5354:
5338:
5336:
5335:
5330:
5305:
5303:
5302:
5297:
5281:
5279:
5278:
5273:
5255:
5253:
5252:
5247:
5242:
5241:
5239:
5238:
5237:
5227:
5222:
5219:
5218:
5202:
5200:
5199:
5194:
5176:
5174:
5173:
5168:
5142:
5141:
5125:
5123:
5122:
5117:
5102:
5100:
5099:
5094:
5092:
5091:
5079:
5078:
5059:
5057:
5056:
5051:
5033:
5031:
5030:
5025:
5020:
5019:
5017:
5012:
5007:
5004:
5003:
4987:
4985:
4984:
4979:
4977:
4976:
4954:
4952:
4951:
4946:
4928:
4926:
4925:
4920:
4903:
4899:
4898:
4875:
4873:
4872:
4867:
4854:converges weakly
4852:
4850:
4849:
4844:
4832:
4830:
4829:
4824:
4822:
4821:
4801:
4799:
4798:
4793:
4776:weak convergence
4765:
4763:
4762:
4757:
4745:
4743:
4742:
4737:
4732:
4724:
4722:
4721:
4705:
4703:
4702:
4697:
4695:
4694:
4693:
4677:
4676:
4657:
4655:
4654:
4649:
4614:
4612:
4611:
4606:
4594:
4592:
4591:
4586:
4584:
4583:
4582:
4566:
4565:
4546:
4544:
4543:
4538:
4522:
4520:
4519:
4514:
4512:
4511:
4510:
4494:
4493:
4464:
4462:
4461:
4456:
4444:
4443:
4431:
4430:
4414:
4413:
4397:
4395:
4394:
4389:
4378:
4377:
4355:
4353:
4352:
4347:
4334:converges weakly
4332:
4330:
4329:
4324:
4322:
4321:
4320:
4308:
4304:
4303:
4279:Weak Convergence
4274:
4272:
4271:
4266:
4254:
4253:
4241:
4240:
4224:
4223:
4207:
4205:
4204:
4199:
4188:
4187:
4165:
4163:
4162:
4157:
4139:
4137:
4136:
4131:
4129:
4128:
4127:
4115:
4111:
4110:
4081:
4079:
4078:
4073:
4071:
4070:
4054:
4052:
4051:
4046:
4044:
4043:
4027:
4025:
4024:
4019:
4011:
4010:
3998:
3997:
3985:
3984:
3964:
3962:
3961:
3956:
3945:
3944:
3927:
3925:
3924:
3919:
3898:
3891:
3883:
3866:
3864:
3863:
3858:
3846:
3844:
3843:
3838:
3827:
3826:
3808:
3806:
3805:
3800:
3788:
3786:
3785:
3780:
3769:
3768:
3749:
3747:
3746:
3743:{\displaystyle }
3741:
3717:
3715:
3714:
3709:
3707:
3695:
3693:
3692:
3687:
3633:
3598:random variables
3595:
3574:
3572:
3571:
3566:
3561:
3553:
3552:
3516:
3514:
3513:
3508:
3501:
3492:
3491:
3490:
3474:
3472:
3471:
3466:
3452:
3451:
3450:
3434:
3432:
3431:
3426:
3418:
3417:
3401:
3399:
3398:
3393:
3385:
3384:
3353:
3351:
3350:
3345:
3334:
3333:
3320:
3318:
3317:
3312:
3297:
3295:
3294:
3289:
3278:
3277:
3260:
3258:
3257:
3252:
3236:
3234:
3233:
3228:
3217:
3216:
3199:
3197:
3196:
3191:
3176:
3174:
3173:
3168:
3163:
3159:
3158:
3155:
3152:
3145:
3141:
3131:
3122:
3121:
3099:
3089:
3088:
3074:
3065:
3064:
3030:
3028:
3027:
3022:
3017:
3013:
3012:
3009:
3006:
2996:
2993:
2991:
2980:
2977:
2975:
2950:
2949:
2948:
2925:
2909:
2907:
2906:
2901:
2877:
2875:
2874:
2869:
2858:
2857:
2837:
2835:
2834:
2829:
2814:
2812:
2811:
2806:
2804:
2792:
2790:
2789:
2784:
2772:
2770:
2769:
2764:
2759:
2744:
2742:
2741:
2736:
2734:
2722:
2720:
2719:
2714:
2709:
2690:
2688:
2687:
2682:
2680:
2679:
2660:
2658:
2657:
2652:
2640:
2638:
2637:
2632:
2630:
2612:
2610:
2609:
2604:
2578:
2577:
2567:
2545:
2543:
2542:
2537:
2525:
2523:
2522:
2517:
2515:
2514:
2498:
2496:
2495:
2490:
2478:
2476:
2475:
2470:
2468:
2467:
2452:of the measures
2447:
2445:
2444:
2439:
2427:
2425:
2424:
2419:
2417:
2416:
2400:
2398:
2397:
2392:
2390:
2368:
2366:
2365:
2360:
2348:
2346:
2345:
2340:
2325:
2323:
2322:
2317:
2291:
2290:
2269:
2267:
2266:
2261:
2249:
2247:
2246:
2241:
2226:
2224:
2223:
2218:
2192:
2191:
2170:
2168:
2167:
2162:
2150:
2148:
2147:
2142:
2127:
2125:
2124:
2119:
2093:
2092:
2071:
2069:
2068:
2063:
2047:
2045:
2044:
2039:
2007:
2006:
1985:
1983:
1982:
1977:
1961:
1959:
1958:
1953:
1921:
1920:
1899:
1897:
1896:
1891:
1876:
1874:
1873:
1868:
1836:
1835:
1817:
1815:
1814:
1809:
1790:
1788:
1787:
1782:
1750:
1749:
1730:
1728:
1727:
1722:
1710:
1708:
1707:
1702:
1700:
1699:
1683:
1681:
1680:
1675:
1663:
1661:
1660:
1655:
1653:
1652:
1636:
1634:
1633:
1628:
1626:
1625:
1609:
1607:
1606:
1601:
1599:
1598:
1582:
1580:
1579:
1574:
1566:
1565:
1549:
1547:
1546:
1541:
1526:
1524:
1523:
1518:
1494:
1492:
1491:
1486:
1456:
1455:
1436:
1434:
1433:
1428:
1415:
1413:
1412:
1407:
1389:
1387:
1386:
1381:
1347:weak convergence
1324:
1303:
1299:
1283:
1281:
1280:
1275:
1261:
1260:
1259:
1243:
1241:
1240:
1235:
1218:
1217:
1216:
1197:
1195:
1194:
1189:
1187:
1186:
1164:
1162:
1161:
1156:
1130:
1129:
1119:
1094:
1090:
1080:measurable space
1077:
1075:
1074:
1069:
1064:
1063:
1033:
1031:
1030:
1025:
1014:
1013:
1010:
998:
997:
975:
965:
961:
954:
946:
931:
929:
928:
923:
921:
916:
915:
914:
911:
886:
874:
866:
862:
858:
854:
850:
846:
842:
828:
826:
825:
820:
815:
783:
777:
776:
775:
749:
748:
745:
743:
739:
709:
705:
694:
690:
686:
682:
670:
666:
662:
654:
650:
642:
635:
633:
632:
627:
622:
618:
607:
606:
584:
583:
568:
556:
555:
552:
550:
546:
520:
516:
507:measurable space
504:
502:
501:
496:
491:
490:
460:
456:
446:
444:
443:
438:
430:
401:
400:
391:
379:
375:
357:
353:
349:
347:
346:
341:
323:
322:
291:
285:
283:
282:
277:
251:
250:
227:
223:
213:
209:
205:
200:weak convergence
192:
190:
189:
184:
166:
165:
132:
107:
104:is smaller than
103:
99:
90:
80:
76:
65:
61:
21:
6882:
6881:
6877:
6876:
6875:
6873:
6872:
6871:
6852:
6851:
6846:
6835:
6829:
6826:
6816:Please help to
6815:
6799:
6795:
6788:
6783:
6779:Spectral theory
6759:Convex analysis
6743:
6700:
6695:
6648:
6548:
6496:in distribution
6441:
6334:
6164:Logarithmically
6103:
6059:
6042:Essential range
5976:
5917:
5912:
5877:
5854:
5826:
5825:
5821:
5805:
5803:Further reading
5800:
5799:
5792:
5776:
5767:
5760:
5746:
5742:
5701:
5697:
5692:
5665:
5634:
5630:
5612:
5608:
5606:
5603:
5602:
5586:
5583:
5582:
5556:
5552:
5544:
5541:
5540:
5523:
5519:
5506:
5502:
5484:
5480:
5478:
5475:
5474:
5457:
5453:
5441:
5437:
5428:
5424:
5422:
5419:
5418:
5386:
5382:
5380:
5377:
5376:
5350:
5346:
5344:
5341:
5340:
5315:
5312:
5311:
5291:
5288:
5287:
5261:
5258:
5257:
5233:
5229:
5228:
5223:
5221:
5220:
5214:
5210:
5208:
5205:
5204:
5182:
5179:
5178:
5137:
5133:
5131:
5128:
5127:
5111:
5108:
5107:
5087:
5083:
5074:
5070:
5068:
5065:
5064:
5039:
5036:
5035:
5013:
5008:
5006:
5005:
4999:
4995:
4993:
4990:
4989:
4972:
4968:
4960:
4957:
4956:
4934:
4931:
4930:
4894:
4890:
4886:
4881:
4878:
4877:
4861:
4858:
4857:
4838:
4835:
4834:
4817:
4813:
4811:
4808:
4807:
4787:
4784:
4783:
4772:
4751:
4748:
4747:
4723:
4717:
4713:
4711:
4708:
4707:
4689:
4682:
4678:
4672:
4668:
4663:
4660:
4659:
4628:
4625:
4624:
4600:
4597:
4596:
4578:
4571:
4567:
4561:
4557:
4552:
4549:
4548:
4547:if and only if
4532:
4529:
4528:
4506:
4499:
4495:
4489:
4485:
4480:
4477:
4476:
4439:
4435:
4426:
4422:
4409:
4405:
4403:
4400:
4399:
4373:
4369:
4361:
4358:
4357:
4341:
4338:
4337:
4316:
4309:
4299:
4295:
4291:
4290:
4288:
4285:
4284:
4281:
4249:
4245:
4236:
4232:
4219:
4215:
4213:
4210:
4209:
4183:
4179:
4171:
4168:
4167:
4151:
4148:
4147:
4123:
4116:
4106:
4102:
4098:
4097:
4095:
4092:
4091:
4088:
4066:
4062:
4060:
4057:
4056:
4039:
4035:
4033:
4030:
4029:
4006:
4002:
3993:
3989:
3980:
3976:
3974:
3971:
3970:
3940:
3936:
3934:
3931:
3930:
3887:
3879:
3878:
3872:
3869:
3868:
3852:
3849:
3848:
3822:
3818:
3816:
3813:
3812:
3794:
3791:
3790:
3764:
3760:
3758:
3755:
3754:
3723:
3720:
3719:
3703:
3701:
3698:
3697:
3681:
3678:
3677:
3674:
3659:
3648:
3643:
3628:
3613:in distribution
3609:converge weakly
3605:
3589:
3584:
3557:
3548:
3547:
3536:
3533:
3532:
3529:
3523:
3497:
3486:
3482:
3480:
3477:
3476:
3446:
3442:
3440:
3437:
3436:
3413:
3409:
3407:
3404:
3403:
3380:
3376:
3374:
3371:
3370:
3329:
3328:
3326:
3323:
3322:
3306:
3303:
3302:
3273:
3272:
3270:
3267:
3266:
3246:
3243:
3242:
3212:
3211:
3209:
3206:
3205:
3185:
3182:
3181:
3127:
3117:
3113:
3112:
3108:
3106:
3102:
3084:
3083:
3073:
3069:
3048:
3044:
3042:
3039:
3038:
2994: and
2992:
2987:
2976:
2971:
2957:
2953:
2932:
2928:
2924:
2920:
2918:
2915:
2914:
2883:
2880:
2879:
2853:
2852:
2850:
2847:
2846:
2823:
2820:
2819:
2800:
2798:
2795:
2794:
2778:
2775:
2774:
2755:
2750:
2747:
2746:
2730:
2728:
2725:
2724:
2705:
2700:
2697:
2696:
2675:
2671:
2669:
2666:
2665:
2661:is continuous.
2646:
2643:
2642:
2626:
2618:
2615:
2614:
2613:for all points
2573:
2569:
2557:
2551:
2548:
2547:
2546:if and only if
2531:
2528:
2527:
2510:
2506:
2504:
2501:
2500:
2484:
2481:
2480:
2463:
2459:
2457:
2454:
2453:
2433:
2430:
2429:
2412:
2408:
2406:
2403:
2402:
2386:
2378:
2375:
2374:
2354:
2351:
2350:
2334:
2331:
2330:
2328:continuity sets
2286:
2282:
2277:
2274:
2273:
2255:
2252:
2251:
2235:
2232:
2231:
2187:
2183:
2178:
2175:
2174:
2156:
2153:
2152:
2136:
2133:
2132:
2088:
2084:
2079:
2076:
2075:
2057:
2054:
2053:
2002:
1998:
1993:
1990:
1989:
1971:
1968:
1967:
1916:
1912:
1907:
1904:
1903:
1885:
1882:
1881:
1831:
1827:
1825:
1822:
1821:
1803:
1800:
1799:
1745:
1741:
1739:
1736:
1735:
1716:
1713:
1712:
1695:
1691:
1689:
1686:
1685:
1669:
1666:
1665:
1648:
1644:
1642:
1639:
1638:
1621:
1617:
1615:
1612:
1611:
1594:
1590:
1588:
1585:
1584:
1561:
1557:
1555:
1552:
1551:
1535:
1532:
1531:
1500:
1497:
1496:
1451:
1447:
1445:
1442:
1441:
1422:
1419:
1418:
1401:
1398:
1397:
1375:
1372:
1371:
1335:
1310:
1305:
1302:[â1, 1]
1301:
1297:
1292:
1291:, the sequence
1255:
1251:
1249:
1246:
1245:
1212:
1208:
1206:
1203:
1202:
1182:
1181:
1173:
1170:
1169:
1125:
1121:
1109:
1103:
1100:
1099:
1092:
1088:
1083:
1059:
1058:
1047:
1044:
1043:
1040:
1009:
1005:
993:
989:
984:
981:
980:
976:, one has that
967:
963:
956:
952:
944:
939:
910:
906:
887:
885:
883:
880:
879:
872:
864:
860:
856:
855:has law either
852:
851:. We know that
848:
844:
840:
811:
779:
771:
770:
763:
744:
729:
725:
724:
722:
719:
718:
707:
703:
692:
689:[â1, 1]
688:
684:
680:
669:[â1, 1]
668:
664:
660:
653:[â1, 1]
652:
648:
640:
602:
598:
579:
575:
574:
570:
564:
551:
536:
532:
531:
529:
526:
525:
518:
514:
511:total variation
486:
485:
474:
471:
470:
467:
458:
448:
426:
396:
392:
387:
385:
382:
381:
377:
370:
355:
351:
318:
314:
302:
299:
298:
289:
246:
242:
240:
237:
236:
225:
215:
211:
207:
203:
161:
157:
145:
142:
141:
130:
125:
117:
105:
101:
97:
92:
82:
78:
71:
63:
59:
54:
35:
28:
23:
22:
15:
12:
11:
5:
6880:
6870:
6869:
6864:
6862:Measure theory
6848:
6847:
6802:
6800:
6793:
6785:
6784:
6782:
6781:
6776:
6771:
6766:
6761:
6755:
6753:
6749:
6748:
6745:
6744:
6742:
6741:
6736:
6731:
6726:
6725:
6724:
6714:
6708:
6706:
6697:
6696:
6694:
6693:
6688:
6686:Sard's theorem
6683:
6678:
6673:
6672:
6671:
6669:Lifting theory
6660:
6654:
6650:
6649:
6647:
6646:
6641:
6636:
6631:
6626:
6625:
6624:
6622:FubiniâTonelli
6614:
6609:
6604:
6603:
6602:
6597:
6592:
6584:
6583:
6582:
6577:
6572:
6564:
6558:
6556:
6550:
6549:
6547:
6546:
6541:
6536:
6531:
6526:
6521:
6516:
6510:
6505:
6504:
6503:
6501:in probability
6498:
6488:
6483:
6478:
6472:
6471:
6470:
6465:
6460:
6449:
6447:
6443:
6442:
6440:
6439:
6434:
6429:
6424:
6419:
6414:
6413:
6412:
6402:
6397:
6396:
6395:
6385:
6380:
6375:
6370:
6365:
6360:
6355:
6350:
6344:
6342:
6336:
6335:
6333:
6332:
6327:
6322:
6317:
6312:
6307:
6302:
6297:
6292:
6287:
6282:
6281:
6280:
6275:
6270:
6260:
6255:
6250:
6245:
6235:
6230:
6225:
6220:
6215:
6210:
6208:Locally finite
6205:
6195:
6190:
6185:
6180:
6175:
6170:
6160:
6155:
6150:
6145:
6140:
6135:
6130:
6125:
6120:
6114:
6112:
6105:
6104:
6102:
6101:
6096:
6091:
6086:
6081:
6080:
6079:
6069:
6064:
6056:
6051:
6050:
6049:
6039:
6034:
6033:
6032:
6022:
6017:
6012:
6011:
6010:
6000:
5995:
5990:
5984:
5982:
5978:
5977:
5975:
5974:
5965:
5964:
5963:
5953:
5948:
5940:
5935:
5925:
5923:
5922:Basic concepts
5919:
5918:
5915:Measure theory
5911:
5910:
5903:
5896:
5888:
5882:
5881:
5875:
5858:
5852:
5839:
5819:
5804:
5801:
5798:
5797:
5790:
5765:
5758:
5740:
5711:(3): 882â908.
5694:
5693:
5691:
5688:
5687:
5686:
5681:
5676:
5671:
5664:
5661:
5648:
5645:
5642:
5637:
5633:
5629:
5626:
5623:
5620:
5615:
5611:
5590:
5570:
5567:
5564:
5559:
5555:
5551:
5548:
5526:
5522:
5518:
5514:
5509:
5505:
5501:
5498:
5495:
5492:
5487:
5483:
5460:
5456:
5452:
5449:
5444:
5440:
5436:
5431:
5427:
5406:
5403:
5400:
5397:
5394:
5389:
5385:
5364:
5361:
5358:
5353:
5349:
5328:
5325:
5322:
5319:
5295:
5284:
5283:
5271:
5268:
5265:
5245:
5236:
5232:
5226:
5217:
5213:
5192:
5189:
5186:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5140:
5136:
5126:provided that
5115:
5090:
5086:
5082:
5077:
5073:
5063:A sequence of
5061:
5049:
5046:
5043:
5023:
5016:
5011:
5002:
4998:
4975:
4971:
4967:
4964:
4944:
4941:
4938:
4918:
4915:
4912:
4909:
4906:
4902:
4897:
4893:
4889:
4885:
4865:
4842:
4820:
4816:
4791:
4771:
4768:
4755:
4735:
4730:
4727:
4720:
4716:
4692:
4688:
4685:
4681:
4675:
4671:
4667:
4647:
4644:
4641:
4638:
4635:
4632:
4604:
4581:
4577:
4574:
4570:
4564:
4560:
4556:
4536:
4509:
4505:
4502:
4498:
4492:
4488:
4484:
4454:
4451:
4447:
4442:
4438:
4434:
4429:
4425:
4421:
4417:
4412:
4408:
4387:
4384:
4381:
4376:
4372:
4368:
4365:
4345:
4319:
4315:
4312:
4307:
4302:
4298:
4294:
4280:
4277:
4264:
4261:
4257:
4252:
4248:
4244:
4239:
4235:
4231:
4227:
4222:
4218:
4197:
4194:
4191:
4186:
4182:
4178:
4175:
4155:
4126:
4122:
4119:
4114:
4109:
4105:
4101:
4087:
4084:
4069:
4065:
4042:
4038:
4017:
4014:
4009:
4005:
4001:
3996:
3992:
3988:
3983:
3979:
3967:
3966:
3954:
3951:
3948:
3943:
3939:
3928:
3917:
3914:
3911:
3908:
3905:
3902:
3897:
3894:
3890:
3886:
3882:
3877:
3856:
3836:
3833:
3830:
3825:
3821:
3810:
3798:
3778:
3775:
3772:
3767:
3763:
3739:
3736:
3733:
3730:
3727:
3706:
3685:
3673:
3670:
3657:
3646:
3641:
3603:
3587:
3564:
3560:
3556:
3551:
3546:
3543:
3540:
3525:Main article:
3522:
3519:
3506:
3500:
3495:
3489:
3485:
3464:
3459:
3455:
3449:
3445:
3424:
3421:
3416:
3412:
3391:
3388:
3383:
3379:
3356:Dirac measures
3343:
3340:
3337:
3332:
3310:
3287:
3284:
3281:
3276:
3250:
3226:
3223:
3220:
3215:
3189:
3178:
3177:
3166:
3162:
3157:
3151:
3148:
3144:
3140:
3137:
3134:
3130:
3125:
3120:
3116:
3111:
3105:
3098:
3095:
3092:
3087:
3082:
3079:
3072:
3068:
3063:
3060:
3057:
3054:
3051:
3047:
3032:
3031:
3020:
3016:
3011:
3005:
3002:
2999:
2990:
2986:
2983:
2974:
2970:
2967:
2964:
2961:
2956:
2947:
2944:
2941:
2938:
2935:
2931:
2923:
2899:
2896:
2893:
2890:
2887:
2867:
2864:
2861:
2856:
2827:
2803:
2782:
2773:is "close" to
2762:
2758:
2754:
2733:
2712:
2708:
2704:
2678:
2674:
2650:
2629:
2625:
2622:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2576:
2572:
2566:
2563:
2560:
2556:
2535:
2513:
2509:
2488:
2466:
2462:
2437:
2415:
2411:
2389:
2385:
2382:
2371:
2370:
2358:
2338:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2289:
2285:
2281:
2271:
2259:
2239:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2190:
2186:
2182:
2181:lim inf
2172:
2160:
2140:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2091:
2087:
2083:
2082:lim sup
2073:
2061:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2005:
2001:
1997:
1996:lim inf
1987:
1975:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1919:
1915:
1911:
1910:lim sup
1901:
1889:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1834:
1830:
1819:
1807:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1748:
1744:
1720:
1698:
1694:
1673:
1651:
1647:
1624:
1620:
1597:
1593:
1572:
1569:
1564:
1560:
1539:
1516:
1513:
1510:
1507:
1504:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1454:
1450:
1426:
1405:
1379:
1334:
1331:
1308:
1295:
1273:
1268:
1264:
1258:
1254:
1233:
1228:
1225:
1221:
1215:
1211:
1185:
1180:
1177:
1168:for every set
1166:
1165:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1128:
1124:
1118:
1115:
1112:
1108:
1086:
1067:
1062:
1057:
1054:
1051:
1039:
1036:
1035:
1034:
1023:
1020:
1017:
1008:
1004:
1001:
996:
992:
988:
942:
933:
932:
919:
909:
905:
902:
899:
896:
893:
890:
830:
829:
818:
814:
810:
807:
804:
801:
798:
795:
792:
789:
786:
782:
774:
769:
766:
762:
758:
755:
752:
742:
738:
735:
732:
728:
637:
636:
625:
621:
617:
614:
610:
605:
601:
597:
594:
591:
587:
582:
578:
573:
567:
563:
559:
549:
545:
542:
539:
535:
494:
489:
484:
481:
478:
466:
463:
436:
433:
429:
425:
422:
419:
416:
413:
410:
407:
404:
399:
395:
390:
361:The notion of
339:
336:
332:
329:
326:
321:
317:
313:
309:
306:
275:
272:
269:
266:
263:
260:
257:
254:
249:
245:
231:The notion of
198:The notion of
182:
179:
175:
172:
169:
164:
160:
156:
152:
149:
128:
116:
113:
95:
57:
43:measure theory
26:
9:
6:
4:
3:
2:
6879:
6868:
6865:
6863:
6860:
6859:
6857:
6844:
6841:
6833:
6830:February 2010
6823:
6819:
6813:
6812:
6806:
6801:
6792:
6791:
6780:
6777:
6775:
6774:Real analysis
6772:
6770:
6767:
6765:
6762:
6760:
6757:
6756:
6754:
6750:
6740:
6737:
6735:
6732:
6730:
6727:
6723:
6720:
6719:
6718:
6715:
6713:
6710:
6709:
6707:
6704:
6698:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6670:
6667:
6666:
6665:
6662:
6661:
6658:
6655:
6653:Other results
6651:
6645:
6642:
6640:
6639:RadonâNikodym
6637:
6635:
6632:
6630:
6627:
6623:
6620:
6619:
6618:
6615:
6613:
6612:Fatou's lemma
6610:
6608:
6605:
6601:
6598:
6596:
6593:
6591:
6588:
6587:
6585:
6581:
6578:
6576:
6573:
6571:
6568:
6567:
6565:
6563:
6560:
6559:
6557:
6555:
6551:
6545:
6542:
6540:
6537:
6535:
6532:
6530:
6527:
6525:
6522:
6520:
6517:
6515:
6511:
6509:
6506:
6502:
6499:
6497:
6494:
6493:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:Convergence:
6473:
6469:
6466:
6464:
6461:
6459:
6456:
6455:
6454:
6451:
6450:
6448:
6444:
6438:
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6418:
6415:
6411:
6408:
6407:
6406:
6403:
6401:
6398:
6394:
6391:
6390:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6364:
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6345:
6343:
6341:
6337:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6301:
6298:
6296:
6293:
6291:
6288:
6286:
6283:
6279:
6278:Outer regular
6276:
6274:
6273:Inner regular
6271:
6269:
6268:Borel regular
6266:
6265:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6240:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6200:
6196:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6174:
6171:
6169:
6165:
6161:
6159:
6156:
6154:
6151:
6149:
6146:
6144:
6141:
6139:
6136:
6134:
6131:
6129:
6126:
6124:
6121:
6119:
6116:
6115:
6113:
6111:
6106:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6078:
6075:
6074:
6073:
6070:
6068:
6065:
6063:
6057:
6055:
6052:
6048:
6045:
6044:
6043:
6040:
6038:
6035:
6031:
6028:
6027:
6026:
6023:
6021:
6018:
6016:
6013:
6009:
6006:
6005:
6004:
6001:
5999:
5996:
5994:
5991:
5989:
5986:
5985:
5983:
5979:
5973:
5969:
5966:
5962:
5959:
5958:
5957:
5956:Measure space
5954:
5952:
5949:
5947:
5945:
5941:
5939:
5936:
5934:
5930:
5927:
5926:
5924:
5920:
5916:
5909:
5904:
5902:
5897:
5895:
5890:
5889:
5886:
5878:
5876:0-471-19745-9
5872:
5867:
5866:
5859:
5855:
5853:0-471-00710-2
5849:
5845:
5840:
5836:
5830:
5822:
5820:3-7643-2428-7
5816:
5812:
5807:
5806:
5793:
5787:
5783:
5782:
5774:
5772:
5770:
5761:
5755:
5751:
5744:
5736:
5732:
5728:
5724:
5719:
5714:
5710:
5706:
5699:
5695:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5666:
5660:
5643:
5635:
5631:
5627:
5621:
5613:
5609:
5588:
5565:
5557:
5553:
5549:
5546:
5524:
5520:
5516:
5512:
5507:
5503:
5499:
5493:
5485:
5481:
5458:
5450:
5442:
5438:
5434:
5429:
5425:
5401:
5395:
5392:
5387:
5383:
5359:
5351:
5347:
5323:
5317:
5309:
5293:
5263:
5243:
5234:
5230:
5215:
5211:
5190:
5187:
5184:
5161:
5155:
5146:
5138:
5134:
5113:
5105:
5088:
5084:
5080:
5075:
5071:
5062:
5041:
5021:
5014:
5000:
4996:
4988:. One writes
4973:
4969:
4965:
4962:
4936:
4913:
4907:
4900:
4895:
4891:
4887:
4883:
4863:
4855:
4840:
4818:
4814:
4805:
4804:
4803:
4789:
4780:
4777:
4767:
4753:
4733:
4728:
4718:
4714:
4686:
4683:
4673:
4669:
4645:
4642:
4636:
4630:
4622:
4616:
4602:
4575:
4572:
4562:
4558:
4534:
4526:
4503:
4500:
4490:
4486:
4474:
4469:
4466:
4452:
4449:
4445:
4440:
4436:
4427:
4423:
4419:
4415:
4410:
4406:
4382:
4374:
4370:
4366:
4363:
4343:
4336:to a measure
4335:
4313:
4310:
4305:
4300:
4296:
4292:
4276:
4262:
4259:
4255:
4250:
4246:
4237:
4233:
4229:
4225:
4220:
4216:
4192:
4184:
4180:
4176:
4173:
4153:
4146:to a measure
4145:
4144:
4120:
4117:
4112:
4107:
4103:
4099:
4083:
4067:
4063:
4040:
4036:
4015:
4012:
4007:
4003:
3999:
3994:
3990:
3986:
3981:
3977:
3949:
3941:
3937:
3929:
3915:
3912:
3906:
3900:
3884:
3854:
3831:
3823:
3819:
3811:
3796:
3773:
3765:
3761:
3753:
3752:
3751:
3734:
3731:
3728:
3683:
3669:
3667:
3663:
3656:
3652:
3644:
3637:
3631:
3626:
3622:
3618:
3614:
3610:
3606:
3599:
3594:
3590:
3582:
3578:
3554:
3544:
3528:
3518:
3504:
3493:
3487:
3483:
3462:
3457:
3453:
3447:
3443:
3422:
3414:
3410:
3389:
3381:
3377:
3367:
3365:
3361:
3357:
3338:
3308:
3299:
3282:
3264:
3248:
3240:
3221:
3203:
3187:
3164:
3160:
3149:
3146:
3142:
3138:
3135:
3132:
3123:
3118:
3114:
3109:
3103:
3093:
3080:
3077:
3070:
3066:
3061:
3058:
3055:
3052:
3049:
3045:
3037:
3036:
3035:
3018:
3014:
3003:
3000:
2997:
2984:
2981:
2965:
2962:
2959:
2954:
2945:
2942:
2939:
2936:
2933:
2929:
2921:
2913:
2912:
2911:
2891:
2888:
2862:
2844:
2841:
2825:
2816:
2780:
2760:
2756:
2752:
2710:
2706:
2702:
2694:
2693:Dirac measure
2676:
2672:
2662:
2648:
2623:
2620:
2597:
2591:
2588:
2582:
2574:
2570:
2558:
2533:
2511:
2507:
2486:
2464:
2460:
2451:
2435:
2413:
2409:
2383:
2380:
2356:
2336:
2329:
2310:
2304:
2301:
2295:
2287:
2283:
2272:
2257:
2237:
2230:
2211:
2205:
2202:
2196:
2188:
2184:
2173:
2158:
2138:
2131:
2112:
2106:
2103:
2097:
2089:
2085:
2074:
2059:
2051:
2032:
2026:
2020:
2014:
2008:
2003:
1988:
1973:
1965:
1946:
1940:
1934:
1928:
1922:
1917:
1902:
1887:
1880:
1861:
1855:
1843:
1837:
1832:
1820:
1805:
1798:
1794:
1775:
1769:
1757:
1751:
1746:
1734:
1733:
1732:
1718:
1696:
1692:
1649:
1645:
1622:
1618:
1595:
1570:
1562:
1558:
1537:
1530:
1508:
1505:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1452:
1448:
1440:
1417:
1403:
1393:
1377:
1369:
1365:
1363:
1359:
1354:
1352:
1348:
1344:
1340:
1330:
1326:
1323:
1319:
1316:) = (1 + sin(
1315:
1311:
1298:
1290:
1285:
1271:
1266:
1262:
1256:
1252:
1231:
1226:
1223:
1219:
1213:
1209:
1199:
1178:
1175:
1149:
1143:
1140:
1134:
1126:
1122:
1110:
1098:
1097:
1096:
1089:
1082:, a sequence
1081:
1055:
1052:
1021:
1018:
1015:
1002:
999:
994:
990:
979:
978:
977:
974:
970:
959:
951:to a measure
950:
945:
936:
917:
903:
900:
897:
891:
888:
878:
877:
876:
870:
837:
835:
816:
805:
799:
796:
790:
784:
767:
764:
756:
753:
750:
736:
733:
730:
717:
716:
715:
713:
700:
698:
678:
674:
658:
646:
623:
619:
615:
612:
608:
603:
599:
595:
592:
589:
585:
580:
576:
571:
565:
557:
543:
540:
537:
524:
523:
522:
512:
508:
482:
479:
462:
455:
451:
434:
431:
420:
414:
411:
405:
397:
393:
376:there exists
373:
368:
364:
359:
337:
334:
330:
327:
319:
315:
311:
307:
304:
296:
286:
270:
264:
255:
247:
243:
234:
229:
222:
218:
201:
196:
193:
180:
177:
173:
170:
162:
158:
154:
150:
147:
138:
136:
131:
122:
112:
109:
98:
89:
85:
74:
69:
60:
52:
48:
44:
40:
33:
19:
6836:
6827:
6808:
6554:Main results
6480:
6290:Set function
6218:Metric outer
6173:Decomposable
6030:Cylinder set
5943:
5864:
5843:
5810:
5780:
5749:
5743:
5708:
5704:
5698:
5310:, the space
5285:
5103:
4853:
4781:
4773:
4620:
4617:
4470:
4467:
4333:
4282:
4140:
4089:
4028:. Moreover,
3968:
3675:
3665:
3661:
3654:
3650:
3639:
3629:
3624:
3623:: Ω â
3620:
3616:
3612:
3608:
3601:
3592:
3591:: Ω â
3585:
3580:
3530:
3368:
3300:
3179:
3033:
2817:
2663:
2373:In the case
2372:
1528:
1392:metric space
1367:
1366:
1361:
1355:
1346:
1336:
1327:
1321:
1317:
1313:
1306:
1293:
1286:
1200:
1167:
1084:
1041:
972:
968:
957:
948:
940:
937:
934:
868:
838:
831:
701:
697:Polish space
677:Radon metric
638:
468:
453:
449:
371:
366:
362:
360:
295:Polish space
287:
232:
230:
220:
216:
199:
197:
194:
139:
135:Polish space
126:
118:
110:
93:
87:
83:
72:
55:
50:
46:
36:
6822:introducing
6514:compact set
6481:of measures
6417:Pushforward
6410:Projections
6400:Logarithmic
6243:Probability
6233:Pre-measure
6015:Borel space
5933:of measures
4806:A sequence
4356:if for all
4166:if for all
3607:is said to
3360:convex hull
2695:located at
2448:denote the
2349:of measure
2130:closed sets
1527:is said to
1368:Definition.
1358:definitions
1339:mathematics
671:which have
39:mathematics
6856:Categories
6805:references
6486:in measure
6213:Maximising
6183:Equivalent
6077:Vitali set
5473:such that
4706:such that
4523:converges
4141:converges
3867:such that
3358:, and its
2840:metrizable
2048:for every
1962:for every
1343:statistics
447:for every
380:such that
6600:Maharam's
6570:Dominated
6383:Intensity
6378:Hausdorff
6285:Saturated
6203:Invariant
6108:Types of
6067:Ï-algebra
6037:đ-system
6003:Borel set
5998:Baire set
5829:cite book
5718:1102.5245
5705:Bernoulli
5550:∈
5521:μ
5504:∫
5482:φ
5459:∗
5435:∈
5426:φ
5393:∈
5384:μ
5270:∞
5267:→
5244:φ
5235:∗
5225:→
5212:φ
5188:∈
5156:φ
5153:→
5135:φ
5114:φ
5089:∗
5081:∈
5072:φ
5048:∞
5045:→
5010:→
4974:∗
4966:∈
4963:φ
4943:∞
4940:→
4908:φ
4905:→
4884:φ
4754:μ
4734:μ
4726:→
4715:μ
4687:∈
4670:μ
4643:≤
4631:μ
4603:μ
4576:∈
4559:μ
4535:μ
4504:∈
4487:μ
4473:tightness
4453:μ
4437:∫
4433:→
4424:μ
4407:∫
4367:∈
4344:μ
4314:∈
4297:μ
4263:μ
4247:∫
4243:→
4234:μ
4217:∫
4177:∈
4154:μ
4121:∈
4104:μ
4013:⊂
4000:⊂
3987:⊂
3896:∞
3893:→
3542:Ω
3420:⇀
3387:⇒
3202:separable
3150:δ
3136:−
3133:μ
3124:φ
3115:∫
3081:∈
3078:μ
3062:δ
3050:φ
2998:δ
2985:∈
2969:→
2960:φ
2946:δ
2934:φ
2895:Σ
2641:at which
2624:∈
2565:∞
2562:→
2384:≡
2250:of space
2229:open sets
2203:≥
2151:of space
2104:≤
2052:function
2027:
2021:≥
2009:
1966:function
1941:
1935:≤
1923:
1856:
1850:→
1838:
1770:
1764:→
1752:
1568:⇒
1550:(denoted
1512:Σ
1480:…
1425:Σ
1404:σ
1394:with its
1304:given by
1272:μ
1253:μ
1232:μ
1210:μ
1179:∈
1144:μ
1123:μ
1117:∞
1114:→
1019:ε
1007:‖
1003:μ
1000:−
991:μ
987:‖
908:‖
904:ν
901:−
898:μ
895:‖
800:ν
797:−
785:μ
768:∈
757:⋅
737:ν
734:−
731:μ
710:are both
616:ν
600:∫
596:−
593:μ
577:∫
544:ν
541:−
538:μ
435:ε
415:μ
412:−
394:μ
367:uniformly
338:μ
328:∫
325:→
316:μ
305:∫
265:μ
262:→
244:μ
181:μ
171:∫
168:→
159:μ
148:∫
6617:Fubini's
6607:Egorov's
6575:Monotone
6534:variable
6512:Random:
6463:Strongly
6388:Lebesgue
6373:Harmonic
6363:Gaussian
6348:Counting
6315:Spectral
6310:Singular
6300:s-finite
6295:Ï-finite
6178:Discrete
6153:Complete
6110:Measures
6084:Null set
5972:function
5735:88518773
5663:See also
5539:for all
5177:for all
4955:for all
3969:We have
3494:→
3454:→
3265:, so is
3200:is also
2326:for all
2227:for all
2128:for all
1791:for all
1664:, while
1416:-algebra
1351:measures
1263:→
1220:→
949:converge
741:‖
727:‖
548:‖
534:‖
219:≤
121:calculus
6818:improve
6529:process
6524:measure
6519:element
6458:Bochner
6432:Trivial
6427:Tangent
6405:Product
6263:Regular
6241:)
6228:Perfect
6201:)
6166:)
6158:Content
6148:Complex
6089:Support
6062:-system
5951:Measure
4621:vaguely
4525:vaguely
4143:vaguely
3204:, then
2691:is the
1793:bounded
6807:, but
6595:Jordan
6580:Vitali
6539:vector
6468:Weakly
6330:Vector
6305:Signed
6258:Random
6199:Quasi-
6188:Finite
6168:Convex
6128:Banach
6118:Atomic
5946:spaces
5931:
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5850:
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3617:in law
3263:Polish
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3007:
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1396:Borel
960:> 0
509:. The
374:> 0
75:> 0
68:limits
6437:Young
6358:Euler
6353:Dirac
6325:Tight
6253:Radon
6223:Outer
6193:Inner
6143:Brown
6138:Borel
6133:Besov
6123:Baire
5731:S2CID
5713:arXiv
3600:then
3575:be a
3364:dense
3241:. If
1390:be a
971:>
695:is a
647:from
505:be a
452:>
6701:For
6590:Hahn
6446:Maps
6368:Haar
6239:Sub-
5993:Atom
5981:Sets
5871:ISBN
5848:ISBN
5835:link
5815:ISBN
5786:ISBN
5754:ISBN
4782:Let
4615:.
3676:Let
3611:(or
3579:and
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2838:any
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1042:For
1016:<
843:and
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517:and
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100:and
5723:doi
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6858::
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