6785:
3018:
3164:
112:
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
1317:
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.
623:
2905:
816:
3029:
281:
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
4607:
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
5243:
2035:
1949:
4452:
4262:
4916:
4767:
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:
1021:
5021:
1864:
1778:
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4127:
4015:
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5164:
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919:
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180:
2600:
4733:
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4693:
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1231:
1271:
5460:
3013:{\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\},}
2214:
2115:
516:
3462:
97:. 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.
3422:
1597:
273:
5090:
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6632:
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2388:
1185:
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1570:
1065:
492:
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4975:
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3285:
3224:
2865:
2628:
450:. 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.
5404:
709:
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5568:
4645:
4385:
4195:
51:
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
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5047:
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3705:
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2732:
6727:
2897:
1514:
5823:
5362:
5113:
3952:
3834:
3776:
3159:{\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\}.}
1424:
1403:
1349:
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
5190:
113:
exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if
4820:
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2415:
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1624:
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2760:
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5293:
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3308:
3248:
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2825:
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2533:
2486:
2435:
2356:
2336:
2257:
2237:
2158:
2138:
2059:
1973:
1887:
1805:
1718:
1537:
1377:
5570:. Applying the definition of weak-* convergence in terms of linear functionals, the characterization of vague convergence of measures is obtained. For compact
3737:
828:
To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that we are given two probability measures
5275:
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
5195:
6035:
5894:
6550:
6381:
5778:
5746:
5921:
1980:
1894:
6806:
184:
To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.
129:
The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge:
5692:
Madras, Neil; Sezer, Deniz (25 Feb 2011). "Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances".
825:. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.
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4390:
4200:
822:
132:
6722:
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6855:
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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
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4082:
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108:
This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in
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that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking
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1236:
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5409:
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6236:
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In a probability setting, vague convergence and weak convergence of probability measures are equivalent assuming
618:{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=\sup _{f}\left\{\int _{X}f\,d\mu -\int _{X}f\,d\nu \right\}.}
6810:
6617:
6700:
6146:
6077:
2165:
2066:
6013:
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6850:
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6187:
5880:
3394:
1575:
5055:
1432:
6752:
6652:
6474:
6196:
6042:
5667:
3739:). The following spaces of test functions are commonly used in the convergence of probability measures.
2365:
20:
2734:
with the usual topology), but it does not converge setwise. This is intuitively clear: we only know that
1160:
811:{\displaystyle \left\|\mu -\nu \right\|_{\text{TV}}=2\cdot \sup _{A\in {\mathcal {F}}}|\mu (A)-\nu (A)|.}
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6313:
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and that we are then asked to guess which one of the two distributions describes that law. The quantity
461:
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3313:
3257:
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5367:
1342:. It depends on a topology on the underlying space and thus is not a purely measure-theoretic notion.
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700:
633:
56:
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5328:
of Radon measures is isomorphic to a subspace of the space of continuous linear functionals on
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2831:
458:
This is the strongest notion of convergence shown on this page and is defined as follows. Let
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to a true probability measure, but rather to a sub-probability measure (a measure such that
3358:
There are many "arrow notations" for this kind of convergence: the most frequently used are
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then provides a sharp upper bound on the prior probability that our guess will be correct.
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is not specified to be a probability measure is not guaranteed to imply weak convergence.
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4587:
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4138:
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3190:
499:
2737:
2687:
5648:, so in this case weak convergence of measures is a special case of weak-* convergence.
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6116:
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5903:
5852:
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5278:
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4825:
4774:
4513:
4131:
3839:
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3668:
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3233:
3172:
2810:
2765:
2633:
2518:
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2316:
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2222:
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1522:
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31:
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The equivalence between these two definitions can be seen as a particular case of the
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converges weakly to the Dirac measure located at 0 (if we view these as measures on
1314:
converges setwise to
Lebesgue measure, but it does not converge in total variation.
6356:
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3251:
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685:
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283:
123:
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formalizes the assertion that the measure of each measurable set should converge:
354:
formalizes the assertion that the measure of all measurable sets should converge
6221:
5192:. That is, convergence occurs in the point-wise sense. In this case, one writes
3348:
1327:
27:
5238:{\displaystyle \varphi _{n}\mathrel {\stackrel {w^{*}}{\rightarrow }} \varphi }
6065:
2828:
2118:
1346:
1331:
191:
requires this convergence to take place for every continuous bounded function
6047:
5991:
5986:
3352:
648:, where the definition is of the same form, but the supremum is taken over
6072:
5931:
2217:
109:
5800:
Gradient Flows in Metric Spaces and in the Space of
Probability Measures
2899:. The weak topology is generated by the following basis of open sets:
5715:
2030:{\displaystyle \liminf \operatorname {E} _{n}\geq \operatorname {E} }
1944:{\displaystyle \limsup \operatorname {E} _{n}\leq \operatorname {E} }
213:
to be approximated equally well (thus, convergence is non-uniform in
100:
Three of the most common notions of convergence are described below.
3485:
3445:
1254:
1211:
1338:
is one of many types of convergence relating to the convergence of
927:
Given the above definition of total variation distance, a sequence
5706:
4447:{\displaystyle \int _{X}f\,d\mu _{n}\rightarrow \int _{X}f\,d\mu }
4257:{\displaystyle \int _{X}f\,d\mu _{n}\rightarrow \int _{X}f\,d\mu }
5872:
4911:{\displaystyle \varphi \left(x_{n}\right)\rightarrow \varphi (x)}
4759:
Weak convergence of measures as an example of weak-* convergence
5876:
4457:
In general, these two convergence notions are not equivalent.
6778:
5773:. Internet Archive. New York, Academic Press. pp. 84â99.
1572:) if any of the following equivalent conditions is true (here
1016:{\displaystyle \|\mu _{n}-\mu \|_{\text{TV}}<\varepsilon .}
688:, the total variation metric coincides with the Radon metric.
5016:{\displaystyle x_{n}\mathrel {\stackrel {w}{\rightarrow }} x}
3538:
3487:
3319:
3263:
3202:
3074:
2843:
1859:{\displaystyle \operatorname {E} _{n}\to \operatorname {E} }
1773:{\displaystyle \operatorname {E} _{n}\to \operatorname {E} }
1172:
1049:
761:
476:
3145:
2999:
195:. This notion treats convergence for different functions
4315:{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}
4122:{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}
936:
of measures defined on the same measure space is said to
4010:{\displaystyle C_{c}\subset C_{0}\subset C_{B}\subset C}
2807:
This definition of weak convergence can be extended for
199:
independently of one another, i.e., different functions
1147:{\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}
5159:{\displaystyle \varphi _{n}(x)\rightarrow \varphi (x)}
292:
5596:
5576:
5534:
5521:{\displaystyle \varphi _{n}(f)=\int _{X}f\,d\mu _{n}}
5468:
5412:
5370:
5334:
5305:
5281:
5251:
5198:
5172:
5121:
5101:
5058:
5029:
4983:
4950:
4924:
4871:
4851:
4828:
4801:
4777:
4741:
4701:
4653:
4618:
4590:
4542:
4522:
4470:
4393:
4351:
4331:
4278:
4203:
4161:
4141:
4085:
4050:
4023:
3964:
3924:
3862:
3842:
3806:
3784:
3748:
3713:
3691:
3671:
3557:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
3526:
3470:
3430:
3397:
3364:
3316:
3296:
3260:
3236:
3199:
3175:
3032:
2908:
2873:
2840:
2813:
2788:
2768:
2740:
2718:
2690:
2659:
2636:
2608:
2541:
2521:
2494:
2474:
2447:
2423:
2396:
2368:
2344:
2324:
2267:
2245:
2225:
2168:
2146:
2126:
2069:
2047:
1983:
1961:
1897:
1875:
1815:
1793:
1729:
1706:
1679:
1659:
1632:
1605:
1578:
1545:
1525:
1490:
1435:
1412:
1391:
1365:
1239:
1196:
1163:
1093:
1037:
974:
914:{\displaystyle {2+\|\mu -\nu \|_{\text{TV}} \over 4}}
873:
712:
703:, then the total variation distance is also given by
519:
464:
429:{\displaystyle |\mu _{n}(A)-\mu (A)|<\varepsilon }
375:
230:
135:
38:. For an intuitive general sense of what is meant by
4464:. That is, a tight sequence of probability measures
3910:{\displaystyle \lim _{|x|\rightarrow \infty }f(x)=0}
6740:
6688:
6641:
6541:
6434:
6327:
6096:
5969:
5910:
5851:
5640:
5582:
5562:
5520:
5454:
5398:
5356:
5320:
5287:
5263:
5237:
5184:
5158:
5107:
5084:
5041:
5015:
4969:
4936:
4910:
4857:
4834:
4814:
4783:
4747:
4727:
4687:
4639:
4596:
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4528:
4504:
4446:
4379:
4337:
4314:
4256:
4189:
4147:
4121:
4063:
4036:
4009:
3946:
3909:
3848:
3828:
3790:
3770:
3731:
3699:
3677:
3653:) in the sense of weak convergence of measures on
3556:
3499:{\displaystyle P_{n}\xrightarrow {\mathcal {D}} P}
3498:
3456:
3416:
3383:
3335:
3302:
3279:
3242:
3218:
3181:
3158:
3012:
2891:
2859:
2819:
2796:
2774:
2754:
2726:
2704:
2672:
2642:
2622:
2594:
2527:
2507:
2480:
2460:
2429:
2409:
2382:
2350:
2330:
2307:
2251:
2231:
2208:
2152:
2132:
2109:
2053:
2029:
1967:
1943:
1881:
1858:
1799:
1772:
1712:
1692:
1665:
1645:
1618:
1591:
1564:
1531:
1508:
1476:
1418:
1397:
1371:
1265:
1225:
1179:
1146:
1059:
1015:
913:
860:single sample distributed according to the law of
810:
672:ranging over the set of continuous functions from
652:ranging over the set of measurable functions from
617:
486:
428:
331:
267:
175:{\displaystyle \int f\,d\mu _{n}\to \int f\,d\mu }
174:
5798:Ambrosio, L., Gigli, N. & Savaré, G. (2005).
2867:, the set of all probability measures defined on
2595:{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}
4791:be a topological vector space or Banach space.
4728:{\displaystyle \mu _{n}{\overset {v}{\to }}\mu }
3864:
3226:is metrizable and separable, for example by the
2543:
2268:
2169:
2070:
1984:
1898:
1095:
749:
550:
343:. As before, this convergence is non-uniform in
332:{\textstyle \int f\,d\mu _{n}\to \int f\,d\mu }
286:, setwise convergence implies the convergence
4688:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
4577:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
4505:{\displaystyle (\mu _{n})_{n\in \mathbb {N} }}
1226:{\displaystyle \mu _{n}\xrightarrow {sw} \mu }
5888:
5295:be a locally compact Hausdorff space. By the
1266:{\displaystyle \mu _{n}\xrightarrow {s} \mu }
8:
6633:RieszâMarkovâKakutani representation theorem
5858:. New York, NY: John Wiley & Sons, Inc.
5835:. New York, NY: John Wiley & Sons, Inc.
5822:: CS1 maint: multiple names: authors list (
5455:{\displaystyle \varphi _{n}\in C_{0}(X)^{*}}
4647:). Thus, a sequence of probability measures
995:
975:
896:
883:
664:at most 1; and also in contrast to the
644:. This is in contrast, for example, to the
122:is a sequence of probability measures on a
6728:Vitale's random BrunnâMinkowski inequality
6645:
5895:
5881:
5873:
6829:Learn how and when to remove this message
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4428:
4415:
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3954:the class of continuous bounded functions
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3235:
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3031:
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2812:
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2787:
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2744:
2739:
2719:
2717:
2694:
2689:
2664:
2658:
2635:
2615:
2607:
2562:
2546:
2540:
2520:
2499:
2493:
2473:
2452:
2446:
2422:
2401:
2395:
2375:
2367:
2343:
2323:
2275:
2266:
2244:
2224:
2209:{\displaystyle \liminf P_{n}(U)\geq P(U)}
2176:
2167:
2145:
2125:
2110:{\displaystyle \limsup P_{n}(C)\leq P(C)}
2077:
2068:
2046:
1991:
1982:
1960:
1905:
1896:
1874:
1820:
1814:
1792:
1734:
1728:
1705:
1684:
1678:
1658:
1637:
1631:
1610:
1604:
1583:
1577:
1550:
1544:
1524:
1489:
1446:
1440:
1434:
1411:
1390:
1364:
1244:
1238:
1201:
1195:
1171:
1170:
1162:
1114:
1098:
1092:
1048:
1047:
1036:
998:
982:
973:
944:in total variation distance if for every
899:
874:
872:
800:
768:
760:
759:
752:
733:
711:
600:
591:
577:
568:
553:
540:
518:
502:distance between two (positive) measures
475:
474:
463:
415:
385:
376:
374:
322:
307:
299:
291:
235:
229:
165:
150:
142:
134:
6792:This article includes a list of general
5802:. Basel: ETH ZĂŒrich, BirkhĂ€user Verlag.
1518:converge weakly to a probability measure
5684:
3457:{\displaystyle P_{n}\xrightarrow {w} P}
3310:is separable, it naturally embeds into
1080:is said to converge setwise to a limit
454:Total variation convergence of measures
5815:
3417:{\displaystyle P_{n}\rightharpoonup P}
2967: is bounded and continuous,
1592:{\displaystyle \operatorname {E} _{n}}
268:{\displaystyle \mu _{n}(A)\to \mu (A)}
5085:{\displaystyle \varphi _{n}\in V^{*}}
4071:with respect to uniform convergence.
3798:each vanishing outside a compact set.
2834:. It also defines a weak topology on
1477:{\displaystyle P_{n}\,(n=1,2,\dots )}
1276:For example, as a consequence of the
7:
6741:Applications & related
5762:
5760:
5758:
5364:. Therefore, for each Radon measure
3510:Weak convergence of random variables
2383:{\displaystyle S\equiv \mathbf {R} }
339:for any bounded measurable function
5854:Convergence of Probability Measures
1180:{\displaystyle A\in {\mathcal {F}}}
668:, where the supremum is taken over
80:to ensure the 'difference' between
34:, there are various notions of the
6798:it lacks sufficient corresponding
5258:
5036:
4931:
3884:
3836:the class of continuous functions
3778:the class of continuous functions
3530:
3384:{\displaystyle P_{n}\Rightarrow P}
3117:
2883:
2553:
2308:{\displaystyle \lim P_{n}(A)=P(A)}
2012:
1988:
1926:
1902:
1841:
1817:
1755:
1731:
1666:{\displaystyle \operatorname {E} }
1660:
1580:
1565:{\displaystyle P_{n}\Rightarrow P}
1500:
1413:
1105:
1060:{\displaystyle (X,{\mathcal {F}})}
487:{\displaystyle (X,{\mathcal {F}})}
277:Again, no uniformity over the set
42:, consider a sequence of measures
14:
5641:{\displaystyle C_{0}(X)=C_{B}(X)}
4970:{\displaystyle \varphi \in V^{*}}
3661:Comparison with vague convergence
3336:{\displaystyle {\mathcal {P}}(S)}
3280:{\displaystyle {\mathcal {P}}(S)}
3219:{\displaystyle {\mathcal {P}}(S)}
2860:{\displaystyle {\mathcal {P}}(S)}
2623:{\displaystyle x\in \mathbf {R} }
2439:cumulative distribution functions
1426:. A bounded sequence of positive
6783:
6670:Lebesgue differentiation theorem
6551:Carathéodory's extension theorem
5399:{\displaystyle \mu _{n}\in M(X)}
5093:converges in the weak-* topology
4763:Despite having the same name as
2977:
2961:
2790:
2720:
2653:For example, the sequence where
2616:
2376:
628:Here the supremum is taken over
203:may require different values of
5658:Convergence of random variables
5406:, there is a linear functional
3685:be a metric space (for example
3516:Convergence of random variables
3096:
2947:
1027:Setwise convergence of measures
856:. Assume now that we are given
836:, as well as a random variable
5770:A course in probability theory
5635:
5629:
5613:
5607:
5557:
5551:
5485:
5479:
5443:
5436:
5393:
5387:
5351:
5345:
5315:
5309:
5255:
5213:
5153:
5147:
5141:
5138:
5132:
5033:
4998:
4928:
4905:
4899:
4893:
4714:
4668:
4654:
4628:
4622:
4557:
4543:
4485:
4471:
4421:
4374:
4368:
4231:
4184:
4178:
3941:
3935:
3898:
3892:
3881:
3877:
3869:
3823:
3817:
3765:
3759:
3726:
3714:
3551:
3527:
3408:
3375:
3330:
3324:
3274:
3268:
3213:
3207:
3085:
3079:
2957:
2886:
2874:
2854:
2848:
2589:
2583:
2574:
2568:
2550:
2302:
2296:
2287:
2281:
2203:
2197:
2188:
2182:
2104:
2098:
2089:
2083:
2024:
2018:
2006:
2000:
1938:
1932:
1920:
1914:
1853:
1847:
1838:
1835:
1829:
1767:
1761:
1752:
1749:
1743:
1556:
1503:
1491:
1471:
1447:
1141:
1135:
1126:
1120:
1102:
1054:
1038:
801:
797:
791:
782:
776:
769:
729:
715:
536:
522:
481:
465:
416:
412:
406:
397:
391:
377:
313:
262:
256:
250:
247:
241:
156:
1:
5850:Billingsley, Patrick (1999).
5831:Billingsley, Patrick (1995).
5563:{\displaystyle f\in C_{0}(X)}
4640:{\displaystyle \mu (X)\leq 1}
4380:{\displaystyle f\in C_{B}(X)}
4190:{\displaystyle f\in C_{c}(X)}
1345:There are several equivalent
446:and for every measurable set
5297:Riesz-Representation theorem
5264:{\displaystyle n\to \infty }
5042:{\displaystyle n\to \infty }
4937:{\displaystyle n\to \infty }
3700:{\displaystyle \mathbb {R} }
2797:{\displaystyle \mathbf {R} }
2727:{\displaystyle \mathbf {R} }
2390:with its usual topology, if
1322:Weak convergence of measures
1289:of measures on the interval
1190:Typical arrow notations are
632:ranging over the set of all
6723:PrĂ©kopaâLeindler inequality
2892:{\displaystyle (S,\Sigma )}
2782:because of the topology of
1673:denotes expectation or the
1599:denotes expectation or the
1509:{\displaystyle (S,\Sigma )}
352:total variation convergence
6872:
6665:Lebesgue's density theorem
3513:
59:; for any error tolerance
18:
6856:Convergence (mathematics)
6718:MinkowskiâSteiner formula
6648:
6533:Projection-valued measure
4516:to a probability measure
3608:) to the random variable
823:MongeâKantorovich duality
6701:Isoperimetric inequality
6680:VitaliâHahnâSaks theorem
6009:Carathéodory's criterion
5357:{\displaystyle C_{0}(X)}
5108:{\displaystyle \varphi }
3947:{\displaystyle C_{B}(X)}
3829:{\displaystyle C_{0}(X)}
3771:{\displaystyle C_{c}(X)}
19:Not to be confused with
6813:more precise citations.
6706:BrunnâMinkowski theorem
6575:Decomposition theorems
5833:Probability and Measure
5767:Chung, Kai Lai (1974).
4272:A sequence of measures
4079:A sequence of measures
3343:as the (closed) set of
1419:{\displaystyle \Sigma }
1398:{\displaystyle \sigma }
70:sufficiently large for
40:convergence of measures
36:convergence of measures
6753:Descriptive set theory
6653:Disintegration theorem
6088:Universally measurable
5737:Klenke, Achim (2006).
5642:
5584:
5564:
5522:
5456:
5400:
5358:
5322:
5289:
5265:
5239:
5186:
5185:{\displaystyle x\in V}
5160:
5109:
5086:
5043:
5017:
4971:
4938:
4912:
4859:
4836:
4816:
4785:
4749:
4729:
4689:
4641:
4598:
4578:
4530:
4506:
4448:
4381:
4339:
4316:
4258:
4191:
4149:
4123:
4065:
4038:
4011:
3948:
3911:
3850:
3830:
3792:
3772:
3733:
3701:
3679:
3642:) converges weakly to
3572:be a metric space. If
3558:
3500:
3458:
3418:
3385:
3337:
3304:
3281:
3244:
3220:
3183:
3160:
3014:
2893:
2861:
2821:
2798:
2776:
2756:
2728:
2706:
2674:
2644:
2624:
2596:
2529:
2509:
2482:
2462:
2431:
2411:
2384:
2352:
2332:
2309:
2253:
2233:
2210:
2154:
2134:
2111:
2055:
2031:
1969:
1945:
1883:
1860:
1801:
1774:
1714:
1694:
1667:
1647:
1620:
1593:
1566:
1533:
1510:
1478:
1420:
1399:
1373:
1278:RiemannâLebesgue lemma
1267:
1227:
1181:
1148:
1061:
1017:
915:
812:
619:
488:
430:
333:
269:
176:
21:Convergence in measure
6555:Convergence theorems
6014:Cylindrical Ï-algebra
5673:Tightness of measures
5663:LĂ©vyâProkhorov metric
5643:
5585:
5565:
5523:
5457:
5401:
5359:
5323:
5290:
5266:
5240:
5187:
5161:
5110:
5087:
5044:
5018:
4972:
4939:
4913:
4860:
4837:
4817:
4815:{\displaystyle x_{n}}
4786:
4750:
4730:
4690:
4642:
4599:
4579:
4531:
4507:
4449:
4382:
4340:
4317:
4259:
4192:
4150:
4124:
4066:
4064:{\displaystyle C_{c}}
4039:
4037:{\displaystyle C_{0}}
4012:
3949:
3912:
3851:
3831:
3793:
3773:
3734:
3702:
3680:
3559:
3501:
3459:
3419:
3386:
3338:
3305:
3282:
3245:
3228:LĂ©vyâProkhorov metric
3221:
3184:
3161:
3015:
2894:
2862:
2822:
2799:
2777:
2757:
2729:
2707:
2675:
2673:{\displaystyle P_{n}}
2645:
2625:
2597:
2530:
2510:
2508:{\displaystyle P_{n}}
2488:, respectively, then
2483:
2463:
2461:{\displaystyle P_{n}}
2432:
2412:
2410:{\displaystyle F_{n}}
2385:
2353:
2333:
2310:
2254:
2234:
2211:
2155:
2135:
2112:
2056:
2039:lower semi-continuous
2032:
1970:
1953:upper semi-continuous
1946:
1884:
1861:
1802:
1775:
1715:
1700:norm with respect to
1695:
1693:{\displaystyle L^{1}}
1668:
1648:
1646:{\displaystyle P_{n}}
1626:norm with respect to
1621:
1619:{\displaystyle L^{1}}
1594:
1567:
1534:
1511:
1479:
1421:
1400:
1374:
1268:
1228:
1182:
1149:
1062:
1018:
916:
813:
620:
489:
431:
334:
270:
177:
104:Informal descriptions
6623:Minkowski inequality
6497:Cylinder set measure
6382:Infinite-dimensional
5997:equivalence relation
5927:Lebesgue integration
5679:Notes and references
5594:
5574:
5532:
5466:
5410:
5368:
5332:
5321:{\displaystyle M(X)}
5303:
5279:
5249:
5196:
5170:
5119:
5099:
5056:
5027:
4981:
4948:
4922:
4869:
4849:
4826:
4799:
4775:
4748:{\displaystyle \mu }
4739:
4699:
4651:
4616:
4597:{\displaystyle \mu }
4588:
4584:converges weakly to
4540:
4529:{\displaystyle \mu }
4520:
4468:
4391:
4349:
4338:{\displaystyle \mu }
4329:
4276:
4201:
4159:
4148:{\displaystyle \mu }
4139:
4083:
4048:
4021:
3962:
3922:
3860:
3840:
3804:
3782:
3746:
3711:
3689:
3669:
3657:, as defined above.
3625:pushforward measures
3524:
3468:
3428:
3395:
3362:
3314:
3294:
3258:
3234:
3197:
3173:
3030:
2906:
2871:
2838:
2811:
2786:
2766:
2738:
2716:
2688:
2657:
2634:
2606:
2539:
2519:
2515:converges weakly to
2492:
2472:
2445:
2421:
2394:
2366:
2342:
2322:
2265:
2243:
2223:
2166:
2144:
2124:
2067:
2045:
1981:
1959:
1895:
1873:
1866:for all bounded and
1813:
1791:
1786:continuous functions
1727:
1704:
1677:
1657:
1630:
1603:
1576:
1543:
1523:
1488:
1433:
1428:probability measures
1410:
1389:
1363:
1237:
1194:
1161:
1091:
1035:
972:
871:
710:
701:probability measures
680:. In the case where
634:measurable functions
517:
462:
373:
290:
228:
133:
66:we require there be
30:, more specifically
16:Mathematical concept
6618:Hölder's inequality
6480:of random variables
6442:Measurable function
6329:Particular measures
5918:Absolute continuity
5741:. Springer-Verlag.
5668:Prokhorov's theorem
3623:if the sequence of
3491:
3449:
3250:is also compact or
2755:{\displaystyle 1/n}
2705:{\displaystyle 1/n}
2061:bounded from below;
1975:bounded from above;
1868:Lipschitz functions
1351:Portmanteau theorem
1258:
1218:
222:setwise convergence
6758:Probability theory
6083:Transverse measure
6061:Non-measurable set
6043:Locally measurable
5739:Probability Theory
5638:
5580:
5560:
5518:
5452:
5396:
5354:
5318:
5285:
5261:
5235:
5182:
5156:
5105:
5082:
5039:
5013:
4967:
4934:
4908:
4855:
4832:
4812:
4781:
4745:
4725:
4685:
4637:
4594:
4574:
4526:
4502:
4444:
4377:
4335:
4312:
4254:
4187:
4145:
4119:
4061:
4044:is the closure of
4034:
4007:
3944:
3907:
3888:
3846:
3826:
3788:
3768:
3729:
3697:
3675:
3554:
3496:
3454:
3414:
3381:
3333:
3300:
3277:
3240:
3216:
3179:
3156:
3010:
2889:
2857:
2817:
2794:
2772:
2752:
2724:
2702:
2670:
2640:
2620:
2592:
2557:
2525:
2505:
2478:
2458:
2427:
2407:
2380:
2348:
2328:
2305:
2249:
2229:
2206:
2150:
2130:
2107:
2051:
2027:
1965:
1941:
1879:
1856:
1797:
1770:
1710:
1690:
1663:
1643:
1616:
1589:
1562:
1529:
1506:
1474:
1416:
1395:
1369:
1263:
1223:
1177:
1144:
1109:
1057:
1013:
955:such that for all
951:, there exists an
911:
808:
767:
662:Lipschitz constant
646:Wasserstein metric
615:
558:
484:
426:
329:
265:
172:
6839:
6838:
6831:
6776:
6775:
6736:
6735:
6465:almost everywhere
6411:Spherical measure
6309:Strictly positive
6237:Projection-valued
5977:Almost everywhere
5950:Probability space
5780:978-0-12-174151-8
5748:978-1-84800-047-6
5716:10.3150/09-BEJ238
5583:{\displaystyle X}
5288:{\displaystyle X}
5229:
5007:
4858:{\displaystyle x}
4835:{\displaystyle V}
4784:{\displaystyle V}
4720:
4075:Vague Convergence
3863:
3849:{\displaystyle f}
3791:{\displaystyle f}
3678:{\displaystyle X}
3585:is a sequence of
3566:probability space
3492:
3450:
3303:{\displaystyle S}
3243:{\displaystyle S}
3182:{\displaystyle S}
3143:
3090:
3065:
2997:
2984:
2968:
2941:
2916:
2832:topological space
2820:{\displaystyle S}
2775:{\displaystyle 0}
2643:{\displaystyle F}
2542:
2528:{\displaystyle P}
2481:{\displaystyle P}
2430:{\displaystyle F}
2351:{\displaystyle P}
2331:{\displaystyle A}
2252:{\displaystyle S}
2232:{\displaystyle U}
2153:{\displaystyle S}
2133:{\displaystyle C}
2054:{\displaystyle f}
1968:{\displaystyle f}
1882:{\displaystyle f}
1800:{\displaystyle f}
1713:{\displaystyle P}
1532:{\displaystyle P}
1372:{\displaystyle S}
1259:
1219:
1094:
1001:
909:
902:
748:
736:
549:
543:
510:is then given by
358:, i.e. for every
6863:
6834:
6827:
6823:
6820:
6814:
6809:this article by
6800:inline citations
6787:
6786:
6779:
6711:Milman's reverse
6694:
6692:Lebesgue measure
6646:
6050:
6036:infimum/supremum
5957:Measurable space
5897:
5890:
5883:
5874:
5869:
5857:
5846:
5827:
5821:
5813:
5785:
5784:
5764:
5753:
5752:
5734:
5728:
5727:
5709:
5689:
5647:
5645:
5644:
5639:
5628:
5627:
5606:
5605:
5589:
5587:
5586:
5581:
5569:
5567:
5566:
5561:
5550:
5549:
5527:
5525:
5524:
5519:
5517:
5516:
5500:
5499:
5478:
5477:
5461:
5459:
5458:
5453:
5451:
5450:
5435:
5434:
5422:
5421:
5405:
5403:
5402:
5397:
5380:
5379:
5363:
5361:
5360:
5355:
5344:
5343:
5327:
5325:
5324:
5319:
5294:
5292:
5291:
5286:
5270:
5268:
5267:
5262:
5244:
5242:
5241:
5236:
5231:
5230:
5228:
5227:
5226:
5216:
5211:
5208:
5207:
5191:
5189:
5188:
5183:
5165:
5163:
5162:
5157:
5131:
5130:
5114:
5112:
5111:
5106:
5091:
5089:
5088:
5083:
5081:
5080:
5068:
5067:
5048:
5046:
5045:
5040:
5022:
5020:
5019:
5014:
5009:
5008:
5006:
5001:
4996:
4993:
4992:
4976:
4974:
4973:
4968:
4966:
4965:
4943:
4941:
4940:
4935:
4917:
4915:
4914:
4909:
4892:
4888:
4887:
4864:
4862:
4861:
4856:
4843:converges weakly
4841:
4839:
4838:
4833:
4821:
4819:
4818:
4813:
4811:
4810:
4790:
4788:
4787:
4782:
4765:weak convergence
4754:
4752:
4751:
4746:
4734:
4732:
4731:
4726:
4721:
4713:
4711:
4710:
4694:
4692:
4691:
4686:
4684:
4683:
4682:
4666:
4665:
4646:
4644:
4643:
4638:
4603:
4601:
4600:
4595:
4583:
4581:
4580:
4575:
4573:
4572:
4571:
4555:
4554:
4535:
4533:
4532:
4527:
4511:
4509:
4508:
4503:
4501:
4500:
4499:
4483:
4482:
4453:
4451:
4450:
4445:
4433:
4432:
4420:
4419:
4403:
4402:
4386:
4384:
4383:
4378:
4367:
4366:
4344:
4342:
4341:
4336:
4323:converges weakly
4321:
4319:
4318:
4313:
4311:
4310:
4309:
4297:
4293:
4292:
4268:Weak Convergence
4263:
4261:
4260:
4255:
4243:
4242:
4230:
4229:
4213:
4212:
4196:
4194:
4193:
4188:
4177:
4176:
4154:
4152:
4151:
4146:
4128:
4126:
4125:
4120:
4118:
4117:
4116:
4104:
4100:
4099:
4070:
4068:
4067:
4062:
4060:
4059:
4043:
4041:
4040:
4035:
4033:
4032:
4016:
4014:
4013:
4008:
4000:
3999:
3987:
3986:
3974:
3973:
3953:
3951:
3950:
3945:
3934:
3933:
3916:
3914:
3913:
3908:
3887:
3880:
3872:
3855:
3853:
3852:
3847:
3835:
3833:
3832:
3827:
3816:
3815:
3797:
3795:
3794:
3789:
3777:
3775:
3774:
3769:
3758:
3757:
3738:
3736:
3735:
3732:{\displaystyle }
3730:
3706:
3704:
3703:
3698:
3696:
3684:
3682:
3681:
3676:
3622:
3587:random variables
3584:
3563:
3561:
3560:
3555:
3550:
3542:
3541:
3505:
3503:
3502:
3497:
3490:
3481:
3480:
3479:
3463:
3461:
3460:
3455:
3441:
3440:
3439:
3423:
3421:
3420:
3415:
3407:
3406:
3390:
3388:
3387:
3382:
3374:
3373:
3342:
3340:
3339:
3334:
3323:
3322:
3309:
3307:
3306:
3301:
3286:
3284:
3283:
3278:
3267:
3266:
3249:
3247:
3246:
3241:
3225:
3223:
3222:
3217:
3206:
3205:
3188:
3186:
3185:
3180:
3165:
3163:
3162:
3157:
3152:
3148:
3147:
3144:
3141:
3134:
3130:
3120:
3111:
3110:
3088:
3078:
3077:
3063:
3054:
3053:
3019:
3017:
3016:
3011:
3006:
3002:
3001:
2998:
2995:
2985:
2982:
2980:
2969:
2966:
2964:
2939:
2938:
2937:
2914:
2898:
2896:
2895:
2890:
2866:
2864:
2863:
2858:
2847:
2846:
2826:
2824:
2823:
2818:
2803:
2801:
2800:
2795:
2793:
2781:
2779:
2778:
2773:
2761:
2759:
2758:
2753:
2748:
2733:
2731:
2730:
2725:
2723:
2711:
2709:
2708:
2703:
2698:
2679:
2677:
2676:
2671:
2669:
2668:
2649:
2647:
2646:
2641:
2629:
2627:
2626:
2621:
2619:
2601:
2599:
2598:
2593:
2567:
2566:
2556:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2504:
2503:
2487:
2485:
2484:
2479:
2467:
2465:
2464:
2459:
2457:
2456:
2441:of the measures
2436:
2434:
2433:
2428:
2416:
2414:
2413:
2408:
2406:
2405:
2389:
2387:
2386:
2381:
2379:
2357:
2355:
2354:
2349:
2337:
2335:
2334:
2329:
2314:
2312:
2311:
2306:
2280:
2279:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2215:
2213:
2212:
2207:
2181:
2180:
2159:
2157:
2156:
2151:
2139:
2137:
2136:
2131:
2116:
2114:
2113:
2108:
2082:
2081:
2060:
2058:
2057:
2052:
2036:
2034:
2033:
2028:
1996:
1995:
1974:
1972:
1971:
1966:
1950:
1948:
1947:
1942:
1910:
1909:
1888:
1886:
1885:
1880:
1865:
1863:
1862:
1857:
1825:
1824:
1806:
1804:
1803:
1798:
1779:
1777:
1776:
1771:
1739:
1738:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1689:
1688:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1642:
1641:
1625:
1623:
1622:
1617:
1615:
1614:
1598:
1596:
1595:
1590:
1588:
1587:
1571:
1569:
1568:
1563:
1555:
1554:
1538:
1536:
1535:
1530:
1515:
1513:
1512:
1507:
1483:
1481:
1480:
1475:
1445:
1444:
1425:
1423:
1422:
1417:
1404:
1402:
1401:
1396:
1378:
1376:
1375:
1370:
1336:weak convergence
1313:
1292:
1288:
1272:
1270:
1269:
1264:
1250:
1249:
1248:
1232:
1230:
1229:
1224:
1207:
1206:
1205:
1186:
1184:
1183:
1178:
1176:
1175:
1153:
1151:
1150:
1145:
1119:
1118:
1108:
1083:
1079:
1069:measurable space
1066:
1064:
1063:
1058:
1053:
1052:
1022:
1020:
1019:
1014:
1003:
1002:
999:
987:
986:
964:
954:
950:
943:
935:
920:
918:
917:
912:
910:
905:
904:
903:
900:
875:
863:
855:
851:
847:
843:
839:
835:
831:
817:
815:
814:
809:
804:
772:
766:
765:
764:
738:
737:
734:
732:
728:
698:
694:
683:
679:
675:
671:
659:
655:
651:
643:
639:
631:
624:
622:
621:
616:
611:
607:
596:
595:
573:
572:
557:
545:
544:
541:
539:
535:
509:
505:
496:measurable space
493:
491:
490:
485:
480:
479:
449:
445:
435:
433:
432:
427:
419:
390:
389:
380:
368:
364:
346:
342:
338:
336:
335:
330:
312:
311:
280:
274:
272:
271:
266:
240:
239:
216:
212:
202:
198:
194:
189:weak convergence
181:
179:
178:
173:
155:
154:
121:
96:
93:is smaller than
92:
88:
79:
69:
65:
54:
50:
6871:
6870:
6866:
6865:
6864:
6862:
6861:
6860:
6841:
6840:
6835:
6824:
6818:
6815:
6805:Please help to
6804:
6788:
6784:
6777:
6772:
6768:Spectral theory
6748:Convex analysis
6732:
6689:
6684:
6637:
6537:
6485:in distribution
6430:
6323:
6153:Logarithmically
6092:
6048:
6031:Essential range
5965:
5906:
5901:
5866:
5849:
5843:
5830:
5814:
5810:
5797:
5794:
5792:Further reading
5789:
5788:
5781:
5766:
5765:
5756:
5749:
5736:
5735:
5731:
5691:
5690:
5686:
5681:
5654:
5619:
5597:
5592:
5591:
5572:
5571:
5541:
5530:
5529:
5508:
5491:
5469:
5464:
5463:
5442:
5426:
5413:
5408:
5407:
5371:
5366:
5365:
5335:
5330:
5329:
5301:
5300:
5277:
5276:
5247:
5246:
5218:
5199:
5194:
5193:
5168:
5167:
5122:
5117:
5116:
5097:
5096:
5072:
5059:
5054:
5053:
5025:
5024:
4984:
4979:
4978:
4957:
4946:
4945:
4920:
4919:
4879:
4875:
4867:
4866:
4847:
4846:
4824:
4823:
4802:
4797:
4796:
4773:
4772:
4761:
4737:
4736:
4702:
4697:
4696:
4667:
4657:
4649:
4648:
4614:
4613:
4586:
4585:
4556:
4546:
4538:
4537:
4536:if and only if
4518:
4517:
4484:
4474:
4466:
4465:
4424:
4411:
4394:
4389:
4388:
4358:
4347:
4346:
4327:
4326:
4284:
4280:
4279:
4274:
4273:
4270:
4234:
4221:
4204:
4199:
4198:
4168:
4157:
4156:
4137:
4136:
4091:
4087:
4086:
4081:
4080:
4077:
4051:
4046:
4045:
4024:
4019:
4018:
3991:
3978:
3965:
3960:
3959:
3925:
3920:
3919:
3858:
3857:
3838:
3837:
3807:
3802:
3801:
3780:
3779:
3749:
3744:
3743:
3709:
3708:
3687:
3686:
3667:
3666:
3663:
3648:
3637:
3632:
3617:
3602:in distribution
3598:converge weakly
3594:
3578:
3573:
3522:
3521:
3518:
3512:
3471:
3466:
3465:
3431:
3426:
3425:
3398:
3393:
3392:
3365:
3360:
3359:
3312:
3311:
3292:
3291:
3256:
3255:
3232:
3231:
3195:
3194:
3171:
3170:
3102:
3101:
3097:
3095:
3091:
3062:
3058:
3033:
3028:
3027:
2983: and
2946:
2942:
2917:
2913:
2909:
2904:
2903:
2869:
2868:
2836:
2835:
2809:
2808:
2784:
2783:
2764:
2763:
2736:
2735:
2714:
2713:
2686:
2685:
2660:
2655:
2654:
2650:is continuous.
2632:
2631:
2604:
2603:
2602:for all points
2558:
2537:
2536:
2535:if and only if
2517:
2516:
2495:
2490:
2489:
2470:
2469:
2448:
2443:
2442:
2419:
2418:
2397:
2392:
2391:
2364:
2363:
2340:
2339:
2320:
2319:
2317:continuity sets
2271:
2263:
2262:
2241:
2240:
2221:
2220:
2172:
2164:
2163:
2142:
2141:
2122:
2121:
2073:
2065:
2064:
2043:
2042:
1987:
1979:
1978:
1957:
1956:
1901:
1893:
1892:
1871:
1870:
1816:
1811:
1810:
1789:
1788:
1730:
1725:
1724:
1702:
1701:
1680:
1675:
1674:
1655:
1654:
1633:
1628:
1627:
1606:
1601:
1600:
1579:
1574:
1573:
1546:
1541:
1540:
1521:
1520:
1486:
1485:
1436:
1431:
1430:
1408:
1407:
1387:
1386:
1361:
1360:
1324:
1299:
1294:
1291:[â1, 1]
1290:
1286:
1281:
1280:, the sequence
1240:
1235:
1234:
1197:
1192:
1191:
1159:
1158:
1110:
1089:
1088:
1081:
1077:
1072:
1033:
1032:
1029:
994:
978:
970:
969:
965:, one has that
956:
952:
945:
941:
933:
928:
895:
876:
869:
868:
861:
853:
849:
845:
844:has law either
841:
840:. We know that
837:
833:
829:
718:
714:
713:
708:
707:
696:
692:
681:
678:[â1, 1]
677:
673:
669:
658:[â1, 1]
657:
653:
649:
642:[â1, 1]
641:
637:
629:
587:
564:
563:
559:
525:
521:
520:
515:
514:
507:
503:
500:total variation
460:
459:
456:
447:
437:
381:
371:
370:
366:
359:
344:
340:
303:
288:
287:
278:
231:
226:
225:
214:
204:
200:
196:
192:
146:
131:
130:
119:
114:
106:
94:
90:
86:
81:
71:
67:
60:
52:
48:
43:
24:
17:
12:
11:
5:
6869:
6867:
6859:
6858:
6853:
6851:Measure theory
6843:
6842:
6837:
6836:
6791:
6789:
6782:
6774:
6773:
6771:
6770:
6765:
6760:
6755:
6750:
6744:
6742:
6738:
6737:
6734:
6733:
6731:
6730:
6725:
6720:
6715:
6714:
6713:
6703:
6697:
6695:
6686:
6685:
6683:
6682:
6677:
6675:Sard's theorem
6672:
6667:
6662:
6661:
6660:
6658:Lifting theory
6649:
6643:
6639:
6638:
6636:
6635:
6630:
6625:
6620:
6615:
6614:
6613:
6611:FubiniâTonelli
6603:
6598:
6593:
6592:
6591:
6586:
6581:
6573:
6572:
6571:
6566:
6561:
6553:
6547:
6545:
6539:
6538:
6536:
6535:
6530:
6525:
6520:
6515:
6510:
6505:
6499:
6494:
6493:
6492:
6490:in probability
6487:
6477:
6472:
6467:
6461:
6460:
6459:
6454:
6449:
6438:
6436:
6432:
6431:
6429:
6428:
6423:
6418:
6413:
6408:
6403:
6402:
6401:
6391:
6386:
6385:
6384:
6374:
6369:
6364:
6359:
6354:
6349:
6344:
6339:
6333:
6331:
6325:
6324:
6322:
6321:
6316:
6311:
6306:
6301:
6296:
6291:
6286:
6281:
6276:
6271:
6270:
6269:
6264:
6259:
6249:
6244:
6239:
6234:
6224:
6219:
6214:
6209:
6204:
6199:
6197:Locally finite
6194:
6184:
6179:
6174:
6169:
6164:
6159:
6149:
6144:
6139:
6134:
6129:
6124:
6119:
6114:
6109:
6103:
6101:
6094:
6093:
6091:
6090:
6085:
6080:
6075:
6070:
6069:
6068:
6058:
6053:
6045:
6040:
6039:
6038:
6028:
6023:
6022:
6021:
6011:
6006:
6001:
6000:
5999:
5989:
5984:
5979:
5973:
5971:
5967:
5966:
5964:
5963:
5954:
5953:
5952:
5942:
5937:
5929:
5924:
5914:
5912:
5911:Basic concepts
5908:
5907:
5904:Measure theory
5902:
5900:
5899:
5892:
5885:
5877:
5871:
5870:
5864:
5847:
5841:
5828:
5808:
5793:
5790:
5787:
5786:
5779:
5754:
5747:
5729:
5700:(3): 882â908.
5683:
5682:
5680:
5677:
5676:
5675:
5670:
5665:
5660:
5653:
5650:
5637:
5634:
5631:
5626:
5622:
5618:
5615:
5612:
5609:
5604:
5600:
5579:
5559:
5556:
5553:
5548:
5544:
5540:
5537:
5515:
5511:
5507:
5503:
5498:
5494:
5490:
5487:
5484:
5481:
5476:
5472:
5449:
5445:
5441:
5438:
5433:
5429:
5425:
5420:
5416:
5395:
5392:
5389:
5386:
5383:
5378:
5374:
5353:
5350:
5347:
5342:
5338:
5317:
5314:
5311:
5308:
5284:
5273:
5272:
5260:
5257:
5254:
5234:
5225:
5221:
5215:
5206:
5202:
5181:
5178:
5175:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5129:
5125:
5115:provided that
5104:
5079:
5075:
5071:
5066:
5062:
5052:A sequence of
5050:
5038:
5035:
5032:
5012:
5005:
5000:
4991:
4987:
4964:
4960:
4956:
4953:
4933:
4930:
4927:
4907:
4904:
4901:
4898:
4895:
4891:
4886:
4882:
4878:
4874:
4854:
4831:
4809:
4805:
4780:
4760:
4757:
4744:
4724:
4719:
4716:
4709:
4705:
4681:
4677:
4674:
4670:
4664:
4660:
4656:
4636:
4633:
4630:
4627:
4624:
4621:
4593:
4570:
4566:
4563:
4559:
4553:
4549:
4545:
4525:
4498:
4494:
4491:
4487:
4481:
4477:
4473:
4443:
4440:
4436:
4431:
4427:
4423:
4418:
4414:
4410:
4406:
4401:
4397:
4376:
4373:
4370:
4365:
4361:
4357:
4354:
4334:
4308:
4304:
4301:
4296:
4291:
4287:
4283:
4269:
4266:
4253:
4250:
4246:
4241:
4237:
4233:
4228:
4224:
4220:
4216:
4211:
4207:
4186:
4183:
4180:
4175:
4171:
4167:
4164:
4144:
4115:
4111:
4108:
4103:
4098:
4094:
4090:
4076:
4073:
4058:
4054:
4031:
4027:
4006:
4003:
3998:
3994:
3990:
3985:
3981:
3977:
3972:
3968:
3956:
3955:
3943:
3940:
3937:
3932:
3928:
3917:
3906:
3903:
3900:
3897:
3894:
3891:
3886:
3883:
3879:
3875:
3871:
3866:
3845:
3825:
3822:
3819:
3814:
3810:
3799:
3787:
3767:
3764:
3761:
3756:
3752:
3728:
3725:
3722:
3719:
3716:
3695:
3674:
3662:
3659:
3646:
3635:
3630:
3592:
3576:
3553:
3549:
3545:
3540:
3535:
3532:
3529:
3514:Main article:
3511:
3508:
3495:
3489:
3484:
3478:
3474:
3453:
3448:
3444:
3438:
3434:
3413:
3410:
3405:
3401:
3380:
3377:
3372:
3368:
3345:Dirac measures
3332:
3329:
3326:
3321:
3299:
3276:
3273:
3270:
3265:
3239:
3215:
3212:
3209:
3204:
3178:
3167:
3166:
3155:
3151:
3146:
3140:
3137:
3133:
3129:
3126:
3123:
3119:
3114:
3109:
3105:
3100:
3094:
3087:
3084:
3081:
3076:
3071:
3068:
3061:
3057:
3052:
3049:
3046:
3043:
3040:
3036:
3021:
3020:
3009:
3005:
3000:
2994:
2991:
2988:
2979:
2975:
2972:
2963:
2959:
2956:
2953:
2950:
2945:
2936:
2933:
2930:
2927:
2924:
2920:
2912:
2888:
2885:
2882:
2879:
2876:
2856:
2853:
2850:
2845:
2816:
2792:
2771:
2762:is "close" to
2751:
2747:
2743:
2722:
2701:
2697:
2693:
2667:
2663:
2639:
2618:
2614:
2611:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2565:
2561:
2555:
2552:
2549:
2545:
2524:
2502:
2498:
2477:
2455:
2451:
2426:
2404:
2400:
2378:
2374:
2371:
2360:
2359:
2347:
2327:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2278:
2274:
2270:
2260:
2248:
2228:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2179:
2175:
2171:
2170:lim inf
2161:
2149:
2129:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2080:
2076:
2072:
2071:lim sup
2062:
2050:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1994:
1990:
1986:
1985:lim inf
1976:
1964:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1908:
1904:
1900:
1899:lim sup
1890:
1878:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1823:
1819:
1808:
1796:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1737:
1733:
1709:
1687:
1683:
1662:
1640:
1636:
1613:
1609:
1586:
1582:
1561:
1558:
1553:
1549:
1528:
1505:
1502:
1499:
1496:
1493:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1443:
1439:
1415:
1394:
1368:
1323:
1320:
1297:
1284:
1262:
1257:
1253:
1247:
1243:
1222:
1217:
1214:
1210:
1204:
1200:
1174:
1169:
1166:
1157:for every set
1155:
1154:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1117:
1113:
1107:
1104:
1101:
1097:
1075:
1056:
1051:
1046:
1043:
1040:
1028:
1025:
1024:
1023:
1012:
1009:
1006:
997:
993:
990:
985:
981:
977:
931:
922:
921:
908:
898:
894:
891:
888:
885:
882:
879:
819:
818:
807:
803:
799:
796:
793:
790:
787:
784:
781:
778:
775:
771:
763:
758:
755:
751:
747:
744:
741:
731:
727:
724:
721:
717:
626:
625:
614:
610:
606:
603:
599:
594:
590:
586:
583:
580:
576:
571:
567:
562:
556:
552:
548:
538:
534:
531:
528:
524:
483:
478:
473:
470:
467:
455:
452:
425:
422:
418:
414:
411:
408:
405:
402:
399:
396:
393:
388:
384:
379:
350:The notion of
328:
325:
321:
318:
315:
310:
306:
302:
298:
295:
264:
261:
258:
255:
252:
249:
246:
243:
238:
234:
220:The notion of
187:The notion of
171:
168:
164:
161:
158:
153:
149:
145:
141:
138:
117:
105:
102:
84:
46:
32:measure theory
15:
13:
10:
9:
6:
4:
3:
2:
6868:
6857:
6854:
6852:
6849:
6848:
6846:
6833:
6830:
6822:
6819:February 2010
6812:
6808:
6802:
6801:
6795:
6790:
6781:
6780:
6769:
6766:
6764:
6763:Real analysis
6761:
6759:
6756:
6754:
6751:
6749:
6746:
6745:
6743:
6739:
6729:
6726:
6724:
6721:
6719:
6716:
6712:
6709:
6708:
6707:
6704:
6702:
6699:
6698:
6696:
6693:
6687:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6659:
6656:
6655:
6654:
6651:
6650:
6647:
6644:
6642:Other results
6640:
6634:
6631:
6629:
6628:RadonâNikodym
6626:
6624:
6621:
6619:
6616:
6612:
6609:
6608:
6607:
6604:
6602:
6601:Fatou's lemma
6599:
6597:
6594:
6590:
6587:
6585:
6582:
6580:
6577:
6576:
6574:
6570:
6567:
6565:
6562:
6560:
6557:
6556:
6554:
6552:
6549:
6548:
6546:
6544:
6540:
6534:
6531:
6529:
6526:
6524:
6521:
6519:
6516:
6514:
6511:
6509:
6506:
6504:
6500:
6498:
6495:
6491:
6488:
6486:
6483:
6482:
6481:
6478:
6476:
6473:
6471:
6468:
6466:
6463:Convergence:
6462:
6458:
6455:
6453:
6450:
6448:
6445:
6444:
6443:
6440:
6439:
6437:
6433:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6407:
6404:
6400:
6397:
6396:
6395:
6392:
6390:
6387:
6383:
6380:
6379:
6378:
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6334:
6332:
6330:
6326:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6285:
6282:
6280:
6277:
6275:
6272:
6268:
6267:Outer regular
6265:
6263:
6262:Inner regular
6260:
6258:
6257:Borel regular
6255:
6254:
6253:
6250:
6248:
6245:
6243:
6240:
6238:
6235:
6233:
6229:
6225:
6223:
6220:
6218:
6215:
6213:
6210:
6208:
6205:
6203:
6200:
6198:
6195:
6193:
6189:
6185:
6183:
6180:
6178:
6175:
6173:
6170:
6168:
6165:
6163:
6160:
6158:
6154:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6133:
6130:
6128:
6125:
6123:
6120:
6118:
6115:
6113:
6110:
6108:
6105:
6104:
6102:
6100:
6095:
6089:
6086:
6084:
6081:
6079:
6076:
6074:
6071:
6067:
6064:
6063:
6062:
6059:
6057:
6054:
6052:
6046:
6044:
6041:
6037:
6034:
6033:
6032:
6029:
6027:
6024:
6020:
6017:
6016:
6015:
6012:
6010:
6007:
6005:
6002:
5998:
5995:
5994:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5974:
5972:
5968:
5962:
5958:
5955:
5951:
5948:
5947:
5946:
5945:Measure space
5943:
5941:
5938:
5936:
5934:
5930:
5928:
5925:
5923:
5919:
5916:
5915:
5913:
5909:
5905:
5898:
5893:
5891:
5886:
5884:
5879:
5878:
5875:
5867:
5865:0-471-19745-9
5861:
5856:
5855:
5848:
5844:
5842:0-471-00710-2
5838:
5834:
5829:
5825:
5819:
5811:
5809:3-7643-2428-7
5805:
5801:
5796:
5795:
5791:
5782:
5776:
5772:
5771:
5763:
5761:
5759:
5755:
5750:
5744:
5740:
5733:
5730:
5725:
5721:
5717:
5713:
5708:
5703:
5699:
5695:
5688:
5685:
5678:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5655:
5651:
5649:
5632:
5624:
5620:
5616:
5610:
5602:
5598:
5577:
5554:
5546:
5542:
5538:
5535:
5513:
5509:
5505:
5501:
5496:
5492:
5488:
5482:
5474:
5470:
5447:
5439:
5431:
5427:
5423:
5418:
5414:
5390:
5384:
5381:
5376:
5372:
5348:
5340:
5336:
5312:
5306:
5298:
5282:
5252:
5232:
5223:
5219:
5204:
5200:
5179:
5176:
5173:
5150:
5144:
5135:
5127:
5123:
5102:
5094:
5077:
5073:
5069:
5064:
5060:
5051:
5030:
5010:
5003:
4989:
4985:
4977:. One writes
4962:
4958:
4954:
4951:
4925:
4902:
4896:
4889:
4884:
4880:
4876:
4872:
4852:
4844:
4829:
4807:
4803:
4794:
4793:
4792:
4778:
4769:
4766:
4758:
4756:
4742:
4722:
4717:
4707:
4703:
4675:
4672:
4662:
4658:
4634:
4631:
4625:
4619:
4611:
4605:
4591:
4564:
4561:
4551:
4547:
4523:
4515:
4492:
4489:
4479:
4475:
4463:
4458:
4455:
4441:
4438:
4434:
4429:
4425:
4416:
4412:
4408:
4404:
4399:
4395:
4371:
4363:
4359:
4355:
4352:
4332:
4325:to a measure
4324:
4302:
4299:
4294:
4289:
4285:
4281:
4267:
4265:
4251:
4248:
4244:
4239:
4235:
4226:
4222:
4218:
4214:
4209:
4205:
4181:
4173:
4169:
4165:
4162:
4142:
4135:to a measure
4134:
4133:
4109:
4106:
4101:
4096:
4092:
4088:
4074:
4072:
4056:
4052:
4029:
4025:
4004:
4001:
3996:
3992:
3988:
3983:
3979:
3975:
3970:
3966:
3938:
3930:
3926:
3918:
3904:
3901:
3895:
3889:
3873:
3843:
3820:
3812:
3808:
3800:
3785:
3762:
3754:
3750:
3742:
3741:
3740:
3723:
3720:
3717:
3672:
3660:
3658:
3656:
3652:
3645:
3641:
3633:
3626:
3620:
3615:
3611:
3607:
3603:
3599:
3595:
3588:
3583:
3579:
3571:
3567:
3543:
3533:
3517:
3509:
3507:
3493:
3482:
3476:
3472:
3451:
3446:
3442:
3436:
3432:
3411:
3403:
3399:
3378:
3370:
3366:
3356:
3354:
3350:
3346:
3327:
3297:
3288:
3271:
3253:
3237:
3229:
3210:
3192:
3176:
3153:
3149:
3138:
3135:
3131:
3127:
3124:
3121:
3112:
3107:
3103:
3098:
3092:
3082:
3069:
3066:
3059:
3055:
3050:
3047:
3044:
3041:
3038:
3034:
3026:
3025:
3024:
3007:
3003:
2992:
2989:
2986:
2973:
2970:
2954:
2951:
2948:
2943:
2934:
2931:
2928:
2925:
2922:
2918:
2910:
2902:
2901:
2900:
2880:
2877:
2851:
2833:
2830:
2814:
2805:
2769:
2749:
2745:
2741:
2699:
2695:
2691:
2683:
2682:Dirac measure
2665:
2661:
2651:
2637:
2612:
2609:
2586:
2580:
2577:
2571:
2563:
2559:
2547:
2522:
2500:
2496:
2475:
2453:
2449:
2440:
2424:
2402:
2398:
2372:
2369:
2345:
2325:
2318:
2299:
2293:
2290:
2284:
2276:
2272:
2261:
2246:
2226:
2219:
2200:
2194:
2191:
2185:
2177:
2173:
2162:
2147:
2127:
2120:
2101:
2095:
2092:
2086:
2078:
2074:
2063:
2048:
2040:
2021:
2015:
2009:
2003:
1997:
1992:
1977:
1962:
1954:
1935:
1929:
1923:
1917:
1911:
1906:
1891:
1876:
1869:
1850:
1844:
1832:
1826:
1821:
1809:
1794:
1787:
1783:
1764:
1758:
1746:
1740:
1735:
1723:
1722:
1721:
1707:
1685:
1681:
1638:
1634:
1611:
1607:
1584:
1559:
1551:
1547:
1526:
1519:
1497:
1494:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1441:
1437:
1429:
1406:
1392:
1382:
1366:
1358:
1354:
1352:
1348:
1343:
1341:
1337:
1333:
1329:
1321:
1319:
1315:
1312:
1308:
1305:) = (1 + sin(
1304:
1300:
1287:
1279:
1274:
1260:
1255:
1251:
1245:
1241:
1220:
1215:
1212:
1208:
1202:
1198:
1188:
1167:
1164:
1138:
1132:
1129:
1123:
1115:
1111:
1099:
1087:
1086:
1085:
1078:
1071:, a sequence
1070:
1044:
1041:
1026:
1010:
1007:
1004:
991:
988:
983:
979:
968:
967:
966:
963:
959:
948:
940:to a measure
939:
934:
925:
906:
892:
889:
886:
880:
877:
867:
866:
865:
859:
826:
824:
805:
794:
788:
785:
779:
773:
756:
753:
745:
742:
739:
725:
722:
719:
706:
705:
704:
702:
689:
687:
667:
663:
647:
635:
612:
608:
604:
601:
597:
592:
588:
584:
581:
578:
574:
569:
565:
560:
554:
546:
532:
529:
526:
513:
512:
511:
501:
497:
471:
468:
453:
451:
444:
440:
423:
420:
409:
403:
400:
394:
386:
382:
365:there exists
362:
357:
353:
348:
326:
323:
319:
316:
308:
304:
300:
296:
293:
285:
275:
259:
253:
244:
236:
232:
223:
218:
211:
207:
190:
185:
182:
169:
166:
162:
159:
151:
147:
143:
139:
136:
127:
125:
120:
111:
103:
101:
98:
87:
78:
74:
63:
58:
49:
41:
37:
33:
29:
22:
6825:
6816:
6797:
6543:Main results
6469:
6279:Set function
6207:Metric outer
6162:Decomposable
6019:Cylinder set
5932:
5853:
5832:
5799:
5769:
5738:
5732:
5697:
5693:
5687:
5299:, the space
5274:
5092:
4842:
4770:
4762:
4609:
4606:
4459:
4456:
4322:
4271:
4129:
4078:
4017:. Moreover,
3957:
3664:
3654:
3650:
3643:
3639:
3628:
3618:
3613:
3612:: Ω â
3609:
3605:
3601:
3597:
3590:
3581:
3580:: Ω â
3574:
3569:
3519:
3357:
3289:
3168:
3022:
2806:
2652:
2362:In the case
2361:
1517:
1381:metric space
1356:
1355:
1350:
1344:
1335:
1325:
1316:
1310:
1306:
1302:
1295:
1282:
1275:
1189:
1156:
1073:
1030:
961:
957:
946:
937:
929:
926:
923:
857:
827:
820:
690:
686:Polish space
666:Radon metric
627:
457:
442:
438:
360:
355:
351:
349:
284:Polish space
276:
221:
219:
209:
205:
188:
186:
183:
128:
124:Polish space
115:
107:
99:
82:
76:
72:
61:
44:
39:
35:
25:
6811:introducing
6503:compact set
6470:of measures
6406:Pushforward
6399:Projections
6389:Logarithmic
6232:Probability
6222:Pre-measure
6004:Borel space
5922:of measures
4795:A sequence
4345:if for all
4155:if for all
3596:is said to
3349:convex hull
2684:located at
2437:denote the
2338:of measure
2119:closed sets
1516:is said to
1357:Definition.
1347:definitions
1328:mathematics
660:which have
28:mathematics
6845:Categories
6794:references
6475:in measure
6202:Maximising
6172:Equivalent
6066:Vitali set
5462:such that
4695:such that
4512:converges
4130:converges
3856:such that
3347:, and its
2829:metrizable
2037:for every
1951:for every
1332:statistics
436:for every
369:such that
6589:Maharam's
6559:Dominated
6372:Intensity
6367:Hausdorff
6274:Saturated
6192:Invariant
6097:Types of
6056:Ï-algebra
6026:đ-system
5992:Borel set
5987:Baire set
5818:cite book
5707:1102.5245
5694:Bernoulli
5539:∈
5510:μ
5493:∫
5471:φ
5448:∗
5424:∈
5415:φ
5382:∈
5373:μ
5259:∞
5256:→
5233:φ
5224:∗
5214:→
5201:φ
5177:∈
5145:φ
5142:→
5124:φ
5103:φ
5078:∗
5070:∈
5061:φ
5037:∞
5034:→
4999:→
4963:∗
4955:∈
4952:φ
4932:∞
4929:→
4897:φ
4894:→
4873:φ
4743:μ
4723:μ
4715:→
4704:μ
4676:∈
4659:μ
4632:≤
4620:μ
4592:μ
4565:∈
4548:μ
4524:μ
4493:∈
4476:μ
4462:tightness
4442:μ
4426:∫
4422:→
4413:μ
4396:∫
4356:∈
4333:μ
4303:∈
4286:μ
4252:μ
4236:∫
4232:→
4223:μ
4206:∫
4166:∈
4143:μ
4110:∈
4093:μ
4002:⊂
3989:⊂
3976:⊂
3885:∞
3882:→
3531:Ω
3409:⇀
3376:⇒
3191:separable
3139:δ
3125:−
3122:μ
3113:φ
3104:∫
3070:∈
3067:μ
3051:δ
3039:φ
2987:δ
2974:∈
2958:→
2949:φ
2935:δ
2923:φ
2884:Σ
2630:at which
2613:∈
2554:∞
2551:→
2373:≡
2239:of space
2218:open sets
2192:≥
2140:of space
2093:≤
2041:function
2016:
2010:≥
1998:
1955:function
1930:
1924:≤
1912:
1845:
1839:→
1827:
1759:
1753:→
1741:
1557:⇒
1539:(denoted
1501:Σ
1469:…
1414:Σ
1393:σ
1383:with its
1293:given by
1261:μ
1242:μ
1221:μ
1199:μ
1168:∈
1133:μ
1112:μ
1106:∞
1103:→
1008:ε
996:‖
992:μ
989:−
980:μ
976:‖
897:‖
893:ν
890:−
887:μ
884:‖
789:ν
786:−
774:μ
757:∈
746:⋅
726:ν
723:−
720:μ
699:are both
605:ν
589:∫
585:−
582:μ
566:∫
533:ν
530:−
527:μ
424:ε
404:μ
401:−
383:μ
356:uniformly
327:μ
317:∫
314:→
305:μ
294:∫
254:μ
251:→
233:μ
170:μ
160:∫
157:→
148:μ
137:∫
6606:Fubini's
6596:Egorov's
6564:Monotone
6523:variable
6501:Random:
6452:Strongly
6377:Lebesgue
6362:Harmonic
6352:Gaussian
6337:Counting
6304:Spectral
6299:Singular
6289:s-finite
6284:Ï-finite
6167:Discrete
6142:Complete
6099:Measures
6073:Null set
5961:function
5724:88518773
5652:See also
5528:for all
5166:for all
4944:for all
3958:We have
3483:→
3443:→
3254:, so is
3189:is also
2315:for all
2216:for all
2117:for all
1780:for all
1653:, while
1405:-algebra
1340:measures
1252:→
1209:→
938:converge
730:‖
716:‖
537:‖
523:‖
208:≤
110:calculus
6807:improve
6518:process
6513:measure
6508:element
6447:Bochner
6421:Trivial
6416:Tangent
6394:Product
6252:Regular
6230:)
6217:Perfect
6190:)
6155:)
6147:Content
6137:Complex
6078:Support
6051:-system
5940:Measure
4610:vaguely
4514:vaguely
4132:vaguely
3193:, then
2680:is the
1782:bounded
6796:, but
6584:Jordan
6569:Vitali
6528:vector
6457:Weakly
6319:Vector
6294:Signed
6247:Random
6188:Quasi-
6177:Finite
6157:Convex
6117:Banach
6107:Atomic
5935:spaces
5920:
5862:
5839:
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3606:in law
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2996:
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949:> 0
498:. The
363:> 0
64:> 0
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6342:Dirac
6314:Tight
6242:Radon
6212:Outer
6182:Inner
6132:Brown
6127:Borel
6122:Besov
6112:Baire
5720:S2CID
5702:arXiv
3589:then
3564:be a
3353:dense
3230:. If
1379:be a
960:>
684:is a
636:from
494:be a
441:>
6690:For
6579:Hahn
6435:Maps
6357:Haar
6228:Sub-
5982:Atom
5970:Sets
5860:ISBN
5837:ISBN
5824:link
5804:ISBN
5775:ISBN
5743:ISBN
4771:Let
4604:.
3665:Let
3600:(or
3568:and
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2990:>
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1031:For
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832:and
695:and
506:and
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5712:doi
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