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10367:
8996:
10100:
11298:
1875:
is involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and
1414:
886:
618:
These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these
9268:{\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})=\sigma \left(\left\{(-\infty ,b_{1}]\times \cdots \times (-\infty ,b_{n}]:b_{i}\in \mathbb {R} \right\}\right)=\sigma \left(\left\{\left(a_{1},b_{1}\right]\times \cdots \times \left(a_{n},b_{n}\right]:a_{i},b_{i}\in \mathbb {R} \right\}\right).}
4350:
2349:
8859:
4492:
10517:
9780:
7880:
9966:
4859:
4703:
9624:
4777:
11133:
5661:
of the two sets. The symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a single
10858:
9372:
1862:
2111:
1259:
731:
10362:{\displaystyle C_{n}\left(B_{1},\dots ,B_{n}\right)=\left(B_{1}\times \cdots \times B_{n}\times \mathbb {R} ^{\infty }\right)\cap X=\left\{\left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\in X:x_{i}\in B_{i},1\leq i\leq n\right\},}
4624:
10531:. Note that the two σ-algebra are equal for separable spaces. For some nonseparable spaces, some maps are ball measurable even though they are not Borel measurable, making use of the ball σ-algebra useful in the analysis of such maps.
2256:
4259:
3986:
2263:
1715:
8722:
4355:
10372:
9631:
8959:
11016:
7701:
6638:
4066:
9843:
6936:
337:
6091:
in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often repeating it until the time
4782:
4629:
2135:
flips of the coin, you may want to determine or revise your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2 possibilities for the first
7310:
552:
this can be thought of as making precise a notion of "size" or "volume" for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of
5196:-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a
5509:
6389:
1163:
10624:
9838:
9463:
4712:
4564:
2805:
6869:
11058:
3726:
5093:
13117:
13022:
12924:
5334:
1631:
5836:
5292:
1467:
962:
13792:
8326:
3638:
One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable
11784:
10905:
10577:
8670:
8620:
432:
4158:
10760:
9409:
9288:
1763:
11718:
11669:
10940:
10028:
8097:
1025:
8389:
8203:
8160:
8051:
7988:
7945:
11751:
2714:
1968:
1758:
1234:
1071:
706:
4986:
2380:
13735:
6144:
6061:
8451:
13355:
11442:
5545:
4569:
2481:
1547:
6810:
13646:
11620:
3003:
2967:
8713:
8521:
7697:
4185:
1495:
10653:
8991:
8554:
8422:
7636:
591:
implies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the
13677:
13611:
6731:
2159:
13759:
13384:
12268:
12137:
12105:
11471:
11361:
6766:
5438:
10965:
9434:
3852:
3771:
3275:) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.
2630:
216:
13580:
13521:
12222:
11591:
11503:
10075:
8256:
6508:
5043:
4215:
3874:
3119:
2842:
1639:
357:
7147:
5642:-algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).
5253:
4938:
4242:
4098:
3634:
110:
14063:
is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that
7665:
7445:
6693:
6418:
6291:
5588:
3602:
3487:
7186:
14021:
7468:
3924:
2935:
2865:
2737:
2597:
13495:
13451:
11562:
11533:
11383:
7396:
7090:
7047:
6027:
5640:
5612:
5414:
5392:
5368:
5148:
3898:
3159:
2908:
2651:
2550:
2528:
11293:{\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in \sigma \left({\mathcal {F}}_{X}\right)\right\}=\sigma \left(\left\{Y^{-1}(A):A\in {\mathcal {F}}_{X}\right\}\right),}
10735:
13296:
13204:
6534:
6110:
6089:
6007:
3078:
14081:
14061:
14041:
13966:
5568:
4524:
3800:
2442:
10705:
13989:
10051:
9993:
8279:
7565:
7229:
7115:
7070:
6665:
6316:
6262:
6199:
5948:
5859:
5779:
5750:
5723:
5178:
5116:
5009:
4009:
3827:
3533:
3458:
3182:
3026:
1964:
1570:
1094:
616:
585:
550:
11128:
11098:
11078:
10755:
10673:
10095:
9458:
8346:
8117:
8008:
7902:
7607:
7587:
7538:
7510:
7490:
7416:
7372:
7352:
7332:
7206:
7027:
7007:
6987:
6967:
6458:
6438:
6339:
6239:
6219:
6176:
5975:
5921:
5901:
5881:
5696:
4906:
4886:
4118:
3746:
3656:
3573:
3553:
3510:
3435:
3415:
3387:
3367:
3347:
3320:
3300:
3245:
3139:
3046:
2888:
2574:
2508:
2413:
2154:
2133:
1941:
1917:
1897:
1254:
906:
726:
519:
489:
There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
456:
236:
4021:
146:, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of
6064:
8395:(for example, a separable complete metric space with its associated Borel sets), then the converse is also true. Examples of standard Borel spaces include
3936:
1409:{\displaystyle \liminf _{n\to \infty }A_{n}=\bigcup _{n=1}^{\infty }\bigcap _{m=n}^{\infty }A_{m}=\bigcup _{n=1}^{\infty }A_{n}\cap A_{n+1}\cap \cdots .}
881:{\displaystyle \limsup _{n\to \infty }A_{n}=\bigcap _{n=1}^{\infty }\bigcup _{m=n}^{\infty }A_{m}=\bigcap _{n=1}^{\infty }A_{n}\cup A_{n+1}\cup \cdots .}
8866:
7234:
10970:
14174:
van der Vaart, A. W., & Wellner, J. A. (1996). Weak
Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York.
5670:
can be properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.
4345:{\displaystyle \bigvee _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }=\sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}
6539:
2344:{\displaystyle {\mathcal {G}}_{1}\subseteq {\mathcal {G}}_{2}\subseteq {\mathcal {G}}_{3}\subseteq \cdots \subseteq {\mathcal {G}}_{\infty },}
134:; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in
8854:{\displaystyle \Sigma _{1}\times \Sigma _{2}=\sigma \left(\left\{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\right\}\right).}
4487:{\displaystyle {\mathcal {P}}=\left\{\bigcap _{i=1}^{n}A_{i}:A_{i}\in \Sigma _{\alpha _{i}},\alpha _{i}\in {\mathcal {A}},\ n\geq 1\right\}.}
10512:{\displaystyle \Sigma _{n}=\sigma \left(\{C_{n}\left(B_{1},\dots ,B_{n}\right):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\}\right)}
6874:
3665:
241:
9775:{\displaystyle \left\{C_{t_{1},\dots ,t_{n}}\left(B_{1},\dots ,B_{n}\right):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\right\}}
11403:
1879:
Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads (
3575:
of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets in
14229:
4018:
The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by:
7875:{\displaystyle \sigma (f)=\sigma \left(\left\{f^{-1}(\left\times \cdots \times \left):a_{i},b_{i}\in \mathbb {R} \right\}\right).}
5454:
9961:{\displaystyle {\mathcal {F}}_{X}=\bigcup _{n=1}^{\infty }\bigcup _{t_{i}\in \mathbb {T} ,i\leq n}\Sigma _{t_{1},\dots ,t_{n}}}
6344:
1099:
10586:
9785:
4529:
2742:
13913:
13856:
6815:
11021:
1919:). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the
635:. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.
4854:{\displaystyle \sigma ({\mathcal {P}})\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}
162:
10527:
The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than the
5048:
4698:{\displaystyle \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)\subseteq \sigma ({\mathcal {P}})}
13092:
12997:
12899:
5297:
1575:
4706:
4253:
3489:
5791:
5258:
1423:
918:
13764:
8477:). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the
469:
and adding in all countable unions, countable intersections, and relative complements and continuing this process (by
14159:
13888:
8284:
11759:
10870:
10542:
8625:
8575:
369:
4123:
3803:
9379:
6068:
11677:
11628:
10914:
10002:
9619:{\displaystyle C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n})=\left\{f\in X:f(t_{i})\in B_{i},1\leq i\leq n\right\}.}
8056:
4772:{\displaystyle {\mathcal {P}}\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)}
970:
628:
8351:
8165:
8122:
8013:
7950:
7907:
11726:
11396:
5883:
which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of
2660:
1720:
1180:
1030:
652:
13832:
Clearly a σ-algebra of subsets is also an algebra of subsets, so the basic results for algebras in still hold.
4947:
2354:
14201:
13682:
6115:
6032:
17:
8427:
13528:
13333:
11790:
11420:
5514:
2451:
1500:
14196:
11321:
6771:
3659:
13625:
11599:
2976:
2940:
8678:
8494:
7670:
4163:
1472:
10629:
8967:
8530:
8398:
7612:
6941:
There are many families of subsets that generate useful σ-algebras. Some of these are presented here.
14234:
14191:
13655:
13589:
11389:
10853:{\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in {\mathcal {B}}\left(\mathbb {R} ^{n}\right)\right\}.}
9367:{\displaystyle X\subseteq \mathbb {R} ^{\mathbb {T} }=\{f:f(t)\in \mathbb {R} ,\ t\in \mathbb {T} \}}
6698:
6241:
may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing
3535:
Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in
2811:
1857:{\displaystyle \lim _{n\to \infty }A_{n}:=\liminf _{n\to \infty }A_{n}=\limsup _{n\to \infty }A_{n}.}
1416:
It consists of all points that are in all but finitely many of these sets (or equivalently, that are
458:
then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
77:
13815:
13740:
13365:
12243:
12112:
12080:
11452:
11342:
6736:
5419:
14224:
10945:
9414:
5204:-ring, since the real line can be obtained by their countable union yet its measure is not finite.
3832:
3751:
2606:
171:
13559:
13506:
12201:
11570:
11487:
10058:
8208:
6466:
5016:
4190:
3857:
3089:
2827:
2106:{\displaystyle \Omega =\{H,T\}^{\infty }=\{(x_{1},x_{2},x_{3},\dots ):x_{i}\in \{H,T\},i\geq 1\}.}
342:
14239:
13942:
7120:
5924:
5226:
4911:
4252:
The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it
4220:
4076:
3607:
2600:
1872:
147:
83:
69:
7641:
7421:
6669:
6394:
6267:
5573:
3578:
3463:
7152:
3272:
3197:
1876:
determined only by the partial information. A simple example suffices to illustrate this idea.
522:
13994:
7450:
4619:{\displaystyle \bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\subseteq {\mathcal {P}}.}
3906:
2917:
2847:
2719:
2579:
13480:
13436:
13386:
11541:
11512:
11473:
11368:
7381:
7075:
7032:
6012:
5625:
5597:
5548:
5399:
5377:
5353:
5133:
3883:
3555:
enjoy the property under consideration while, on the other hand, showing that the collection
3209:
3144:
2893:
2636:
2535:
2513:
909:
498:
470:
135:
131:
123:
of subsets; elements of the latter only need to be closed under the union or intersection of
65:
38:
10714:
5188:-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a
13281:
13229:
13189:
13130:
10097:
is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets
8392:
6513:
6095:
6074:
5992:
5658:
5591:
5449:
3051:
158:
139:
14066:
14046:
14026:
13951:
7667:
is generated by the family of subsets which are inverse images of intervals/rectangles in
5553:
4500:
3776:
2418:
8:
14151:
13844:
12842:
11309:
10684:
7375:
3877:
3201:
3189:
2815:
632:
13971:
10033:
9975:
8261:
7547:
7211:
7097:
7052:
6647:
6298:
6244:
6181:
5930:
5841:
5761:
5732:
5705:
5160:
5098:
4991:
3991:
3809:
3515:
3440:
3164:
3008:
2251:{\displaystyle {\mathcal {G}}_{n}=\{A\times \{H,T\}^{\infty }:A\subseteq \{H,T\}^{n}\}.}
1946:
1552:
1076:
598:
567:
532:
14115:
13617:
11956:
11113:
11102:
11083:
11063:
10740:
10658:
10080:
9443:
8331:
8102:
7993:
7887:
7592:
7572:
7523:
7495:
7475:
7401:
7357:
7337:
7317:
7191:
7012:
6992:
6972:
6952:
6443:
6423:
6324:
6224:
6204:
6161:
5960:
5954:
5906:
5886:
5866:
5681:
5654:
5646:
5220:
4891:
4871:
4103:
3731:
3641:
3558:
3538:
3495:
3420:
3400:
3372:
3352:
3332:
3305:
3285:
3215:
3124:
3031:
2873:
2559:
2493:
2398:
2156:
flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra
2139:
2118:
1926:
1902:
1882:
1239:
891:
711:
595:. For this reason, one considers instead a smaller collection of privileged subsets of
504:
441:
435:
221:
143:
73:
42:
161:, particularly when the statistic is a function or a random process and the notion of
14155:
13909:
13884:
13852:
13394:
10908:
10580:
8466:
7517:
6641:
5667:
5663:
5619:
3981:{\displaystyle \textstyle \left\{\Sigma _{\alpha }:\alpha \in {\mathcal {A}}\right\}}
1710:{\displaystyle \liminf _{n\to \infty }A_{n}~\subseteq ~\limsup _{n\to \infty }A_{n}.}
474:
31:
13544:
11881:
11318: – Set of all possible outcomes or results of a statistical trial or experiment
8482:
2810:
From these properties, it follows that the σ-algebra is also closed under countable
14208:
14125:
9996:
8561:
8524:
3266:
3196:
of every measurable set is measurable. The collection of measurable spaces forms a
113:
477:) until the relevant closure properties are achieved (a construction known as the
14143:
13934:
10677:
8489:
5650:
5441:
588:
478:
8993:
is generated by half-infinite rectangles and by finite rectangles. For example,
14103:
14087:
13649:
13583:
13035:
12035:
11338:
8954:{\displaystyle \{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\}}
3083:
641:
14129:
14104:"On simple representations of stopping times and stopping time sigma-algebras"
11011:{\displaystyle \textstyle Y:\Omega \to X\subseteq \mathbb {R} ^{\mathbb {T} }}
14218:
13930:
12607:
12273:
12164:
8673:
8462:
5987:
3604:
enjoy the property, avoiding the task of checking it for an arbitrary set in
3326:
3254:
1169:
554:
466:
14175:
30:
For an algebraic structure admitting a given signature Σ of operations, see
13872:
12379:
12297:
11481:
11315:
9437:
7513:
5445:
5216:
5197:
3901:
3257:(λ-system). The converse is true as well, by Dynkin's theorem (see below).
1920:
13822:. University of Alabama in Huntsville, Department of Mathematical Sciences
6201:
Then there exists a unique smallest σ-algebra which contains every set in
157:, (sub) σ-algebras are needed for the formal mathematical definition of a
13880:
11312: – Function for which the preimage of a measurable set is measurable
965:
526:
120:
9995:
This σ-algebra is a subalgebra of the Borel σ-algebra determined by the
6633:{\displaystyle \sigma (\{1\})=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\}\}.}
6460:
by a countable number of complement, union and intersection operations.
5622:
is separable, but the converse need not hold. For example, the
Lebesgue
4061:{\displaystyle \bigwedge _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }.}
908:
that are in infinitely many of these sets (or equivalently, that are in
12763:
12532:
12457:
11327:
8478:
8474:
5151:
592:
154:
6931:{\displaystyle \sigma \left(\left\{A_{1},A_{2},\ldots \right\}\right)}
4160:
is not empty. Closure under complement and countable unions for every
13550:
12937:
11799:
10528:
9277:
For each of these two examples, the generating family is a π-system.
7541:
5755:
5615:
3280:
3250:
2822:
2445:
587:
but in many natural settings, this is not possible. For example, the
462:
360:
332:{\displaystyle \Sigma =\{\varnothing ,\{a,b\},\{c,d\},\{a,b,c,d\}\},}
8527:
sets. This σ-algebra contains more sets than the Borel σ-algebra on
10534:
8557:
8470:
6264:(See intersections of σ-algebras above.) This σ-algebra is denoted
3205:
3193:
14120:
13820:
Random: Probability, Mathematical
Statistics, Stochastic Processes
5657:. The distance between two sets is defined as the measure of the
11300:
the σ-algebra generated by the inverse images of cylinder sets.
7305:{\displaystyle \sigma (f)=\left\{f^{-1}(S)\,:\,S\in B\right\}.}
58:
13272:
13180:
10862:
8672:
be two measurable spaces. The σ-algebra for the corresponding
3212:
are defined as certain types of functions from a σ-algebra to
13935:"Properties of the class of measure separable compact spaces"
6153:
3389:
and is closed under complement and under countable unions of
2487:
if and only if it satisfies the following three properties:
6148:
6644:, when a collection of subsets contains only one element,
5504:{\displaystyle \rho (A,B)=\mu (A{\mathbin {\triangle }}B)}
4256:
a σ-algebra known as the join which typically is denoted
6384:{\displaystyle \sigma (\varnothing )=\{\varnothing ,X\}.}
1158:{\displaystyle x\in A_{n_{1}}\cap A_{n_{2}}\cap \cdots .}
627:
Many uses of measure, such as the probability concept of
10619:{\displaystyle \textstyle Y:\Omega \to \mathbb {R} ^{n}}
9833:{\displaystyle \textstyle \Sigma _{t_{1},\dots ,t_{n}}.}
5923:
is uncountable). This is the σ-algebra generated by the
5702:
The family consisting only of the empty set and the set
4559:{\displaystyle \Sigma _{\alpha }\subset {\mathcal {P}},}
2800:{\displaystyle A=A_{1}\cup A_{2}\cup A_{3}\cup \cdots .}
9280:
6864:{\displaystyle \sigma \left(A_{1},A_{2},\ldots \right)}
3188:. A function between two measurable spaces is called a
2914:
asserts that its complement, the empty set, is also in
1636:
The inner limit is always a subset of the outer limit:
14090:
this metric space is separable as a topological space.
11053:{\displaystyle \sigma \left({\mathcal {F}}_{X}\right)}
10974:
10590:
9789:
5301:
5262:
3940:
363:. In general, a finite algebra is always a σ-algebra.
14069:
14049:
14029:
13997:
13974:
13954:
13767:
13743:
13685:
13658:
13628:
13592:
13562:
13509:
13483:
13439:
13368:
13336:
13284:
13192:
13095:
13000:
12902:
12246:
12204:
12115:
12083:
11762:
11729:
11680:
11631:
11602:
11573:
11544:
11515:
11490:
11455:
11423:
11371:
11345:
11136:
11116:
11086:
11066:
11024:
11018:
is measurable with respect to the cylinder σ-algebra
10973:
10948:
10917:
10873:
10763:
10743:
10717:
10687:
10661:
10632:
10626:
is measurable with respect to the Borel σ-algebra on
10589:
10545:
10375:
10103:
10083:
10061:
10036:
10005:
9978:
9846:
9788:
9634:
9466:
9446:
9417:
9382:
9291:
8999:
8970:
8869:
8725:
8681:
8628:
8578:
8533:
8497:
8430:
8401:
8354:
8334:
8287:
8264:
8211:
8168:
8125:
8105:
8059:
8016:
7996:
7953:
7910:
7890:
7704:
7673:
7644:
7615:
7595:
7575:
7550:
7526:
7498:
7478:
7453:
7424:
7404:
7384:
7360:
7340:
7320:
7237:
7214:
7194:
7155:
7123:
7100:
7078:
7055:
7035:
7015:
6995:
6975:
6955:
6877:
6818:
6774:
6739:
6701:
6672:
6650:
6542:
6516:
6469:
6446:
6426:
6397:
6347:
6327:
6301:
6270:
6247:
6227:
6207:
6184:
6164:
6118:
6098:
6077:
6035:
6015:
5995:
5963:
5933:
5909:
5889:
5869:
5844:
5794:
5764:
5735:
5708:
5684:
5628:
5600:
5576:
5556:
5517:
5457:
5422:
5402:
5380:
5356:
5300:
5261:
5229:
5163:
5136:
5101:
5051:
5019:
4994:
4950:
4914:
4894:
4874:
4785:
4715:
4632:
4572:
4532:
4503:
4358:
4262:
4223:
4193:
4166:
4126:
4106:
4079:
4024:
3994:
3939:
3909:
3886:
3860:
3835:
3812:
3779:
3754:
3734:
3721:{\displaystyle \mathbb {P} (X\in A)=\int _{A}\,F(dx)}
3668:
3662:
typically associated with computing the probability:
3644:
3610:
3581:
3561:
3541:
3518:
3498:
3466:
3443:
3423:
3403:
3375:
3355:
3335:
3322:
that is closed under finitely many intersections, and
3308:
3288:
3218:
3167:
3147:
3127:
3092:
3054:
3034:
3011:
2979:
2943:
2920:
2896:
2876:
2850:
2830:
2745:
2722:
2663:
2639:
2609:
2582:
2562:
2538:
2516:
2496:
2454:
2421:
2401:
2382:
is the smallest σ-algebra containing all the others.
2357:
2266:
2162:
2142:
2121:
1971:
1949:
1929:
1923:Ω must consist of all possible infinite sequences of
1905:
1885:
1766:
1723:
1642:
1578:
1555:
1503:
1475:
1426:
1262:
1242:
1183:
1102:
1079:
1033:
973:
921:
894:
734:
714:
655:
601:
570:
535:
507:
444:
372:
345:
244:
224:
174:
86:
11330: – Family of sets closed under countable unions
8348:
is finite or countably infinite or, more generally,
6944:
5953:
The collection of all unions of sets in a countable
5088:{\displaystyle \{A\subseteq X:A\cap Y\in \Lambda \}}
13112:{\displaystyle \varnothing \not \in {\mathcal {F}}}
13017:{\displaystyle \varnothing \not \in {\mathcal {F}}}
12919:{\displaystyle \varnothing \not \in {\mathcal {F}}}
7472:One common situation, and understood by default if
5329:{\displaystyle \scriptstyle (X,\,{\mathfrak {F}}).}
1626:{\displaystyle x\in A_{N}\cap A_{N+1}\cap \cdots .}
461:A more useful example is the set of subsets of the
130:The main use of σ-algebras is in the definition of
14075:
14055:
14035:
14015:
13983:
13960:
13786:
13753:
13729:
13671:
13640:
13605:
13574:
13515:
13489:
13445:
13378:
13349:
13290:
13198:
13111:
13016:
12918:
12262:
12216:
12131:
12099:
11778:
11745:
11712:
11663:
11614:
11585:
11556:
11527:
11497:
11465:
11436:
11377:
11355:
11292:
11122:
11092:
11072:
11052:
11010:
10959:
10934:
10899:
10852:
10749:
10729:
10699:
10667:
10647:
10618:
10571:
10511:
10361:
10089:
10069:
10045:
10022:
9987:
9960:
9832:
9774:
9618:
9452:
9428:
9403:
9366:
9267:
8985:
8953:
8853:
8707:
8664:
8614:
8548:
8515:
8445:
8416:
8383:
8340:
8320:
8273:
8250:
8197:
8154:
8111:
8091:
8045:
8002:
7982:
7939:
7896:
7874:
7691:
7659:
7630:
7601:
7581:
7559:
7532:
7504:
7484:
7462:
7439:
7410:
7390:
7366:
7346:
7326:
7304:
7223:
7200:
7180:
7141:
7109:
7084:
7064:
7041:
7021:
7001:
6981:
6961:
6930:
6863:
6804:
6760:
6725:
6687:
6659:
6632:
6528:
6510:Then the σ-algebra generated by the single subset
6502:
6452:
6432:
6412:
6383:
6333:
6310:
6285:
6256:
6233:
6213:
6193:
6170:
6138:
6104:
6083:
6055:
6021:
6001:
5969:
5942:
5915:
5895:
5875:
5853:
5830:
5773:
5744:
5717:
5690:
5634:
5606:
5582:
5562:
5539:
5503:
5432:
5408:
5386:
5362:
5328:
5286:
5247:
5172:
5142:
5110:
5087:
5037:
5003:
4980:
4932:
4900:
4880:
4853:
4771:
4697:
4618:
4558:
4518:
4486:
4344:
4236:
4209:
4179:
4152:
4112:
4092:
4060:
4003:
3980:
3918:
3892:
3868:
3846:
3821:
3794:
3765:
3740:
3720:
3650:
3628:
3596:
3567:
3547:
3527:
3504:
3481:
3452:
3429:
3409:
3381:
3361:
3341:
3314:
3294:
3239:
3176:
3153:
3133:
3113:
3072:
3040:
3020:
2997:
2961:
2929:
2902:
2882:
2859:
2836:
2799:
2731:
2708:
2645:
2624:
2591:
2568:
2544:
2522:
2502:
2475:
2436:
2407:
2374:
2343:
2250:
2148:
2127:
2105:
1958:
1935:
1911:
1891:
1856:
1752:
1709:
1625:
1564:
1541:
1489:
1461:
1408:
1248:
1228:
1157:
1088:
1065:
1019:
956:
900:
880:
720:
700:
610:
579:
544:
513:
450:
426:
351:
331:
230:
210:
104:
5831:{\displaystyle \{\varnothing ,A,X\setminus A,X\}}
5287:{\displaystyle \scriptstyle (X,\,{\mathcal {F}})}
1462:{\displaystyle x\in \liminf _{n\to \infty }A_{n}}
957:{\displaystyle x\in \limsup _{n\to \infty }A_{n}}
14216:
14023:is the Boolean algebra of all Borel sets modulo
13787:{\displaystyle {\mathcal {F}}\neq \varnothing .}
10535:σ-algebra generated by random variable or vector
8523:another σ-algebra is of importance: that of all
8456:
6071:describes the information up to the random time
5338:
1826:
1797:
1768:
1725:
1679:
1644:
1434:
1264:
929:
736:
8321:{\displaystyle \sigma (f)\subseteq \sigma (g).}
5981:
4709:by a collection of subsets. On the other hand,
11779:{\displaystyle \varnothing \in {\mathcal {F}}}
10900:{\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}
10572:{\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}
8665:{\displaystyle \left(X_{2},\Sigma _{2}\right)}
8615:{\displaystyle \left(X_{1},\Sigma _{1}\right)}
5838:is a simple σ-algebra generated by the subset
427:{\displaystyle \{A_{1},A_{2},A_{3},\ldots \},}
14095:
13928:
11397:
8453:with the cylinder σ-algebra described below.
4153:{\displaystyle \Sigma _{\alpha },\Sigma ^{*}}
1717:If these two sets are equal then their limit
10519:is a non-decreasing sequence of σ-algebras.
10501:
10397:
9404:{\displaystyle {\mathcal {B}}(\mathbb {R} )}
9361:
9315:
8948:
8870:
6793:
6790:
6784:
6781:
6752:
6746:
6714:
6708:
6624:
6621:
6603:
6597:
6585:
6579:
6573:
6564:
6555:
6549:
6523:
6517:
6494:
6476:
6375:
6363:
5825:
5795:
5673:
5082:
5052:
4975:
4951:
2992:
2980:
2956:
2944:
2385:
2242:
2233:
2220:
2202:
2189:
2180:
2097:
2082:
2070:
2003:
1991:
1978:
418:
373:
323:
320:
296:
290:
278:
272:
260:
251:
205:
181:
13843:
11713:{\displaystyle A_{1}\cup A_{2}\cup \cdots }
11664:{\displaystyle A_{1}\cap A_{2}\cap \cdots }
10935:{\displaystyle \mathbb {R} ^{\mathbb {T} }}
10863:σ-algebra generated by a stochastic process
10023:{\displaystyle \mathbb {R} ^{\mathbb {T} }}
8092:{\displaystyle \left(T,\Sigma _{T}\right).}
7884:A useful property is the following. Assume
5950:Note: "countable" includes finite or empty.
4863:
1020:{\displaystyle A_{n_{1}},A_{n_{2}},\ldots }
127:many subsets, which is a weaker condition.
14142:
13897:
13837:
11404:
11390:
8384:{\displaystyle \left(S,\Sigma _{S}\right)}
8198:{\displaystyle \left(S,\Sigma _{S}\right)}
8155:{\displaystyle \left(T,\Sigma _{T}\right)}
8046:{\displaystyle \left(X,\Sigma _{X}\right)}
7983:{\displaystyle \left(S,\Sigma _{S}\right)}
7940:{\displaystyle \left(X,\Sigma _{X}\right)}
6154:σ-algebra generated by an arbitrary family
14176:https://doi.org/10.1007/978-1-4757-2545-2
14119:
11746:{\displaystyle \Omega \in {\mathcal {F}}}
11491:
11001:
10995:
10950:
10926:
10920:
10890:
10828:
10635:
10605:
10562:
10476:
10195:
10063:
10014:
10008:
9904:
9782:is a π-system that generates a σ-algebra
9742:
9419:
9394:
9357:
9340:
9306:
9300:
9249:
9116:
9012:
8973:
8536:
8500:
8433:
8404:
7856:
7676:
7618:
7284:
7280:
5311:
5272:
3988:is a collection of σ-algebras on a space
3862:
3837:
3756:
3702:
3670:
2709:{\displaystyle A_{1},A_{2},A_{3},\ldots }
1760:exists and is equal to this common set:
1753:{\displaystyle \lim _{n\to \infty }A_{n}}
1483:
1229:{\displaystyle A_{1},A_{2},A_{3},\ldots }
1066:{\displaystyle n_{1}<n_{2}<\cdots }
701:{\displaystyle A_{1},A_{2},A_{3},\ldots }
9460:is a finitely restricted set defined as
7149:is the collection of all inverse images
6149:σ-algebras generated by families of sets
5645:A separable measure space has a natural
4981:{\displaystyle \{Y\cap B:B\in \Sigma \}}
4779:which, by Dynkin's π-λ theorem, implies
2375:{\displaystyle {\mathcal {G}}_{\infty }}
1871:In much of probability, especially when
964:if and only if there exists an infinite
14101:
13903:
13730:{\displaystyle A,B,A_{1},A_{2},\ldots }
10942:is the set of real-valued functions on
9376:is a set of real-valued functions. Let
6463:For a simple example, consider the set
6139:{\displaystyle {\mathcal {F}}_{\tau }.}
6056:{\displaystyle {\mathcal {F}}_{\tau },}
5343:
4352:A π-system that generates the join is
3928:
3260:
14:
14217:
13906:The Theory of Measures and Integration
8446:{\displaystyle \mathbb {R} ^{\infty }}
5215:-algebras are sometimes denoted using
5045:is a measurable space. The collection
3005:is the smallest possible σ-algebra on
13908:. John Wiley & Sons. p. 12.
13871:
13622:is a semiring where every complement
13350:{\displaystyle {\mathcal {F}}\colon }
11437:{\displaystyle {\mathcal {F}}\colon }
7492:is not specified explicitly, is when
6178:be an arbitrary family of subsets of
5540:{\displaystyle A,B\in {\mathcal {F}}}
5121:
3082:Elements of the σ-algebra are called
2476:{\displaystyle \Sigma \subseteq P(X)}
1542:{\displaystyle A_{N},A_{N+1},\ldots }
1469:if and only if there exists an index
9281:σ-algebra generated by cylinder sets
8567:
13315:
13308:
13270:
13249:
13242:
13235:
13216:
13209:
13178:
13150:
13143:
13136:
13120:
13025:
12946:
12927:
12889:
12882:
12862:
12855:
12848:
12818:
12811:
12783:
12776:
12769:
12750:
12743:
12736:
12729:
12722:
12715:
12708:
12701:
12694:
12672:
12665:
12644:
12637:
12630:
12623:
12616:
12594:
12580:
12573:
12559:
12552:
12545:
12538:
12519:
12498:
12484:
12477:
12470:
12463:
12444:
12409:
12402:
12395:
12388:
12320:
12313:
12306:
12284:
12277:
12239:
12225:
12196:
12175:
12108:
12076:
12022:
11973:
11966:
11942:
11893:
11886:
11814:
11807:
6805:{\displaystyle \sigma (\{\{1\}\}).}
5314:
5207:
560:One would like to assign a size to
24:
14108:Statistics and Probability Letters
13770:
13746:
13661:
13641:{\displaystyle \Omega \setminus A}
13629:
13595:
13484:
13440:
13371:
13339:
13322:
13301:
13263:
13256:
13223:
13171:
13164:
13157:
13104:
13089:
13082:
13075:
13068:
13065:
13062:
13055:
13048:
13041:
13009:
12994:
12987:
12980:
12973:
12970:
12967:
12960:
12953:
12911:
12896:
12875:
12872:
12869:
12832:
12825:
12804:
12797:
12790:
12757:
12679:
12658:
12651:
12601:
12587:
12566:
12526:
12512:
12505:
12491:
12451:
12437:
12430:
12423:
12416:
12369:
12362:
12355:
12348:
12341:
12334:
12327:
12291:
12232:
12189:
12182:
12154:
12147:
12140:
12069:
12062:
12055:
12048:
12041:
12029:
12015:
12008:
12001:
11994:
11987:
11980:
11949:
11935:
11928:
11921:
11914:
11907:
11900:
11870:
11863:
11856:
11849:
11842:
11835:
11828:
11821:
11771:
11738:
11730:
11615:{\displaystyle \Omega \setminus A}
11603:
11458:
11426:
11372:
11348:
11267:
11198:
11035:
10981:
10883:
10877:
10817:
10597:
10555:
10549:
10467:
10377:
10200:
10077:is the set of natural numbers and
10055:An important special case is when
9923:
9880:
9850:
9791:
9733:
9385:
9080:
9046:
9002:
8939:
8913:
8830:
8804:
8740:
8727:
8648:
8598:
8438:
8367:
8181:
8138:
8072:
8029:
7966:
7923:
7454:
7385:
7093:-algebra generated by the function
6440:that can be made from elements of
6122:
6039:
5577:
5532:
5490:
5425:
5275:
5239:
5137:
5079:
5029:
4972:
4924:
4834:
4826:
4794:
4755:
4747:
4718:
4687:
4662:
4654:
4608:
4594:
4586:
4548:
4534:
4456:
4422:
4361:
4325:
4317:
4284:
4276:
4225:
4195:
4187:implies the same must be true for
4168:
4141:
4128:
4081:
4046:
4038:
3967:
3947:
3910:
3887:
3228:
3148:
3102:
3028:The largest possible σ-algebra on
2998:{\displaystyle \{X,\varnothing \}}
2962:{\displaystyle \{X,\varnothing \}}
2921:
2897:
2851:
2723:
2640:
2583:
2539:
2517:
2455:
2367:
2361:
2333:
2327:
2304:
2287:
2270:
2206:
2166:
1995:
1972:
1836:
1807:
1778:
1735:
1689:
1654:
1444:
1363:
1329:
1308:
1274:
939:
835:
801:
780:
746:
619:properties are called σ-algebras.
245:
96:
27:Algebraic structure of set algebra
25:
14251:
14184:
14148:Foundations of Modern Probability
13778:
13632:
13566:
11606:
11577:
10522:
9968:is an algebra that generates the
8708:{\displaystyle X_{1}\times X_{2}}
8516:{\displaystyle \mathbb {R} ^{n},}
8469:: the σ-algebra generated by the
8099:If there exists a measurable map
7692:{\displaystyle \mathbb {R} ^{n}:}
6945:σ-algebra generated by a function
6567:
6366:
6354:
5813:
5798:
4705:by the definition of a σ-algebra
4180:{\displaystyle \Sigma _{\alpha }}
3437:is a Dynkin system that contains
2989:
2953:
2613:
1866:
1490:{\displaystyle N\in \mathbb {N} }
622:
254:
13316:
13309:
13302:
13271:
13264:
13257:
13250:
13243:
13236:
13217:
13210:
13179:
13172:
13165:
13158:
13151:
13144:
13137:
13121:
13083:
13076:
13069:
13056:
13049:
13042:
13026:
12988:
12981:
12974:
12961:
12954:
12947:
12928:
12890:
12883:
12876:
12863:
12856:
12849:
12833:
12826:
12819:
12812:
12805:
12798:
12791:
12784:
12777:
12770:
12751:
12744:
12737:
12730:
12723:
12716:
12709:
12702:
12695:
12673:
12666:
12659:
12652:
12645:
12638:
12631:
12624:
12617:
12595:
12588:
12581:
12574:
12567:
12560:
12553:
12546:
12539:
12520:
12513:
12506:
12499:
12492:
12485:
12478:
12471:
12464:
12445:
12438:
12431:
12424:
12417:
12410:
12403:
12396:
12389:
12370:
12363:
12356:
12349:
12342:
12335:
12328:
12321:
12314:
12307:
12285:
12278:
12233:
12226:
12190:
12183:
12176:
12155:
12148:
12141:
12070:
12063:
12056:
12049:
12042:
12023:
12016:
12009:
12002:
11995:
11988:
11981:
11974:
11967:
11943:
11936:
11929:
11922:
11915:
11908:
11901:
11894:
11887:
11871:
11864:
11857:
11850:
11843:
11836:
11829:
11822:
11815:
11808:
10648:{\displaystyle \mathbb {R} ^{n}}
8986:{\displaystyle \mathbb {R} ^{n}}
8549:{\displaystyle \mathbb {R} ^{n}}
8417:{\displaystyle \mathbb {R} ^{n}}
7631:{\displaystyle \mathbb {R} ^{n}}
5157:that contains the universal set
3804:cumulative distribution function
14230:Experiment (probability theory)
13851:(Anniversary ed.). Wiley.
13672:{\displaystyle {\mathcal {F}}.}
13606:{\displaystyle {\mathcal {F}}.}
6726:{\displaystyle \sigma (\{A\});}
6420:consists of all the subsets of
4100:denote the intersection. Since
14168:
14136:
14010:
13998:
13922:
13865:
13808:
13754:{\displaystyle {\mathcal {F}}}
13379:{\displaystyle {\mathcal {F}}}
12263:{\displaystyle A_{i}\nearrow }
12257:
12132:{\displaystyle A_{i}\nearrow }
12126:
12100:{\displaystyle A_{i}\searrow }
12094:
11466:{\displaystyle {\mathcal {F}}}
11356:{\displaystyle {\mathcal {F}}}
11252:
11246:
11176:
11170:
11146:
11140:
10984:
10894:
10874:
10803:
10797:
10773:
10767:
10737:). The σ-algebra generated by
10600:
10566:
10546:
10480:
10472:
9746:
9738:
9574:
9561:
9535:
9503:
9398:
9390:
9333:
9327:
9096:
9074:
9062:
9040:
9022:
9007:
8312:
8306:
8297:
8291:
8245:
8242:
8236:
8230:
8221:
8215:
7823:
7745:
7714:
7708:
7654:
7648:
7434:
7428:
7277:
7271:
7247:
7241:
7175:
7169:
7133:
7127:
6796:
6778:
6761:{\displaystyle \sigma (\{1\})}
6755:
6743:
6717:
6705:
6682:
6676:
6558:
6546:
6407:
6401:
6357:
6351:
6280:
6274:
5498:
5482:
5473:
5461:
5433:{\displaystyle {\mathcal {F}}}
5319:
5302:
5280:
5263:
5242:
5230:
5032:
5020:
4927:
4915:
4799:
4789:
4692:
4682:
3789:
3783:
3715:
3706:
3686:
3674:
3620:
3614:
3591:
3585:
3476:
3470:
3397:Dynkin's π-λ theorem says, if
3349:is a collection of subsets of
3302:is a collection of subsets of
3231:
3219:
3105:
3093:
3064:
3058:
2470:
2464:
2431:
2425:
2051:
2006:
1833:
1804:
1775:
1732:
1686:
1651:
1441:
1271:
936:
743:
99:
87:
57:is a nonempty collection Σ of
13:
1:
13801:
11110:. The σ-algebra generated by
10960:{\displaystyle \mathbb {T} .}
9429:{\displaystyle \mathbb {R} .}
8457:Borel and Lebesgue σ-algebras
5863:The collection of subsets of
5339:Particular cases and examples
5095:is a σ-algebra of subsets of
4988:is a σ-algebra of subsets of
3847:{\displaystyle \mathbb {R} ,}
3766:{\displaystyle \mathbb {R} ,}
3271:This theorem (or the related
2655:closed under countable unions
2625:{\displaystyle X\setminus A.}
2390:
484:
211:{\displaystyle X=\{a,b,c,d\}}
13575:{\displaystyle B\setminus A}
13516:{\displaystyle \varnothing }
12217:{\displaystyle A\subseteq B}
11586:{\displaystyle B\setminus A}
11498:{\displaystyle \,\supseteq }
10070:{\displaystyle \mathbb {T} }
9411:denote the Borel subsets of
8461:An important example is the
8251:{\displaystyle f(x)=h(g(x))}
7378:with respect to a σ-algebra
6503:{\displaystyle X=\{1,2,3\}.}
5982:Stopping time sigma-algebras
5038:{\displaystyle (Y,\Lambda )}
4210:{\displaystyle \Sigma ^{*}.}
3869:{\displaystyle \mathbb {P} }
3114:{\displaystyle (X,\Sigma ),}
2837:{\displaystyle \varnothing }
2554:closed under complementation
525:that assigns a non-negative
465:formed by starting with all
352:{\displaystyle \varnothing }
119:A σ-algebra of subsets is a
7:
14197:Encyclopedia of Mathematics
13877:Real & Complex Analysis
13317:
13310:
13251:
13244:
13237:
13218:
13211:
13152:
13145:
13138:
13122:
13027:
12948:
12929:
12891:
12884:
12864:
12857:
12850:
12820:
12813:
12785:
12778:
12771:
12752:
12745:
12738:
12731:
12724:
12717:
12710:
12703:
12696:
12674:
12667:
12646:
12639:
12632:
12625:
12618:
12596:
12582:
12575:
12561:
12554:
12547:
12540:
12521:
12500:
12486:
12479:
12472:
12465:
12446:
12411:
12404:
12397:
12390:
12322:
12315:
12308:
12286:
12279:
12227:
12177:
12024:
11975:
11968:
11944:
11895:
11888:
11816:
11809:
11322:Sigma-additive set function
11303:
9840:Then the family of subsets
7142:{\displaystyle \sigma (f),}
6295:the σ-algebra generated by
6065:stopping time sigma-algebra
5248:{\displaystyle (X,\Sigma )}
4933:{\displaystyle (X,\Sigma )}
4237:{\displaystyle \Sigma ^{*}}
4093:{\displaystyle \Sigma ^{*}}
3660:Lebesgue-Stieltjes integral
3629:{\displaystyle \sigma (P).}
1073:) of sets that all contain
633:limits of sequences of sets
105:{\displaystyle (X,\Sigma )}
10:
14256:
13737:are arbitrary elements of
13539:
13303:
13265:
13258:
13173:
13166:
13159:
13084:
13077:
13070:
13057:
13050:
13043:
12989:
12982:
12975:
12962:
12955:
12877:
12834:
12827:
12806:
12799:
12792:
12660:
12653:
12589:
12568:
12514:
12507:
12493:
12439:
12432:
12425:
12418:
12371:
12364:
12357:
12350:
12343:
12336:
12329:
12234:
12191:
12184:
12156:
12149:
12142:
12071:
12064:
12057:
12050:
12043:
12017:
12010:
12003:
11996:
11989:
11982:
11937:
11930:
11923:
11916:
11909:
11902:
11872:
11865:
11858:
11851:
11844:
11837:
11830:
11823:
11336:
8473:(or, equivalently, by the
7660:{\displaystyle \sigma (f)}
7440:{\displaystyle \sigma (f)}
6695:may be written instead of
6688:{\displaystyle \sigma (A)}
6413:{\displaystyle \sigma (F)}
6286:{\displaystyle \sigma (F)}
6069:filtered probability space
5583:{\displaystyle \triangle }
3748:in the Borel σ-algebra on
3597:{\displaystyle \sigma (P)}
3482:{\displaystyle \sigma (P)}
3264:
1420:in all of them). That is,
888:It consists of all points
492:
218:one possible σ-algebra on
29:
14130:10.1016/j.spl.2012.09.024
13904:Vestrup, Eric M. (2009).
8010:is a measurable map from
7904:is a measurable map from
7181:{\displaystyle f^{-1}(S)}
6969:is a function from a set
5674:Simple set-based examples
3880:, defined on a σ-algebra
2973:as well, it follows that
2821:It also follows that the
2386:Definition and properties
14016:{\displaystyle (X,\mu )}
11324: – Mapping function
8964:The Borel σ-algebra for
8424:with its Borel sets and
7463:{\displaystyle \Sigma .}
5614:-algebra generated by a
5219:capital letters, or the
4864:σ-algebras for subspaces
3919:{\displaystyle \Omega .}
2930:{\displaystyle \Sigma .}
2860:{\displaystyle \Sigma ,}
2732:{\displaystyle \Sigma ,}
2592:{\displaystyle \Sigma ,}
13991:the measure algebra of
13943:Fundamenta Mathematicae
13849:Probability and Measure
13816:"11. Measurable Spaces"
13761:and it is assumed that
13556:where every complement
13490:{\displaystyle \Omega }
13446:{\displaystyle \Omega }
13330:Is necessarily true of
11557:{\displaystyle A\cup B}
11528:{\displaystyle A\cap B}
11417:Is necessarily true of
11378:{\displaystyle \Omega }
7391:{\displaystyle \Sigma }
7085:{\displaystyle \sigma }
7049:-algebra of subsets of
7042:{\displaystyle \sigma }
6022:{\displaystyle \sigma }
5635:{\displaystyle \sigma }
5607:{\displaystyle \sigma }
5409:{\displaystyle \sigma }
5387:{\displaystyle \sigma }
5363:{\displaystyle \sigma }
5143:{\displaystyle \Sigma }
4940:be a measurable space.
3893:{\displaystyle \Sigma }
3154:{\displaystyle \Sigma }
2903:{\displaystyle \Sigma }
2646:{\displaystyle \Sigma }
2545:{\displaystyle \Sigma }
2523:{\displaystyle \Sigma }
1873:conditional expectation
629:almost sure convergence
148:conditional expectation
49:("sigma algebra"; also
14077:
14057:
14037:
14017:
13985:
13968:is a Borel measure on
13962:
13788:
13755:
13731:
13673:
13642:
13607:
13576:
13517:
13491:
13447:
13380:
13351:
13292:
13200:
13113:
13018:
12920:
12264:
12218:
12133:
12101:
11780:
11747:
11714:
11665:
11616:
11587:
11558:
11529:
11499:
11467:
11438:
11379:
11357:
11294:
11124:
11094:
11074:
11054:
11012:
10961:
10936:
10901:
10854:
10751:
10731:
10730:{\displaystyle n>1}
10701:
10669:
10649:
10620:
10573:
10513:
10363:
10091:
10071:
10047:
10024:
9989:
9962:
9884:
9834:
9776:
9620:
9454:
9430:
9405:
9368:
9269:
8987:
8955:
8855:
8709:
8666:
8616:
8562:complete measure space
8560:theory, as it gives a
8550:
8517:
8447:
8418:
8385:
8342:
8322:
8275:
8252:
8199:
8156:
8113:
8093:
8047:
8004:
7984:
7941:
7898:
7876:
7693:
7661:
7632:
7603:
7583:
7561:
7534:
7506:
7486:
7464:
7441:
7412:
7392:
7368:
7348:
7328:
7306:
7225:
7202:
7182:
7143:
7111:
7086:
7066:
7043:
7023:
7003:
6983:
6963:
6938:is also quite common.
6932:
6865:
6806:
6762:
6727:
6689:
6661:
6634:
6530:
6504:
6454:
6434:
6414:
6385:
6335:
6312:
6287:
6258:
6235:
6215:
6195:
6172:
6140:
6106:
6085:
6057:
6023:
6003:
5971:
5944:
5917:
5897:
5877:
5855:
5832:
5775:
5746:
5725:called the minimal or
5719:
5692:
5636:
5608:
5584:
5564:
5541:
5505:
5434:
5410:
5388:
5364:
5330:
5288:
5249:
5174:
5144:
5112:
5089:
5039:
5005:
4982:
4934:
4902:
4882:
4855:
4773:
4699:
4620:
4560:
4520:
4488:
4394:
4346:
4238:
4211:
4181:
4154:
4114:
4094:
4062:
4005:
3982:
3920:
3894:
3870:
3848:
3823:
3796:
3767:
3742:
3722:
3652:
3630:
3598:
3569:
3549:
3529:
3506:
3483:
3454:
3431:
3411:
3383:
3363:
3343:
3316:
3296:
3273:monotone class theorem
3249:A σ-algebra is both a
3241:
3178:
3155:
3135:
3115:
3074:
3042:
3022:
2999:
2963:
2931:
2904:
2884:
2861:
2838:
2801:
2733:
2710:
2647:
2626:
2593:
2570:
2546:
2524:
2504:
2477:
2438:
2409:
2376:
2345:
2252:
2150:
2129:
2107:
1960:
1937:
1913:
1893:
1858:
1754:
1711:
1627:
1566:
1543:
1491:
1463:
1410:
1367:
1333:
1312:
1250:
1230:
1159:
1090:
1067:
1021:
958:
902:
882:
839:
805:
784:
722:
702:
612:
581:
546:
515:
452:
428:
353:
333:
232:
212:
138:as the foundation for
106:
14102:Fischer, Tom (2013).
14078:
14058:
14038:
14018:
13986:
13963:
13789:
13756:
13732:
13674:
13648:is equal to a finite
13643:
13608:
13582:is equal to a finite
13577:
13518:
13492:
13448:
13381:
13352:
13293:
13291:{\displaystyle \cap }
13201:
13199:{\displaystyle \cup }
13114:
13019:
12921:
12265:
12219:
12134:
12102:
11781:
11748:
11715:
11666:
11617:
11588:
11559:
11530:
11500:
11468:
11439:
11380:
11358:
11295:
11125:
11095:
11075:
11055:
11013:
10962:
10937:
10902:
10855:
10752:
10732:
10702:
10670:
10650:
10621:
10574:
10514:
10364:
10092:
10072:
10048:
10025:
9990:
9963:
9864:
9835:
9777:
9621:
9455:
9431:
9406:
9369:
9270:
8988:
8956:
8856:
8710:
8667:
8617:
8551:
8518:
8448:
8419:
8386:
8343:
8323:
8276:
8253:
8200:
8157:
8114:
8094:
8048:
8005:
7985:
7942:
7899:
7877:
7694:
7662:
7633:
7604:
7584:
7562:
7540:is the collection of
7535:
7507:
7487:
7465:
7442:
7413:
7393:
7369:
7349:
7329:
7307:
7226:
7203:
7183:
7144:
7112:
7087:
7067:
7044:
7024:
7004:
6984:
6964:
6933:
6866:
6807:
6763:
6733:in the prior example
6728:
6690:
6662:
6635:
6531:
6529:{\displaystyle \{1\}}
6505:
6455:
6435:
6415:
6386:
6336:
6313:
6288:
6259:
6236:
6216:
6196:
6173:
6141:
6107:
6105:{\displaystyle \tau }
6086:
6084:{\displaystyle \tau }
6058:
6024:
6004:
6002:{\displaystyle \tau }
5972:
5945:
5918:
5898:
5878:
5856:
5833:
5776:
5747:
5720:
5693:
5637:
5609:
5585:
5565:
5542:
5506:
5444:when considered as a
5435:
5411:
5389:
5365:
5331:
5289:
5250:
5175:
5145:
5113:
5090:
5040:
5006:
4983:
4935:
4903:
4883:
4856:
4774:
4700:
4621:
4561:
4526:it is seen that each
4521:
4489:
4374:
4347:
4239:
4212:
4182:
4155:
4115:
4095:
4063:
4006:
3983:
3921:
3895:
3871:
3849:
3824:
3797:
3768:
3743:
3723:
3653:
3631:
3599:
3570:
3550:
3530:
3507:
3484:
3455:
3432:
3412:
3384:
3364:
3344:
3317:
3297:
3242:
3179:
3156:
3136:
3116:
3075:
3073:{\displaystyle P(X).}
3043:
3023:
3000:
2964:
2932:
2905:
2885:
2862:
2839:
2802:
2734:
2711:
2648:
2627:
2594:
2571:
2547:
2525:
2505:
2478:
2439:
2415:be some set, and let
2410:
2377:
2346:
2253:
2151:
2130:
2108:
1961:
1938:
1914:
1894:
1859:
1755:
1712:
1628:
1567:
1544:
1492:
1464:
1411:
1347:
1313:
1292:
1251:
1231:
1160:
1091:
1068:
1022:
959:
903:
883:
819:
785:
764:
723:
703:
613:
582:
547:
516:
471:transfinite iteration
453:
429:
354:
334:
233:
213:
136:mathematical analysis
107:
39:mathematical analysis
14076:{\displaystyle \mu }
14067:
14056:{\displaystyle \mu }
14047:
14036:{\displaystyle \mu }
14027:
13995:
13972:
13961:{\displaystyle \mu }
13952:
13845:Billingsley, Patrick
13765:
13741:
13683:
13656:
13626:
13590:
13560:
13507:
13481:
13437:
13366:
13334:
13282:
13190:
13093:
12998:
12900:
12244:
12202:
12113:
12081:
11760:
11727:
11678:
11629:
11600:
11571:
11542:
11513:
11488:
11453:
11421:
11369:
11343:
11134:
11114:
11084:
11064:
11022:
10971:
10946:
10915:
10871:
10761:
10741:
10715:
10685:
10659:
10630:
10587:
10543:
10373:
10101:
10081:
10059:
10034:
10003:
9976:
9844:
9786:
9632:
9464:
9444:
9415:
9380:
9289:
8997:
8968:
8867:
8723:
8679:
8626:
8576:
8556:and is preferred in
8531:
8495:
8428:
8399:
8393:standard Borel space
8352:
8332:
8285:
8262:
8209:
8166:
8123:
8103:
8057:
8014:
7994:
7951:
7908:
7888:
7702:
7671:
7642:
7613:
7593:
7573:
7548:
7524:
7496:
7476:
7451:
7422:
7402:
7382:
7358:
7338:
7318:
7235:
7212:
7192:
7153:
7121:
7098:
7076:
7053:
7033:
7013:
6993:
6973:
6953:
6875:
6816:
6772:
6737:
6699:
6670:
6648:
6540:
6514:
6467:
6444:
6424:
6395:
6345:
6325:
6299:
6268:
6245:
6225:
6205:
6182:
6162:
6116:
6096:
6075:
6033:
6013:
5993:
5961:
5931:
5907:
5887:
5867:
5842:
5792:
5762:
5733:
5706:
5682:
5659:symmetric difference
5626:
5598:
5592:symmetric difference
5574:
5563:{\displaystyle \mu }
5554:
5515:
5455:
5420:
5400:
5378:
5354:
5344:Separable σ-algebras
5298:
5259:
5227:
5184:-ring need not be a
5161:
5134:
5099:
5049:
5017:
4992:
4948:
4912:
4892:
4872:
4783:
4713:
4630:
4570:
4530:
4519:{\displaystyle n=1,}
4501:
4356:
4260:
4221:
4191:
4164:
4124:
4104:
4077:
4022:
3992:
3937:
3929:Combining σ-algebras
3907:
3884:
3858:
3833:
3810:
3795:{\displaystyle F(x)}
3777:
3752:
3732:
3666:
3642:
3608:
3579:
3559:
3539:
3516:
3496:
3464:
3441:
3421:
3401:
3373:
3353:
3333:
3306:
3286:
3261:Dynkin's π-λ theorem
3216:
3202:measurable functions
3165:
3161:is a σ-algebra over
3145:
3125:
3090:
3052:
3032:
3009:
2977:
2969:satisfies condition
2941:
2918:
2894:
2874:
2848:
2828:
2743:
2720:
2661:
2637:
2607:
2580:
2560:
2536:
2514:
2494:
2452:
2437:{\displaystyle P(X)}
2419:
2399:
2355:
2264:
2160:
2140:
2119:
1969:
1947:
1927:
1903:
1883:
1764:
1721:
1640:
1576:
1553:
1501:
1473:
1424:
1260:
1240:
1181:
1100:
1077:
1031:
971:
919:
892:
732:
712:
653:
599:
568:
533:
505:
442:
370:
343:
242:
222:
172:
159:sufficient statistic
140:Lebesgue integration
84:
11310:Measurable function
10700:{\displaystyle n=1}
8525:Lebesgue measurable
7589:is a function from
5547:and a given finite
3900:of subsets of some
3878:probability measure
3460:then the σ-algebra
3190:measurable function
1572:that is, such that
1096:that is, such that
915:of them). That is,
165:is not applicable.
163:conditional density
80:. The ordered pair
14073:
14053:
14033:
14013:
13984:{\displaystyle X,}
13981:
13958:
13784:
13751:
13727:
13669:
13638:
13603:
13572:
13513:
13487:
13443:
13376:
13347:
13288:
13196:
13109:
13014:
12916:
12260:
12214:
12129:
12097:
11776:
11743:
11710:
11661:
11612:
11583:
11554:
11525:
11495:
11463:
11434:
11375:
11353:
11290:
11120:
11103:stochastic process
11090:
11070:
11050:
11008:
11007:
10957:
10932:
10897:
10850:
10747:
10727:
10697:
10665:
10645:
10616:
10615:
10569:
10509:
10359:
10087:
10067:
10046:{\displaystyle X.}
10043:
10020:
9988:{\displaystyle X.}
9985:
9970:cylinder σ-algebra
9958:
9921:
9830:
9829:
9772:
9616:
9450:
9426:
9401:
9364:
9265:
8983:
8951:
8851:
8719:and is defined by
8705:
8662:
8612:
8546:
8513:
8443:
8414:
8381:
8338:
8318:
8274:{\displaystyle x,}
8271:
8248:
8195:
8152:
8109:
8089:
8043:
8000:
7980:
7937:
7894:
7872:
7689:
7657:
7628:
7599:
7579:
7560:{\displaystyle Y.}
7557:
7530:
7502:
7482:
7460:
7437:
7408:
7388:
7364:
7344:
7324:
7302:
7224:{\displaystyle B.}
7221:
7198:
7178:
7139:
7110:{\displaystyle f,}
7107:
7082:
7065:{\displaystyle Y,}
7062:
7039:
7019:
6999:
6979:
6959:
6928:
6861:
6802:
6758:
6723:
6685:
6660:{\displaystyle A,}
6657:
6630:
6526:
6500:
6450:
6430:
6410:
6381:
6331:
6311:{\displaystyle F.}
6308:
6283:
6257:{\displaystyle F.}
6254:
6231:
6211:
6194:{\displaystyle X.}
6191:
6168:
6136:
6102:
6081:
6053:
6019:
5999:
5967:
5943:{\displaystyle X.}
5940:
5913:
5893:
5873:
5854:{\displaystyle A.}
5851:
5828:
5783:discrete σ-algebra
5774:{\displaystyle X,}
5771:
5745:{\displaystyle X.}
5742:
5718:{\displaystyle X,}
5715:
5688:
5655:pseudometric space
5632:
5604:
5580:
5560:
5537:
5501:
5430:
5406:
5384:
5360:
5326:
5325:
5284:
5283:
5255:may be denoted as
5245:
5173:{\displaystyle X.}
5170:
5140:
5122:Relation to σ-ring
5111:{\displaystyle X.}
5108:
5085:
5035:
5004:{\displaystyle Y.}
5001:
4978:
4930:
4898:
4878:
4851:
4832:
4769:
4753:
4695:
4660:
4616:
4592:
4556:
4516:
4484:
4342:
4323:
4282:
4234:
4207:
4177:
4150:
4110:
4090:
4058:
4044:
4004:{\displaystyle X.}
4001:
3978:
3977:
3916:
3890:
3866:
3844:
3822:{\displaystyle X,}
3819:
3792:
3763:
3738:
3718:
3648:
3626:
3594:
3565:
3545:
3528:{\displaystyle D.}
3525:
3502:
3479:
3453:{\displaystyle P,}
3450:
3427:
3417:is a π-system and
3407:
3379:
3359:
3339:
3312:
3292:
3237:
3177:{\displaystyle X,}
3174:
3151:
3131:
3111:
3086:. An ordered pair
3070:
3038:
3021:{\displaystyle X.}
3018:
2995:
2959:
2927:
2900:
2880:
2857:
2834:
2797:
2729:
2706:
2643:
2622:
2589:
2566:
2542:
2520:
2500:
2473:
2434:
2405:
2372:
2341:
2260:Observe that then
2248:
2146:
2125:
2103:
1959:{\displaystyle T:}
1956:
1933:
1909:
1889:
1854:
1840:
1811:
1782:
1750:
1739:
1707:
1693:
1658:
1623:
1565:{\displaystyle x;}
1562:
1539:
1487:
1459:
1448:
1406:
1278:
1246:
1226:
1155:
1089:{\displaystyle x;}
1086:
1063:
1017:
954:
943:
898:
878:
750:
718:
698:
611:{\displaystyle X.}
608:
580:{\displaystyle X,}
577:
545:{\displaystyle X;}
542:
511:
475:countable ordinals
448:
424:
349:
329:
228:
208:
144:probability theory
102:
43:probability theory
14192:"Algebra of sets"
13929:Džamonja, Mirna;
13915:978-0-470-31795-2
13858:978-1-118-12237-2
13799:
13798:
11123:{\displaystyle Y}
11093:{\displaystyle Y}
11073:{\displaystyle X}
10909:probability space
10750:{\displaystyle Y}
10668:{\displaystyle Y}
10581:probability space
10090:{\displaystyle X}
9885:
9453:{\displaystyle X}
9349:
8717:product σ-algebra
8568:Product σ-algebra
8467:topological space
8341:{\displaystyle S}
8112:{\displaystyle h}
8003:{\displaystyle g}
7897:{\displaystyle f}
7602:{\displaystyle X}
7582:{\displaystyle f}
7533:{\displaystyle B}
7518:topological space
7505:{\displaystyle Y}
7485:{\displaystyle B}
7411:{\displaystyle X}
7367:{\displaystyle Y}
7347:{\displaystyle X}
7327:{\displaystyle f}
7201:{\displaystyle S}
7022:{\displaystyle B}
7002:{\displaystyle Y}
6982:{\displaystyle X}
6962:{\displaystyle f}
6642:abuse of notation
6453:{\displaystyle F}
6433:{\displaystyle X}
6334:{\displaystyle F}
6234:{\displaystyle F}
6214:{\displaystyle F}
6171:{\displaystyle F}
5970:{\displaystyle X}
5916:{\displaystyle X}
5896:{\displaystyle X}
5876:{\displaystyle X}
5727:trivial σ-algebra
5691:{\displaystyle X}
5664:equivalence class
5192:-ring, but not a
4901:{\displaystyle X}
4881:{\displaystyle Y}
4813:
4734:
4641:
4573:
4466:
4304:
4263:
4113:{\displaystyle X}
4025:
3741:{\displaystyle A}
3651:{\displaystyle X}
3568:{\displaystyle D}
3548:{\displaystyle P}
3505:{\displaystyle P}
3430:{\displaystyle D}
3410:{\displaystyle P}
3382:{\displaystyle X}
3362:{\displaystyle X}
3342:{\displaystyle D}
3315:{\displaystyle X}
3295:{\displaystyle P}
3240:{\displaystyle .}
3134:{\displaystyle X}
3041:{\displaystyle X}
2883:{\displaystyle X}
2569:{\displaystyle A}
2503:{\displaystyle X}
2408:{\displaystyle X}
2149:{\displaystyle n}
2128:{\displaystyle n}
1936:{\displaystyle H}
1912:{\displaystyle T}
1892:{\displaystyle H}
1825:
1796:
1767:
1724:
1678:
1677:
1671:
1643:
1433:
1263:
1249:{\displaystyle X}
928:
901:{\displaystyle x}
735:
721:{\displaystyle X}
514:{\displaystyle X}
451:{\displaystyle X}
231:{\displaystyle X}
32:Universal algebra
16:(Redirected from
14247:
14235:Families of sets
14211:from PlanetMath.
14205:
14178:
14172:
14166:
14165:
14150:(2nd ed.).
14144:Kallenberg, Olav
14140:
14134:
14133:
14123:
14099:
14093:
14092:
14082:
14080:
14079:
14074:
14062:
14060:
14059:
14054:
14042:
14040:
14039:
14034:
14022:
14020:
14019:
14014:
13990:
13988:
13987:
13982:
13967:
13965:
13964:
13959:
13939:
13926:
13920:
13919:
13901:
13895:
13894:
13869:
13863:
13862:
13841:
13835:
13834:
13829:
13827:
13812:
13793:
13791:
13790:
13785:
13774:
13773:
13760:
13758:
13757:
13752:
13750:
13749:
13736:
13734:
13733:
13728:
13720:
13719:
13707:
13706:
13678:
13676:
13675:
13670:
13665:
13664:
13647:
13645:
13644:
13639:
13612:
13610:
13609:
13604:
13599:
13598:
13581:
13579:
13578:
13573:
13553:
13541:Additionally, a
13535:
13524:
13523:
13522:
13520:
13519:
13514:
13498:
13497:
13496:
13494:
13493:
13488:
13472:
13471:
13463:
13462:
13454:
13453:
13452:
13450:
13449:
13444:
13426:
13425:
13417:
13416:
13408:
13407:
13399:
13390:
13389:
13385:
13383:
13382:
13377:
13375:
13374:
13358:
13357:
13356:
13354:
13353:
13348:
13343:
13342:
13320:
13319:
13313:
13312:
13306:
13305:
13299:
13297:
13295:
13294:
13289:
13278:(even arbitrary
13275:
13268:
13267:
13261:
13260:
13254:
13253:
13247:
13246:
13240:
13239:
13232:
13221:
13220:
13214:
13213:
13207:
13205:
13203:
13202:
13197:
13186:(even arbitrary
13183:
13176:
13175:
13169:
13168:
13162:
13161:
13155:
13154:
13148:
13147:
13141:
13140:
13133:
13125:
13124:
13118:
13116:
13115:
13110:
13108:
13107:
13087:
13086:
13080:
13079:
13073:
13072:
13060:
13059:
13053:
13052:
13046:
13045:
13038:
13030:
13029:
13023:
13021:
13020:
13015:
13013:
13012:
12992:
12991:
12985:
12984:
12978:
12977:
12965:
12964:
12958:
12957:
12951:
12950:
12943:
12940:
12932:
12931:
12925:
12923:
12922:
12917:
12915:
12914:
12894:
12893:
12887:
12886:
12880:
12879:
12867:
12866:
12860:
12859:
12853:
12852:
12845:
12837:
12836:
12830:
12829:
12823:
12822:
12816:
12815:
12809:
12808:
12802:
12801:
12795:
12794:
12788:
12787:
12781:
12780:
12774:
12773:
12766:
12755:
12754:
12748:
12747:
12741:
12740:
12734:
12733:
12727:
12726:
12720:
12719:
12713:
12712:
12706:
12705:
12699:
12698:
12691:
12688:
12677:
12676:
12670:
12669:
12663:
12662:
12656:
12655:
12649:
12648:
12642:
12641:
12635:
12634:
12628:
12627:
12621:
12620:
12613:
12612:
12599:
12598:
12592:
12591:
12585:
12584:
12578:
12577:
12571:
12570:
12564:
12563:
12557:
12556:
12550:
12549:
12543:
12542:
12535:
12524:
12523:
12517:
12516:
12510:
12509:
12503:
12502:
12496:
12495:
12489:
12488:
12482:
12481:
12475:
12474:
12468:
12467:
12460:
12449:
12448:
12442:
12441:
12435:
12434:
12428:
12427:
12421:
12420:
12414:
12413:
12407:
12406:
12400:
12399:
12393:
12392:
12385:
12384:
12383:(Measure theory)
12374:
12373:
12367:
12366:
12360:
12359:
12353:
12352:
12346:
12345:
12339:
12338:
12332:
12331:
12325:
12324:
12318:
12317:
12311:
12310:
12303:
12302:
12289:
12288:
12282:
12281:
12269:
12267:
12266:
12261:
12256:
12255:
12237:
12236:
12230:
12229:
12223:
12221:
12220:
12215:
12194:
12193:
12187:
12186:
12180:
12179:
12172:
12171:
12167:
12159:
12158:
12152:
12151:
12145:
12144:
12138:
12136:
12135:
12130:
12125:
12124:
12106:
12104:
12103:
12098:
12093:
12092:
12074:
12073:
12067:
12066:
12060:
12059:
12053:
12052:
12046:
12045:
12038:
12027:
12026:
12020:
12019:
12013:
12012:
12006:
12005:
11999:
11998:
11992:
11991:
11985:
11984:
11978:
11977:
11971:
11970:
11964:
11962:
11959:
11947:
11946:
11940:
11939:
11933:
11932:
11926:
11925:
11919:
11918:
11912:
11911:
11905:
11904:
11898:
11897:
11891:
11890:
11884:
11875:
11874:
11868:
11867:
11861:
11860:
11854:
11853:
11847:
11846:
11840:
11839:
11833:
11832:
11826:
11825:
11819:
11818:
11812:
11811:
11805:
11802:
11793:
11786:
11785:
11783:
11782:
11777:
11775:
11774:
11753:
11752:
11750:
11749:
11744:
11742:
11741:
11720:
11719:
11717:
11716:
11711:
11703:
11702:
11690:
11689:
11671:
11670:
11668:
11667:
11662:
11654:
11653:
11641:
11640:
11622:
11621:
11619:
11618:
11613:
11593:
11592:
11590:
11589:
11584:
11564:
11563:
11561:
11560:
11555:
11535:
11534:
11532:
11531:
11526:
11506:
11504:
11502:
11501:
11496:
11477:
11476:
11472:
11470:
11469:
11464:
11462:
11461:
11445:
11444:
11443:
11441:
11440:
11435:
11430:
11429:
11406:
11399:
11392:
11385:
11384:
11382:
11381:
11376:
11362:
11360:
11359:
11354:
11352:
11351:
11334:
11333:
11299:
11297:
11296:
11291:
11286:
11282:
11278:
11277:
11276:
11271:
11270:
11245:
11244:
11217:
11213:
11212:
11208:
11207:
11202:
11201:
11169:
11168:
11129:
11127:
11126:
11121:
11099:
11097:
11096:
11091:
11079:
11077:
11076:
11071:
11060:(see above) for
11059:
11057:
11056:
11051:
11049:
11045:
11044:
11039:
11038:
11017:
11015:
11014:
11009:
11006:
11005:
11004:
10998:
10966:
10964:
10963:
10958:
10953:
10941:
10939:
10938:
10933:
10931:
10930:
10929:
10923:
10906:
10904:
10903:
10898:
10893:
10859:
10857:
10856:
10851:
10846:
10842:
10841:
10837:
10836:
10831:
10821:
10820:
10796:
10795:
10756:
10754:
10753:
10748:
10736:
10734:
10733:
10728:
10706:
10704:
10703:
10698:
10674:
10672:
10671:
10666:
10654:
10652:
10651:
10646:
10644:
10643:
10638:
10625:
10623:
10622:
10617:
10614:
10613:
10608:
10578:
10576:
10575:
10570:
10565:
10518:
10516:
10515:
10510:
10508:
10504:
10479:
10471:
10470:
10461:
10460:
10448:
10444:
10443:
10442:
10424:
10423:
10409:
10408:
10385:
10384:
10368:
10366:
10365:
10360:
10355:
10351:
10332:
10331:
10319:
10318:
10300:
10296:
10289:
10288:
10270:
10269:
10251:
10250:
10238:
10237:
10209:
10205:
10204:
10203:
10198:
10189:
10188:
10170:
10169:
10152:
10148:
10147:
10146:
10128:
10127:
10113:
10112:
10096:
10094:
10093:
10088:
10076:
10074:
10073:
10068:
10066:
10052:
10050:
10049:
10044:
10029:
10027:
10026:
10021:
10019:
10018:
10017:
10011:
9997:product topology
9994:
9992:
9991:
9986:
9967:
9965:
9964:
9959:
9957:
9956:
9955:
9954:
9936:
9935:
9920:
9907:
9899:
9898:
9883:
9878:
9860:
9859:
9854:
9853:
9839:
9837:
9836:
9831:
9825:
9824:
9823:
9822:
9804:
9803:
9781:
9779:
9778:
9773:
9771:
9767:
9745:
9737:
9736:
9727:
9726:
9714:
9710:
9709:
9708:
9690:
9689:
9675:
9674:
9673:
9672:
9654:
9653:
9625:
9623:
9622:
9617:
9612:
9608:
9589:
9588:
9573:
9572:
9534:
9533:
9515:
9514:
9502:
9501:
9500:
9499:
9481:
9480:
9459:
9457:
9456:
9451:
9435:
9433:
9432:
9427:
9422:
9410:
9408:
9407:
9402:
9397:
9389:
9388:
9373:
9371:
9370:
9365:
9360:
9347:
9343:
9311:
9310:
9309:
9303:
9274:
9272:
9271:
9266:
9261:
9257:
9253:
9252:
9244:
9243:
9231:
9230:
9218:
9214:
9213:
9212:
9200:
9199:
9176:
9172:
9171:
9170:
9158:
9157:
9128:
9124:
9120:
9119:
9111:
9110:
9095:
9094:
9061:
9060:
9021:
9020:
9015:
9006:
9005:
8992:
8990:
8989:
8984:
8982:
8981:
8976:
8960:
8958:
8957:
8952:
8947:
8946:
8934:
8933:
8921:
8920:
8908:
8907:
8895:
8894:
8882:
8881:
8860:
8858:
8857:
8852:
8847:
8843:
8839:
8838:
8837:
8825:
8824:
8812:
8811:
8799:
8798:
8786:
8785:
8773:
8772:
8748:
8747:
8735:
8734:
8714:
8712:
8711:
8706:
8704:
8703:
8691:
8690:
8671:
8669:
8668:
8663:
8661:
8657:
8656:
8655:
8643:
8642:
8621:
8619:
8618:
8613:
8611:
8607:
8606:
8605:
8593:
8592:
8555:
8553:
8552:
8547:
8545:
8544:
8539:
8522:
8520:
8519:
8514:
8509:
8508:
8503:
8452:
8450:
8449:
8444:
8442:
8441:
8436:
8423:
8421:
8420:
8415:
8413:
8412:
8407:
8390:
8388:
8387:
8382:
8380:
8376:
8375:
8374:
8347:
8345:
8344:
8339:
8327:
8325:
8324:
8319:
8280:
8278:
8277:
8272:
8257:
8255:
8254:
8249:
8204:
8202:
8201:
8196:
8194:
8190:
8189:
8188:
8161:
8159:
8158:
8153:
8151:
8147:
8146:
8145:
8118:
8116:
8115:
8110:
8098:
8096:
8095:
8090:
8085:
8081:
8080:
8079:
8052:
8050:
8049:
8044:
8042:
8038:
8037:
8036:
8009:
8007:
8006:
8001:
7989:
7987:
7986:
7981:
7979:
7975:
7974:
7973:
7946:
7944:
7943:
7938:
7936:
7932:
7931:
7930:
7903:
7901:
7900:
7895:
7881:
7879:
7878:
7873:
7868:
7864:
7860:
7859:
7851:
7850:
7838:
7837:
7822:
7818:
7817:
7816:
7804:
7803:
7780:
7776:
7775:
7774:
7762:
7761:
7744:
7743:
7698:
7696:
7695:
7690:
7685:
7684:
7679:
7666:
7664:
7663:
7658:
7637:
7635:
7634:
7629:
7627:
7626:
7621:
7608:
7606:
7605:
7600:
7588:
7586:
7585:
7580:
7566:
7564:
7563:
7558:
7539:
7537:
7536:
7531:
7511:
7509:
7508:
7503:
7491:
7489:
7488:
7483:
7469:
7467:
7466:
7461:
7446:
7444:
7443:
7438:
7417:
7415:
7414:
7409:
7397:
7395:
7394:
7389:
7373:
7371:
7370:
7365:
7353:
7351:
7350:
7345:
7333:
7331:
7330:
7325:
7311:
7309:
7308:
7303:
7298:
7294:
7270:
7269:
7230:
7228:
7227:
7222:
7207:
7205:
7204:
7199:
7187:
7185:
7184:
7179:
7168:
7167:
7148:
7146:
7145:
7140:
7116:
7114:
7113:
7108:
7091:
7089:
7088:
7083:
7071:
7069:
7068:
7063:
7048:
7046:
7045:
7040:
7028:
7026:
7025:
7020:
7008:
7006:
7005:
7000:
6988:
6986:
6985:
6980:
6968:
6966:
6965:
6960:
6937:
6935:
6934:
6929:
6927:
6923:
6919:
6912:
6911:
6899:
6898:
6870:
6868:
6867:
6862:
6860:
6856:
6849:
6848:
6836:
6835:
6811:
6809:
6808:
6803:
6767:
6765:
6764:
6759:
6732:
6730:
6729:
6724:
6694:
6692:
6691:
6686:
6666:
6664:
6663:
6658:
6639:
6637:
6636:
6631:
6535:
6533:
6532:
6527:
6509:
6507:
6506:
6501:
6459:
6457:
6456:
6451:
6439:
6437:
6436:
6431:
6419:
6417:
6416:
6411:
6390:
6388:
6387:
6382:
6340:
6338:
6337:
6332:
6317:
6315:
6314:
6309:
6292:
6290:
6289:
6284:
6263:
6261:
6260:
6255:
6240:
6238:
6237:
6232:
6220:
6218:
6217:
6212:
6200:
6198:
6197:
6192:
6177:
6175:
6174:
6169:
6145:
6143:
6142:
6137:
6132:
6131:
6126:
6125:
6111:
6109:
6108:
6103:
6090:
6088:
6087:
6082:
6062:
6060:
6059:
6054:
6049:
6048:
6043:
6042:
6028:
6026:
6025:
6020:
6008:
6006:
6005:
6000:
5976:
5974:
5973:
5968:
5949:
5947:
5946:
5941:
5922:
5920:
5919:
5914:
5902:
5900:
5899:
5894:
5882:
5880:
5879:
5874:
5860:
5858:
5857:
5852:
5837:
5835:
5834:
5829:
5780:
5778:
5777:
5772:
5751:
5749:
5748:
5743:
5724:
5722:
5721:
5716:
5697:
5695:
5694:
5689:
5666:, the resulting
5649:that renders it
5641:
5639:
5638:
5633:
5613:
5611:
5610:
5605:
5594:operator). Any
5589:
5587:
5586:
5581:
5569:
5567:
5566:
5561:
5546:
5544:
5543:
5538:
5536:
5535:
5510:
5508:
5507:
5502:
5494:
5493:
5439:
5437:
5436:
5431:
5429:
5428:
5415:
5413:
5412:
5407:
5393:
5391:
5390:
5385:
5369:
5367:
5366:
5361:
5335:
5333:
5332:
5327:
5318:
5317:
5293:
5291:
5290:
5285:
5279:
5278:
5254:
5252:
5251:
5246:
5221:Fraktur typeface
5208:Typographic note
5179:
5177:
5176:
5171:
5149:
5147:
5146:
5141:
5117:
5115:
5114:
5109:
5094:
5092:
5091:
5086:
5044:
5042:
5041:
5036:
5010:
5008:
5007:
5002:
4987:
4985:
4984:
4979:
4939:
4937:
4936:
4931:
4907:
4905:
4904:
4899:
4887:
4885:
4884:
4879:
4860:
4858:
4857:
4852:
4847:
4843:
4842:
4841:
4831:
4830:
4829:
4798:
4797:
4778:
4776:
4775:
4770:
4768:
4764:
4763:
4762:
4752:
4751:
4750:
4722:
4721:
4704:
4702:
4701:
4696:
4691:
4690:
4675:
4671:
4670:
4669:
4659:
4658:
4657:
4625:
4623:
4622:
4617:
4612:
4611:
4602:
4601:
4591:
4590:
4589:
4565:
4563:
4562:
4557:
4552:
4551:
4542:
4541:
4525:
4523:
4522:
4517:
4495:Sketch of Proof:
4493:
4491:
4490:
4485:
4480:
4476:
4464:
4460:
4459:
4450:
4449:
4437:
4436:
4435:
4434:
4417:
4416:
4404:
4403:
4393:
4388:
4365:
4364:
4351:
4349:
4348:
4343:
4338:
4334:
4333:
4332:
4322:
4321:
4320:
4292:
4291:
4281:
4280:
4279:
4244:is a σ-algebra.
4243:
4241:
4240:
4235:
4233:
4232:
4216:
4214:
4213:
4208:
4203:
4202:
4186:
4184:
4183:
4178:
4176:
4175:
4159:
4157:
4156:
4151:
4149:
4148:
4136:
4135:
4119:
4117:
4116:
4111:
4099:
4097:
4096:
4091:
4089:
4088:
4071:Sketch of Proof:
4067:
4065:
4064:
4059:
4054:
4053:
4043:
4042:
4041:
4010:
4008:
4007:
4002:
3987:
3985:
3984:
3979:
3976:
3972:
3971:
3970:
3955:
3954:
3925:
3923:
3922:
3917:
3899:
3897:
3896:
3891:
3875:
3873:
3872:
3867:
3865:
3853:
3851:
3850:
3845:
3840:
3828:
3826:
3825:
3820:
3801:
3799:
3798:
3793:
3772:
3770:
3769:
3764:
3759:
3747:
3745:
3744:
3739:
3727:
3725:
3724:
3719:
3701:
3700:
3673:
3657:
3655:
3654:
3649:
3635:
3633:
3632:
3627:
3603:
3601:
3600:
3595:
3574:
3572:
3571:
3566:
3554:
3552:
3551:
3546:
3534:
3532:
3531:
3526:
3512:is contained in
3511:
3509:
3508:
3503:
3488:
3486:
3485:
3480:
3459:
3457:
3456:
3451:
3436:
3434:
3433:
3428:
3416:
3414:
3413:
3408:
3388:
3386:
3385:
3380:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3321:
3319:
3318:
3313:
3301:
3299:
3298:
3293:
3246:
3244:
3243:
3238:
3186:measurable space
3183:
3181:
3180:
3175:
3160:
3158:
3157:
3152:
3140:
3138:
3137:
3132:
3120:
3118:
3117:
3112:
3079:
3077:
3076:
3071:
3047:
3045:
3044:
3039:
3027:
3025:
3024:
3019:
3004:
3002:
3001:
2996:
2968:
2966:
2965:
2960:
2937:Moreover, since
2936:
2934:
2933:
2928:
2909:
2907:
2906:
2901:
2889:
2887:
2886:
2881:
2866:
2864:
2863:
2858:
2843:
2841:
2840:
2835:
2816:De Morgan's laws
2806:
2804:
2803:
2798:
2787:
2786:
2774:
2773:
2761:
2760:
2738:
2736:
2735:
2730:
2715:
2713:
2712:
2707:
2699:
2698:
2686:
2685:
2673:
2672:
2652:
2650:
2649:
2644:
2631:
2629:
2628:
2623:
2598:
2596:
2595:
2590:
2575:
2573:
2572:
2567:
2551:
2549:
2548:
2543:
2529:
2527:
2526:
2521:
2509:
2507:
2506:
2501:
2482:
2480:
2479:
2474:
2448:. Then a subset
2443:
2441:
2440:
2435:
2414:
2412:
2411:
2406:
2381:
2379:
2378:
2373:
2371:
2370:
2365:
2364:
2350:
2348:
2347:
2342:
2337:
2336:
2331:
2330:
2314:
2313:
2308:
2307:
2297:
2296:
2291:
2290:
2280:
2279:
2274:
2273:
2257:
2255:
2254:
2249:
2241:
2240:
2210:
2209:
2176:
2175:
2170:
2169:
2155:
2153:
2152:
2147:
2134:
2132:
2131:
2126:
2112:
2110:
2109:
2104:
2066:
2065:
2044:
2043:
2031:
2030:
2018:
2017:
1999:
1998:
1965:
1963:
1962:
1957:
1942:
1940:
1939:
1934:
1918:
1916:
1915:
1910:
1898:
1896:
1895:
1890:
1863:
1861:
1860:
1855:
1850:
1849:
1839:
1821:
1820:
1810:
1792:
1791:
1781:
1759:
1757:
1756:
1751:
1749:
1748:
1738:
1716:
1714:
1713:
1708:
1703:
1702:
1692:
1675:
1669:
1668:
1667:
1657:
1632:
1630:
1629:
1624:
1613:
1612:
1594:
1593:
1571:
1569:
1568:
1563:
1548:
1546:
1545:
1540:
1532:
1531:
1513:
1512:
1496:
1494:
1493:
1488:
1486:
1468:
1466:
1465:
1460:
1458:
1457:
1447:
1415:
1413:
1412:
1407:
1396:
1395:
1377:
1376:
1366:
1361:
1343:
1342:
1332:
1327:
1311:
1306:
1288:
1287:
1277:
1255:
1253:
1252:
1247:
1235:
1233:
1232:
1227:
1219:
1218:
1206:
1205:
1193:
1192:
1164:
1162:
1161:
1156:
1145:
1144:
1143:
1142:
1125:
1124:
1123:
1122:
1095:
1093:
1092:
1087:
1072:
1070:
1069:
1064:
1056:
1055:
1043:
1042:
1026:
1024:
1023:
1018:
1010:
1009:
1008:
1007:
990:
989:
988:
987:
963:
961:
960:
955:
953:
952:
942:
907:
905:
904:
899:
887:
885:
884:
879:
868:
867:
849:
848:
838:
833:
815:
814:
804:
799:
783:
778:
760:
759:
749:
727:
725:
724:
719:
707:
705:
704:
699:
691:
690:
678:
677:
665:
664:
617:
615:
614:
609:
586:
584:
583:
578:
551:
549:
548:
543:
520:
518:
517:
512:
457:
455:
454:
449:
433:
431:
430:
425:
411:
410:
398:
397:
385:
384:
358:
356:
355:
350:
338:
336:
335:
330:
237:
235:
234:
229:
217:
215:
214:
209:
114:measurable space
111:
109:
108:
103:
76:, and countable
21:
14255:
14254:
14250:
14249:
14248:
14246:
14245:
14244:
14225:Boolean algebra
14215:
14214:
14190:
14187:
14182:
14181:
14173:
14169:
14162:
14141:
14137:
14100:
14096:
14068:
14065:
14064:
14048:
14045:
14044:
14043:-null sets. If
14028:
14025:
14024:
13996:
13993:
13992:
13973:
13970:
13969:
13953:
13950:
13949:
13937:
13927:
13923:
13916:
13902:
13898:
13891:
13870:
13866:
13859:
13842:
13838:
13825:
13823:
13814:
13813:
13809:
13804:
13794:
13769:
13768:
13766:
13763:
13762:
13745:
13744:
13742:
13739:
13738:
13715:
13711:
13702:
13698:
13684:
13681:
13680:
13679:
13660:
13659:
13657:
13654:
13653:
13627:
13624:
13623:
13613:
13594:
13593:
13591:
13588:
13587:
13561:
13558:
13557:
13551:
13532:
13530:
13527:
13508:
13505:
13504:
13502:
13501:
13482:
13479:
13478:
13476:
13475:
13469:
13467:
13466:
13460:
13458:
13457:
13438:
13435:
13434:
13432:
13430:
13429:
13423:
13421:
13420:
13414:
13412:
13411:
13405:
13403:
13402:
13396:
13393:
13370:
13369:
13367:
13364:
13363:
13361:
13360:
13359:
13338:
13337:
13335:
13332:
13331:
13329:
13328:
13283:
13280:
13279:
13277:
13276:
13231:Closed Topology
13230:
13191:
13188:
13187:
13185:
13184:
13131:
13103:
13102:
13094:
13091:
13090:
13036:
13008:
13007:
12999:
12996:
12995:
12941:
12938:
12910:
12909:
12901:
12898:
12897:
12843:
12764:
12689:
12686:
12610:
12608:
12533:
12458:
12382:
12380:
12300:
12298:
12271:
12251:
12247:
12245:
12242:
12241:
12203:
12200:
12199:
12198:
12170:(Dynkin System)
12169:
12168:
12165:
12120:
12116:
12114:
12111:
12110:
12088:
12084:
12082:
12079:
12078:
12036:
11960:
11957:
11955:
11880:
11800:
11798:
11789:
11770:
11769:
11761:
11758:
11757:
11756:
11737:
11736:
11728:
11725:
11724:
11723:
11698:
11694:
11685:
11681:
11679:
11676:
11675:
11674:
11649:
11645:
11636:
11632:
11630:
11627:
11626:
11625:
11601:
11598:
11597:
11596:
11572:
11569:
11568:
11567:
11543:
11540:
11539:
11538:
11514:
11511:
11510:
11509:
11489:
11486:
11485:
11483:
11480:
11457:
11456:
11454:
11451:
11450:
11448:
11447:
11446:
11425:
11424:
11422:
11419:
11418:
11416:
11415:
11410:
11370:
11367:
11366:
11347:
11346:
11344:
11341:
11340:
11337:
11306:
11272:
11266:
11265:
11264:
11237:
11233:
11232:
11228:
11224:
11203:
11197:
11196:
11195:
11191:
11161:
11157:
11156:
11152:
11135:
11132:
11131:
11115:
11112:
11111:
11085:
11082:
11081:
11065:
11062:
11061:
11040:
11034:
11033:
11032:
11028:
11023:
11020:
11019:
11000:
10999:
10994:
10993:
10972:
10969:
10968:
10949:
10947:
10944:
10943:
10925:
10924:
10919:
10918:
10916:
10913:
10912:
10889:
10872:
10869:
10868:
10865:
10832:
10827:
10826:
10822:
10816:
10815:
10788:
10784:
10783:
10779:
10762:
10759:
10758:
10742:
10739:
10738:
10716:
10713:
10712:
10686:
10683:
10682:
10678:random variable
10660:
10657:
10656:
10639:
10634:
10633:
10631:
10628:
10627:
10609:
10604:
10603:
10588:
10585:
10584:
10561:
10544:
10541:
10540:
10537:
10529:Borel σ-algebra
10525:
10475:
10466:
10465:
10456:
10452:
10438:
10434:
10419:
10415:
10414:
10410:
10404:
10400:
10396:
10392:
10380:
10376:
10374:
10371:
10370:
10327:
10323:
10314:
10310:
10278:
10274:
10265:
10261:
10246:
10242:
10233:
10229:
10228:
10224:
10223:
10219:
10199:
10194:
10193:
10184:
10180:
10165:
10161:
10160:
10156:
10142:
10138:
10123:
10119:
10118:
10114:
10108:
10104:
10102:
10099:
10098:
10082:
10079:
10078:
10062:
10060:
10057:
10056:
10035:
10032:
10031:
10013:
10012:
10007:
10006:
10004:
10001:
10000:
9977:
9974:
9973:
9950:
9946:
9931:
9927:
9926:
9922:
9903:
9894:
9890:
9889:
9879:
9868:
9855:
9849:
9848:
9847:
9845:
9842:
9841:
9818:
9814:
9799:
9795:
9794:
9790:
9787:
9784:
9783:
9741:
9732:
9731:
9722:
9718:
9704:
9700:
9685:
9681:
9680:
9676:
9668:
9664:
9649:
9645:
9644:
9640:
9639:
9635:
9633:
9630:
9629:
9584:
9580:
9568:
9564:
9545:
9541:
9529:
9525:
9510:
9506:
9495:
9491:
9476:
9472:
9471:
9467:
9465:
9462:
9461:
9445:
9442:
9441:
9438:cylinder subset
9418:
9416:
9413:
9412:
9393:
9384:
9383:
9381:
9378:
9377:
9356:
9339:
9305:
9304:
9299:
9298:
9290:
9287:
9286:
9283:
9248:
9239:
9235:
9226:
9222:
9208:
9204:
9195:
9191:
9190:
9186:
9166:
9162:
9153:
9149:
9148:
9144:
9143:
9139:
9135:
9115:
9106:
9102:
9090:
9086:
9056:
9052:
9039:
9035:
9031:
9016:
9011:
9010:
9001:
9000:
8998:
8995:
8994:
8977:
8972:
8971:
8969:
8966:
8965:
8961:is a π-system.
8942:
8938:
8929:
8925:
8916:
8912:
8903:
8899:
8890:
8886:
8877:
8873:
8868:
8865:
8864:
8833:
8829:
8820:
8816:
8807:
8803:
8794:
8790:
8781:
8777:
8768:
8764:
8763:
8759:
8755:
8743:
8739:
8730:
8726:
8724:
8721:
8720:
8699:
8695:
8686:
8682:
8680:
8677:
8676:
8651:
8647:
8638:
8634:
8633:
8629:
8627:
8624:
8623:
8601:
8597:
8588:
8584:
8583:
8579:
8577:
8574:
8573:
8570:
8540:
8535:
8534:
8532:
8529:
8528:
8504:
8499:
8498:
8496:
8493:
8492:
8490:Euclidean space
8459:
8437:
8432:
8431:
8429:
8426:
8425:
8408:
8403:
8402:
8400:
8397:
8396:
8370:
8366:
8359:
8355:
8353:
8350:
8349:
8333:
8330:
8329:
8286:
8283:
8282:
8263:
8260:
8259:
8210:
8207:
8206:
8184:
8180:
8173:
8169:
8167:
8164:
8163:
8141:
8137:
8130:
8126:
8124:
8121:
8120:
8104:
8101:
8100:
8075:
8071:
8064:
8060:
8058:
8055:
8054:
8032:
8028:
8021:
8017:
8015:
8012:
8011:
7995:
7992:
7991:
7969:
7965:
7958:
7954:
7952:
7949:
7948:
7926:
7922:
7915:
7911:
7909:
7906:
7905:
7889:
7886:
7885:
7855:
7846:
7842:
7833:
7829:
7812:
7808:
7799:
7795:
7794:
7790:
7770:
7766:
7757:
7753:
7752:
7748:
7736:
7732:
7731:
7727:
7723:
7703:
7700:
7699:
7680:
7675:
7674:
7672:
7669:
7668:
7643:
7640:
7639:
7622:
7617:
7616:
7614:
7611:
7610:
7594:
7591:
7590:
7574:
7571:
7570:
7549:
7546:
7545:
7525:
7522:
7521:
7497:
7494:
7493:
7477:
7474:
7473:
7452:
7449:
7448:
7447:is a subset of
7423:
7420:
7419:
7418:if and only if
7403:
7400:
7399:
7383:
7380:
7379:
7359:
7356:
7355:
7339:
7336:
7335:
7319:
7316:
7315:
7262:
7258:
7257:
7253:
7236:
7233:
7232:
7213:
7210:
7209:
7193:
7190:
7189:
7160:
7156:
7154:
7151:
7150:
7122:
7119:
7118:
7099:
7096:
7095:
7077:
7074:
7073:
7054:
7051:
7050:
7034:
7031:
7030:
7014:
7011:
7010:
6994:
6991:
6990:
6974:
6971:
6970:
6954:
6951:
6950:
6947:
6907:
6903:
6894:
6890:
6889:
6885:
6881:
6876:
6873:
6872:
6844:
6840:
6831:
6827:
6826:
6822:
6817:
6814:
6813:
6773:
6770:
6769:
6738:
6735:
6734:
6700:
6697:
6696:
6671:
6668:
6667:
6649:
6646:
6645:
6541:
6538:
6537:
6515:
6512:
6511:
6468:
6465:
6464:
6445:
6442:
6441:
6425:
6422:
6421:
6396:
6393:
6392:
6346:
6343:
6342:
6341:is empty, then
6326:
6323:
6322:
6300:
6297:
6296:
6269:
6266:
6265:
6246:
6243:
6242:
6226:
6223:
6222:
6206:
6203:
6202:
6183:
6180:
6179:
6163:
6160:
6159:
6156:
6151:
6127:
6121:
6120:
6119:
6117:
6114:
6113:
6097:
6094:
6093:
6076:
6073:
6072:
6044:
6038:
6037:
6036:
6034:
6031:
6030:
6014:
6011:
6010:
5994:
5991:
5990:
5984:
5977:is a σ-algebra.
5962:
5959:
5958:
5932:
5929:
5928:
5908:
5905:
5904:
5903:if and only if
5888:
5885:
5884:
5868:
5865:
5864:
5843:
5840:
5839:
5793:
5790:
5789:
5788:The collection
5763:
5760:
5759:
5734:
5731:
5730:
5707:
5704:
5703:
5683:
5680:
5679:
5676:
5627:
5624:
5623:
5599:
5596:
5595:
5575:
5572:
5571:
5555:
5552:
5551:
5531:
5530:
5516:
5513:
5512:
5489:
5488:
5456:
5453:
5452:
5442:separable space
5424:
5423:
5421:
5418:
5417:
5401:
5398:
5397:
5379:
5376:
5375:
5355:
5352:
5351:
5346:
5341:
5313:
5312:
5299:
5296:
5295:
5274:
5273:
5260:
5257:
5256:
5228:
5225:
5224:
5210:
5162:
5159:
5158:
5135:
5132:
5131:
5124:
5100:
5097:
5096:
5050:
5047:
5046:
5018:
5015:
5014:
4993:
4990:
4989:
4949:
4946:
4945:
4944:The collection
4913:
4910:
4909:
4893:
4890:
4889:
4888:is a subset of
4873:
4870:
4869:
4866:
4837:
4833:
4825:
4824:
4817:
4812:
4808:
4793:
4792:
4784:
4781:
4780:
4758:
4754:
4746:
4745:
4738:
4733:
4729:
4717:
4716:
4714:
4711:
4710:
4686:
4685:
4665:
4661:
4653:
4652:
4645:
4640:
4636:
4631:
4628:
4627:
4607:
4606:
4597:
4593:
4585:
4584:
4577:
4571:
4568:
4567:
4547:
4546:
4537:
4533:
4531:
4528:
4527:
4502:
4499:
4498:
4455:
4454:
4445:
4441:
4430:
4426:
4425:
4421:
4412:
4408:
4399:
4395:
4389:
4378:
4373:
4369:
4360:
4359:
4357:
4354:
4353:
4328:
4324:
4316:
4315:
4308:
4303:
4299:
4287:
4283:
4275:
4274:
4267:
4261:
4258:
4257:
4228:
4224:
4222:
4219:
4218:
4198:
4194:
4192:
4189:
4188:
4171:
4167:
4165:
4162:
4161:
4144:
4140:
4131:
4127:
4125:
4122:
4121:
4105:
4102:
4101:
4084:
4080:
4078:
4075:
4074:
4049:
4045:
4037:
4036:
4029:
4023:
4020:
4019:
3993:
3990:
3989:
3966:
3965:
3950:
3946:
3945:
3941:
3938:
3935:
3934:
3931:
3908:
3905:
3904:
3885:
3882:
3881:
3861:
3859:
3856:
3855:
3836:
3834:
3831:
3830:
3811:
3808:
3807:
3778:
3775:
3774:
3755:
3753:
3750:
3749:
3733:
3730:
3729:
3696:
3692:
3669:
3667:
3664:
3663:
3643:
3640:
3639:
3609:
3606:
3605:
3580:
3577:
3576:
3560:
3557:
3556:
3540:
3537:
3536:
3517:
3514:
3513:
3497:
3494:
3493:
3465:
3462:
3461:
3442:
3439:
3438:
3422:
3419:
3418:
3402:
3399:
3398:
3374:
3371:
3370:
3354:
3351:
3350:
3334:
3331:
3330:
3307:
3304:
3303:
3287:
3284:
3283:
3269:
3263:
3217:
3214:
3213:
3166:
3163:
3162:
3146:
3143:
3142:
3126:
3123:
3122:
3091:
3088:
3087:
3084:measurable sets
3053:
3050:
3049:
3033:
3030:
3029:
3010:
3007:
3006:
2978:
2975:
2974:
2942:
2939:
2938:
2919:
2916:
2915:
2895:
2892:
2891:
2875:
2872:
2871:
2849:
2846:
2845:
2829:
2826:
2825:
2782:
2778:
2769:
2765:
2756:
2752:
2744:
2741:
2740:
2721:
2718:
2717:
2694:
2690:
2681:
2677:
2668:
2664:
2662:
2659:
2658:
2638:
2635:
2634:
2608:
2605:
2604:
2599:then so is its
2581:
2578:
2577:
2561:
2558:
2557:
2537:
2534:
2533:
2515:
2512:
2511:
2495:
2492:
2491:
2453:
2450:
2449:
2420:
2417:
2416:
2400:
2397:
2396:
2393:
2388:
2366:
2360:
2359:
2358:
2356:
2353:
2352:
2332:
2326:
2325:
2324:
2309:
2303:
2302:
2301:
2292:
2286:
2285:
2284:
2275:
2269:
2268:
2267:
2265:
2262:
2261:
2236:
2232:
2205:
2201:
2171:
2165:
2164:
2163:
2161:
2158:
2157:
2141:
2138:
2137:
2120:
2117:
2116:
2115:However, after
2061:
2057:
2039:
2035:
2026:
2022:
2013:
2009:
1994:
1990:
1970:
1967:
1966:
1948:
1945:
1944:
1928:
1925:
1924:
1904:
1901:
1900:
1884:
1881:
1880:
1869:
1845:
1841:
1829:
1816:
1812:
1800:
1787:
1783:
1771:
1765:
1762:
1761:
1744:
1740:
1728:
1722:
1719:
1718:
1698:
1694:
1682:
1663:
1659:
1647:
1641:
1638:
1637:
1602:
1598:
1589:
1585:
1577:
1574:
1573:
1554:
1551:
1550:
1521:
1517:
1508:
1504:
1502:
1499:
1498:
1482:
1474:
1471:
1470:
1453:
1449:
1437:
1425:
1422:
1421:
1385:
1381:
1372:
1368:
1362:
1351:
1338:
1334:
1328:
1317:
1307:
1296:
1283:
1279:
1267:
1261:
1258:
1257:
1241:
1238:
1237:
1214:
1210:
1201:
1197:
1188:
1184:
1182:
1179:
1178:
1138:
1134:
1133:
1129:
1118:
1114:
1113:
1109:
1101:
1098:
1097:
1078:
1075:
1074:
1051:
1047:
1038:
1034:
1032:
1029:
1028:
1003:
999:
998:
994:
983:
979:
978:
974:
972:
969:
968:
948:
944:
932:
920:
917:
916:
893:
890:
889:
857:
853:
844:
840:
834:
823:
810:
806:
800:
789:
779:
768:
755:
751:
739:
733:
730:
729:
713:
710:
709:
686:
682:
673:
669:
660:
656:
654:
651:
650:
625:
600:
597:
596:
589:axiom of choice
569:
566:
565:
534:
531:
530:
506:
503:
502:
495:
487:
479:Borel hierarchy
443:
440:
439:
434:is a countable
406:
402:
393:
389:
380:
376:
371:
368:
367:
344:
341:
340:
243:
240:
239:
223:
220:
219:
173:
170:
169:
85:
82:
81:
35:
28:
23:
22:
15:
12:
11:
5:
14253:
14243:
14242:
14240:Measure theory
14237:
14232:
14227:
14213:
14212:
14206:
14186:
14185:External links
14183:
14180:
14179:
14167:
14160:
14135:
14114:(1): 345–349.
14094:
14088:if and only if
14072:
14052:
14032:
14012:
14009:
14006:
14003:
14000:
13980:
13977:
13957:
13931:Kunen, Kenneth
13921:
13914:
13896:
13889:
13864:
13857:
13836:
13806:
13805:
13803:
13800:
13797:
13796:
13783:
13780:
13777:
13772:
13748:
13726:
13723:
13718:
13714:
13710:
13705:
13701:
13697:
13694:
13691:
13688:
13668:
13663:
13650:disjoint union
13637:
13634:
13631:
13620:
13602:
13597:
13584:disjoint union
13571:
13568:
13565:
13547:
13537:
13536:
13525:
13512:
13499:
13486:
13473:
13464:
13455:
13442:
13427:
13418:
13409:
13400:
13391:
13373:
13346:
13341:
13325:
13324:
13321:
13314:
13307:
13300:
13287:
13269:
13262:
13255:
13248:
13241:
13234:
13226:
13225:
13222:
13215:
13208:
13195:
13177:
13170:
13163:
13156:
13149:
13142:
13135:
13127:
13126:
13119:
13106:
13101:
13098:
13088:
13081:
13074:
13067:
13064:
13061:
13054:
13047:
13040:
13037:Filter subbase
13032:
13031:
13024:
13011:
13006:
13003:
12993:
12986:
12979:
12972:
12969:
12966:
12959:
12952:
12945:
12934:
12933:
12926:
12913:
12908:
12905:
12895:
12888:
12881:
12874:
12871:
12868:
12861:
12854:
12847:
12839:
12838:
12831:
12824:
12817:
12810:
12803:
12796:
12789:
12782:
12775:
12768:
12760:
12759:
12756:
12749:
12742:
12735:
12728:
12721:
12714:
12707:
12700:
12693:
12682:
12681:
12678:
12671:
12664:
12657:
12650:
12643:
12636:
12629:
12622:
12615:
12604:
12603:
12600:
12593:
12586:
12579:
12572:
12565:
12558:
12551:
12544:
12537:
12529:
12528:
12525:
12518:
12511:
12504:
12497:
12490:
12483:
12476:
12469:
12462:
12454:
12453:
12450:
12443:
12436:
12429:
12422:
12415:
12408:
12401:
12394:
12387:
12376:
12375:
12368:
12361:
12354:
12347:
12340:
12333:
12326:
12319:
12312:
12305:
12301:(Order theory)
12294:
12293:
12290:
12283:
12276:
12259:
12254:
12250:
12238:
12231:
12224:
12213:
12210:
12207:
12195:
12188:
12181:
12174:
12161:
12160:
12153:
12146:
12139:
12128:
12123:
12119:
12107:
12096:
12091:
12087:
12075:
12068:
12061:
12054:
12047:
12040:
12037:Monotone class
12032:
12031:
12028:
12021:
12014:
12007:
12000:
11993:
11986:
11979:
11972:
11965:
11952:
11951:
11948:
11941:
11934:
11927:
11920:
11913:
11906:
11899:
11892:
11885:
11877:
11876:
11869:
11862:
11855:
11848:
11841:
11834:
11827:
11820:
11813:
11806:
11795:
11794:
11787:
11773:
11768:
11765:
11754:
11740:
11735:
11732:
11721:
11709:
11706:
11701:
11697:
11693:
11688:
11684:
11672:
11660:
11657:
11652:
11648:
11644:
11639:
11635:
11623:
11611:
11608:
11605:
11594:
11582:
11579:
11576:
11565:
11553:
11550:
11547:
11536:
11524:
11521:
11518:
11507:
11494:
11478:
11460:
11433:
11428:
11412:
11411:
11409:
11408:
11401:
11394:
11386:
11374:
11350:
11332:
11331:
11325:
11319:
11313:
11305:
11302:
11289:
11285:
11281:
11275:
11269:
11263:
11260:
11257:
11254:
11251:
11248:
11243:
11240:
11236:
11231:
11227:
11223:
11220:
11216:
11211:
11206:
11200:
11194:
11190:
11187:
11184:
11181:
11178:
11175:
11172:
11167:
11164:
11160:
11155:
11151:
11148:
11145:
11142:
11139:
11119:
11108:random process
11089:
11069:
11048:
11043:
11037:
11031:
11027:
11003:
10997:
10992:
10989:
10986:
10983:
10980:
10977:
10956:
10952:
10928:
10922:
10896:
10892:
10888:
10885:
10882:
10879:
10876:
10864:
10861:
10849:
10845:
10840:
10835:
10830:
10825:
10819:
10814:
10811:
10808:
10805:
10802:
10799:
10794:
10791:
10787:
10782:
10778:
10775:
10772:
10769:
10766:
10746:
10726:
10723:
10720:
10696:
10693:
10690:
10664:
10642:
10637:
10612:
10607:
10602:
10599:
10596:
10593:
10568:
10564:
10560:
10557:
10554:
10551:
10548:
10536:
10533:
10524:
10523:Ball σ-algebra
10521:
10507:
10503:
10500:
10497:
10494:
10491:
10488:
10485:
10482:
10478:
10474:
10469:
10464:
10459:
10455:
10451:
10447:
10441:
10437:
10433:
10430:
10427:
10422:
10418:
10413:
10407:
10403:
10399:
10395:
10391:
10388:
10383:
10379:
10358:
10354:
10350:
10347:
10344:
10341:
10338:
10335:
10330:
10326:
10322:
10317:
10313:
10309:
10306:
10303:
10299:
10295:
10292:
10287:
10284:
10281:
10277:
10273:
10268:
10264:
10260:
10257:
10254:
10249:
10245:
10241:
10236:
10232:
10227:
10222:
10218:
10215:
10212:
10208:
10202:
10197:
10192:
10187:
10183:
10179:
10176:
10173:
10168:
10164:
10159:
10155:
10151:
10145:
10141:
10137:
10134:
10131:
10126:
10122:
10117:
10111:
10107:
10086:
10065:
10042:
10039:
10030:restricted to
10016:
10010:
9984:
9981:
9953:
9949:
9945:
9942:
9939:
9934:
9930:
9925:
9919:
9916:
9913:
9910:
9906:
9902:
9897:
9893:
9888:
9882:
9877:
9874:
9871:
9867:
9863:
9858:
9852:
9828:
9821:
9817:
9813:
9810:
9807:
9802:
9798:
9793:
9770:
9766:
9763:
9760:
9757:
9754:
9751:
9748:
9744:
9740:
9735:
9730:
9725:
9721:
9717:
9713:
9707:
9703:
9699:
9696:
9693:
9688:
9684:
9679:
9671:
9667:
9663:
9660:
9657:
9652:
9648:
9643:
9638:
9615:
9611:
9607:
9604:
9601:
9598:
9595:
9592:
9587:
9583:
9579:
9576:
9571:
9567:
9563:
9560:
9557:
9554:
9551:
9548:
9544:
9540:
9537:
9532:
9528:
9524:
9521:
9518:
9513:
9509:
9505:
9498:
9494:
9490:
9487:
9484:
9479:
9475:
9470:
9449:
9425:
9421:
9400:
9396:
9392:
9387:
9363:
9359:
9355:
9352:
9346:
9342:
9338:
9335:
9332:
9329:
9326:
9323:
9320:
9317:
9314:
9308:
9302:
9297:
9294:
9282:
9279:
9264:
9260:
9256:
9251:
9247:
9242:
9238:
9234:
9229:
9225:
9221:
9217:
9211:
9207:
9203:
9198:
9194:
9189:
9185:
9182:
9179:
9175:
9169:
9165:
9161:
9156:
9152:
9147:
9142:
9138:
9134:
9131:
9127:
9123:
9118:
9114:
9109:
9105:
9101:
9098:
9093:
9089:
9085:
9082:
9079:
9076:
9073:
9070:
9067:
9064:
9059:
9055:
9051:
9048:
9045:
9042:
9038:
9034:
9030:
9027:
9024:
9019:
9014:
9009:
9004:
8980:
8975:
8950:
8945:
8941:
8937:
8932:
8928:
8924:
8919:
8915:
8911:
8906:
8902:
8898:
8893:
8889:
8885:
8880:
8876:
8872:
8850:
8846:
8842:
8836:
8832:
8828:
8823:
8819:
8815:
8810:
8806:
8802:
8797:
8793:
8789:
8784:
8780:
8776:
8771:
8767:
8762:
8758:
8754:
8751:
8746:
8742:
8738:
8733:
8729:
8715:is called the
8702:
8698:
8694:
8689:
8685:
8660:
8654:
8650:
8646:
8641:
8637:
8632:
8610:
8604:
8600:
8596:
8591:
8587:
8582:
8569:
8566:
8543:
8538:
8512:
8507:
8502:
8483:Non-Borel sets
8458:
8455:
8440:
8435:
8411:
8406:
8379:
8373:
8369:
8365:
8362:
8358:
8337:
8317:
8314:
8311:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8270:
8267:
8247:
8244:
8241:
8238:
8235:
8232:
8229:
8226:
8223:
8220:
8217:
8214:
8193:
8187:
8183:
8179:
8176:
8172:
8150:
8144:
8140:
8136:
8133:
8129:
8108:
8088:
8084:
8078:
8074:
8070:
8067:
8063:
8041:
8035:
8031:
8027:
8024:
8020:
7999:
7978:
7972:
7968:
7964:
7961:
7957:
7935:
7929:
7925:
7921:
7918:
7914:
7893:
7871:
7867:
7863:
7858:
7854:
7849:
7845:
7841:
7836:
7832:
7828:
7825:
7821:
7815:
7811:
7807:
7802:
7798:
7793:
7789:
7786:
7783:
7779:
7773:
7769:
7765:
7760:
7756:
7751:
7747:
7742:
7739:
7735:
7730:
7726:
7722:
7719:
7716:
7713:
7710:
7707:
7688:
7683:
7678:
7656:
7653:
7650:
7647:
7625:
7620:
7598:
7578:
7556:
7553:
7529:
7501:
7481:
7459:
7456:
7436:
7433:
7430:
7427:
7407:
7398:of subsets of
7387:
7363:
7343:
7323:
7301:
7297:
7293:
7290:
7287:
7283:
7279:
7276:
7273:
7268:
7265:
7261:
7256:
7252:
7249:
7246:
7243:
7240:
7220:
7217:
7197:
7177:
7174:
7171:
7166:
7163:
7159:
7138:
7135:
7132:
7129:
7126:
7106:
7103:
7081:
7061:
7058:
7038:
7018:
6998:
6978:
6958:
6946:
6943:
6926:
6922:
6918:
6915:
6910:
6906:
6902:
6897:
6893:
6888:
6884:
6880:
6859:
6855:
6852:
6847:
6843:
6839:
6834:
6830:
6825:
6821:
6812:Indeed, using
6801:
6798:
6795:
6792:
6789:
6786:
6783:
6780:
6777:
6757:
6754:
6751:
6748:
6745:
6742:
6722:
6719:
6716:
6713:
6710:
6707:
6704:
6684:
6681:
6678:
6675:
6656:
6653:
6629:
6626:
6623:
6620:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6551:
6548:
6545:
6525:
6522:
6519:
6499:
6496:
6493:
6490:
6487:
6484:
6481:
6478:
6475:
6472:
6449:
6429:
6409:
6406:
6403:
6400:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6330:
6307:
6304:
6293:and is called
6282:
6279:
6276:
6273:
6253:
6250:
6230:
6210:
6190:
6187:
6167:
6155:
6152:
6150:
6147:
6135:
6130:
6124:
6101:
6080:
6063:the so-called
6052:
6047:
6041:
6018:
5998:
5983:
5980:
5979:
5978:
5966:
5951:
5939:
5936:
5912:
5892:
5872:
5861:
5850:
5847:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5786:
5770:
5767:
5752:
5741:
5738:
5714:
5711:
5687:
5675:
5672:
5631:
5618:collection of
5603:
5579:
5559:
5534:
5529:
5526:
5523:
5520:
5500:
5497:
5492:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5466:
5463:
5460:
5427:
5405:
5383:
5359:
5345:
5342:
5340:
5337:
5324:
5321:
5316:
5310:
5307:
5304:
5282:
5277:
5271:
5268:
5265:
5244:
5241:
5238:
5235:
5232:
5209:
5206:
5169:
5166:
5139:
5123:
5120:
5119:
5118:
5107:
5104:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5034:
5031:
5028:
5025:
5022:
5011:
5000:
4997:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4929:
4926:
4923:
4920:
4917:
4897:
4877:
4865:
4862:
4850:
4846:
4840:
4836:
4828:
4823:
4820:
4816:
4811:
4807:
4804:
4801:
4796:
4791:
4788:
4767:
4761:
4757:
4749:
4744:
4741:
4737:
4732:
4728:
4725:
4720:
4694:
4689:
4684:
4681:
4678:
4674:
4668:
4664:
4656:
4651:
4648:
4644:
4639:
4635:
4615:
4610:
4605:
4600:
4596:
4588:
4583:
4580:
4576:
4555:
4550:
4545:
4540:
4536:
4515:
4512:
4509:
4506:
4483:
4479:
4475:
4472:
4469:
4463:
4458:
4453:
4448:
4444:
4440:
4433:
4429:
4424:
4420:
4415:
4411:
4407:
4402:
4398:
4392:
4387:
4384:
4381:
4377:
4372:
4368:
4363:
4341:
4337:
4331:
4327:
4319:
4314:
4311:
4307:
4302:
4298:
4295:
4290:
4286:
4278:
4273:
4270:
4266:
4231:
4227:
4206:
4201:
4197:
4174:
4170:
4147:
4143:
4139:
4134:
4130:
4109:
4087:
4083:
4057:
4052:
4048:
4040:
4035:
4032:
4028:
4000:
3997:
3975:
3969:
3964:
3961:
3958:
3953:
3949:
3944:
3930:
3927:
3915:
3912:
3889:
3864:
3843:
3839:
3818:
3815:
3791:
3788:
3785:
3782:
3762:
3758:
3737:
3717:
3714:
3711:
3708:
3705:
3699:
3695:
3691:
3688:
3685:
3682:
3679:
3676:
3672:
3647:
3625:
3622:
3619:
3616:
3613:
3593:
3590:
3587:
3584:
3564:
3544:
3524:
3521:
3501:
3478:
3475:
3472:
3469:
3449:
3446:
3426:
3406:
3395:
3394:
3378:
3369:that contains
3358:
3338:
3329:(or λ-system)
3323:
3311:
3291:
3262:
3259:
3236:
3233:
3230:
3227:
3224:
3221:
3173:
3170:
3150:
3130:
3110:
3107:
3104:
3101:
3098:
3095:
3069:
3066:
3063:
3060:
3057:
3037:
3017:
3014:
2994:
2991:
2988:
2985:
2982:
2958:
2955:
2952:
2949:
2946:
2926:
2923:
2899:
2879:
2856:
2853:
2833:
2808:
2807:
2796:
2793:
2790:
2785:
2781:
2777:
2772:
2768:
2764:
2759:
2755:
2751:
2748:
2728:
2725:
2705:
2702:
2697:
2693:
2689:
2684:
2680:
2676:
2671:
2667:
2642:
2632:
2621:
2618:
2615:
2612:
2588:
2585:
2565:
2556:: If some set
2541:
2531:
2519:
2499:
2472:
2469:
2466:
2463:
2460:
2457:
2444:represent its
2433:
2430:
2427:
2424:
2404:
2392:
2389:
2387:
2384:
2369:
2363:
2340:
2335:
2329:
2323:
2320:
2317:
2312:
2306:
2300:
2295:
2289:
2283:
2278:
2272:
2247:
2244:
2239:
2235:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2208:
2204:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2174:
2168:
2145:
2124:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2064:
2060:
2056:
2053:
2050:
2047:
2042:
2038:
2034:
2029:
2025:
2021:
2016:
2012:
2008:
2005:
2002:
1997:
1993:
1989:
1986:
1983:
1980:
1977:
1974:
1955:
1952:
1932:
1908:
1888:
1868:
1867:Sub σ-algebras
1865:
1853:
1848:
1844:
1838:
1835:
1832:
1828:
1827:lim sup
1824:
1819:
1815:
1809:
1806:
1803:
1799:
1798:lim inf
1795:
1790:
1786:
1780:
1777:
1774:
1770:
1747:
1743:
1737:
1734:
1731:
1727:
1706:
1701:
1697:
1691:
1688:
1685:
1681:
1680:lim sup
1674:
1666:
1662:
1656:
1653:
1650:
1646:
1645:lim inf
1634:
1633:
1622:
1619:
1616:
1611:
1608:
1605:
1601:
1597:
1592:
1588:
1584:
1581:
1561:
1558:
1538:
1535:
1530:
1527:
1524:
1520:
1516:
1511:
1507:
1485:
1481:
1478:
1456:
1452:
1446:
1443:
1440:
1436:
1435:lim inf
1432:
1429:
1419:
1405:
1402:
1399:
1394:
1391:
1388:
1384:
1380:
1375:
1371:
1365:
1360:
1357:
1354:
1350:
1346:
1341:
1337:
1331:
1326:
1323:
1320:
1316:
1310:
1305:
1302:
1299:
1295:
1291:
1286:
1282:
1276:
1273:
1270:
1266:
1265:lim inf
1245:
1236:of subsets of
1225:
1222:
1217:
1213:
1209:
1204:
1200:
1196:
1191:
1187:
1177:of a sequence
1176:
1172:
1165:
1154:
1151:
1148:
1141:
1137:
1132:
1128:
1121:
1117:
1112:
1108:
1105:
1085:
1082:
1062:
1059:
1054:
1050:
1046:
1041:
1037:
1016:
1013:
1006:
1002:
997:
993:
986:
982:
977:
951:
947:
941:
938:
935:
931:
930:lim sup
927:
924:
912:
897:
877:
874:
871:
866:
863:
860:
856:
852:
847:
843:
837:
832:
829:
826:
822:
818:
813:
809:
803:
798:
795:
792:
788:
782:
777:
774:
771:
767:
763:
758:
754:
748:
745:
742:
738:
737:lim sup
717:
708:of subsets of
697:
694:
689:
685:
681:
676:
672:
668:
663:
659:
649:of a sequence
648:
644:
642:limit supremum
624:
623:Limits of sets
621:
607:
604:
576:
573:
563:
541:
538:
529:to subsets of
510:
494:
491:
486:
483:
467:open intervals
447:
423:
420:
417:
414:
409:
405:
401:
396:
392:
388:
383:
379:
375:
348:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
227:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
101:
98:
95:
92:
89:
26:
9:
6:
4:
3:
2:
14252:
14241:
14238:
14236:
14233:
14231:
14228:
14226:
14223:
14222:
14220:
14210:
14209:Sigma Algebra
14207:
14203:
14199:
14198:
14193:
14189:
14188:
14177:
14171:
14163:
14161:0-387-95313-2
14157:
14154:. p. 7.
14153:
14149:
14145:
14139:
14131:
14127:
14122:
14117:
14113:
14109:
14105:
14098:
14091:
14089:
14086:
14070:
14050:
14030:
14007:
14004:
14001:
13978:
13975:
13955:
13945:
13944:
13936:
13932:
13925:
13917:
13911:
13907:
13900:
13892:
13890:0-07-054234-1
13886:
13882:
13878:
13874:
13873:Rudin, Walter
13868:
13860:
13854:
13850:
13846:
13840:
13833:
13821:
13817:
13811:
13807:
13795:
13781:
13775:
13724:
13721:
13716:
13712:
13708:
13703:
13699:
13695:
13692:
13689:
13686:
13666:
13651:
13635:
13621:
13619:
13616:
13600:
13585:
13569:
13563:
13555:
13548:
13546:
13543:
13538:
13534:
13526:
13510:
13500:
13474:
13465:
13461:intersections
13456:
13428:
13419:
13410:
13406:intersections
13401:
13398:
13392:
13388:
13387:closed under:
13344:
13327:
13326:
13285:
13274:
13233:
13228:
13227:
13193:
13182:
13134:
13132:Open Topology
13129:
13128:
13099:
13096:
13039:
13034:
13033:
13004:
13001:
12944:
12942:(Filter base)
12936:
12935:
12906:
12903:
12846:
12841:
12840:
12767:
12762:
12761:
12692:
12684:
12683:
12614:
12606:
12605:
12536:
12531:
12530:
12461:
12456:
12455:
12386:
12378:
12377:
12304:
12296:
12295:
12275:
12252:
12248:
12211:
12208:
12205:
12173:
12163:
12162:
12121:
12117:
12089:
12085:
12039:
12034:
12033:
11963:
11954:
11953:
11883:
11879:
11878:
11804:
11797:
11796:
11792:
11788:
11766:
11763:
11755:
11733:
11722:
11707:
11704:
11699:
11695:
11691:
11686:
11682:
11673:
11658:
11655:
11650:
11646:
11642:
11637:
11633:
11624:
11609:
11595:
11580:
11574:
11566:
11551:
11548:
11545:
11537:
11522:
11519:
11516:
11508:
11505:
11492:
11479:
11475:
11474:closed under:
11431:
11414:
11413:
11407:
11402:
11400:
11395:
11393:
11388:
11387:
11364:
11335:
11329:
11326:
11323:
11320:
11317:
11314:
11311:
11308:
11307:
11301:
11287:
11283:
11279:
11273:
11261:
11258:
11255:
11249:
11241:
11238:
11234:
11229:
11225:
11221:
11218:
11214:
11209:
11204:
11192:
11188:
11185:
11182:
11179:
11173:
11165:
11162:
11158:
11153:
11149:
11143:
11137:
11117:
11109:
11105:
11104:
11087:
11067:
11046:
11041:
11029:
11025:
10990:
10987:
10978:
10975:
10954:
10910:
10886:
10880:
10860:
10847:
10843:
10838:
10833:
10823:
10812:
10809:
10806:
10800:
10792:
10789:
10785:
10780:
10776:
10770:
10764:
10744:
10724:
10721:
10718:
10710:
10709:random vector
10694:
10691:
10688:
10680:
10679:
10662:
10640:
10610:
10594:
10591:
10582:
10558:
10552:
10532:
10530:
10520:
10505:
10498:
10495:
10492:
10489:
10486:
10483:
10462:
10457:
10453:
10449:
10445:
10439:
10435:
10431:
10428:
10425:
10420:
10416:
10411:
10405:
10401:
10393:
10389:
10386:
10381:
10356:
10352:
10348:
10345:
10342:
10339:
10336:
10333:
10328:
10324:
10320:
10315:
10311:
10307:
10304:
10301:
10297:
10293:
10290:
10285:
10282:
10279:
10275:
10271:
10266:
10262:
10258:
10255:
10252:
10247:
10243:
10239:
10234:
10230:
10225:
10220:
10216:
10213:
10210:
10206:
10190:
10185:
10181:
10177:
10174:
10171:
10166:
10162:
10157:
10153:
10149:
10143:
10139:
10135:
10132:
10129:
10124:
10120:
10115:
10109:
10105:
10084:
10053:
10040:
10037:
9998:
9982:
9979:
9971:
9951:
9947:
9943:
9940:
9937:
9932:
9928:
9917:
9914:
9911:
9908:
9900:
9895:
9891:
9886:
9875:
9872:
9869:
9865:
9861:
9856:
9826:
9819:
9815:
9811:
9808:
9805:
9800:
9796:
9768:
9764:
9761:
9758:
9755:
9752:
9749:
9728:
9723:
9719:
9715:
9711:
9705:
9701:
9697:
9694:
9691:
9686:
9682:
9677:
9669:
9665:
9661:
9658:
9655:
9650:
9646:
9641:
9636:
9626:
9613:
9609:
9605:
9602:
9599:
9596:
9593:
9590:
9585:
9581:
9577:
9569:
9565:
9558:
9555:
9552:
9549:
9546:
9542:
9538:
9530:
9526:
9522:
9519:
9516:
9511:
9507:
9496:
9492:
9488:
9485:
9482:
9477:
9473:
9468:
9447:
9439:
9423:
9374:
9353:
9350:
9344:
9336:
9330:
9324:
9321:
9318:
9312:
9295:
9292:
9278:
9275:
9262:
9258:
9254:
9245:
9240:
9236:
9232:
9227:
9223:
9219:
9215:
9209:
9205:
9201:
9196:
9192:
9187:
9183:
9180:
9177:
9173:
9167:
9163:
9159:
9154:
9150:
9145:
9140:
9136:
9132:
9129:
9125:
9121:
9112:
9107:
9103:
9099:
9091:
9087:
9083:
9077:
9071:
9068:
9065:
9057:
9053:
9049:
9043:
9036:
9032:
9028:
9025:
9017:
8978:
8962:
8943:
8935:
8930:
8926:
8922:
8917:
8909:
8904:
8900:
8896:
8891:
8887:
8883:
8878:
8874:
8863:Observe that
8861:
8848:
8844:
8840:
8834:
8826:
8821:
8817:
8813:
8808:
8800:
8795:
8791:
8787:
8782:
8778:
8774:
8769:
8765:
8760:
8756:
8752:
8749:
8744:
8736:
8731:
8718:
8700:
8696:
8692:
8687:
8683:
8675:
8674:product space
8658:
8652:
8644:
8639:
8635:
8630:
8608:
8602:
8594:
8589:
8585:
8580:
8565:
8563:
8559:
8541:
8526:
8510:
8505:
8491:
8486:
8484:
8480:
8476:
8472:
8468:
8464:
8463:Borel algebra
8454:
8409:
8394:
8377:
8371:
8363:
8360:
8356:
8335:
8315:
8309:
8303:
8300:
8294:
8288:
8268:
8265:
8239:
8233:
8227:
8224:
8218:
8212:
8191:
8185:
8177:
8174:
8170:
8148:
8142:
8134:
8131:
8127:
8106:
8086:
8082:
8076:
8068:
8065:
8061:
8039:
8033:
8025:
8022:
8018:
7997:
7976:
7970:
7962:
7959:
7955:
7933:
7927:
7919:
7916:
7912:
7891:
7882:
7869:
7865:
7861:
7852:
7847:
7843:
7839:
7834:
7830:
7826:
7819:
7813:
7809:
7805:
7800:
7796:
7791:
7787:
7784:
7781:
7777:
7771:
7767:
7763:
7758:
7754:
7749:
7740:
7737:
7733:
7728:
7724:
7720:
7717:
7711:
7705:
7686:
7681:
7651:
7645:
7623:
7596:
7576:
7567:
7554:
7551:
7543:
7527:
7519:
7515:
7499:
7479:
7470:
7457:
7431:
7425:
7405:
7377:
7361:
7341:
7321:
7312:
7299:
7295:
7291:
7288:
7285:
7281:
7274:
7266:
7263:
7259:
7254:
7250:
7244:
7238:
7218:
7215:
7195:
7172:
7164:
7161:
7157:
7136:
7130:
7124:
7104:
7101:
7094:
7079:
7059:
7056:
7036:
7016:
6996:
6976:
6956:
6942:
6939:
6924:
6920:
6916:
6913:
6908:
6904:
6900:
6895:
6891:
6886:
6882:
6878:
6857:
6853:
6850:
6845:
6841:
6837:
6832:
6828:
6823:
6819:
6799:
6787:
6775:
6749:
6740:
6720:
6711:
6702:
6679:
6673:
6654:
6651:
6643:
6627:
6618:
6615:
6612:
6609:
6606:
6600:
6594:
6591:
6588:
6582:
6576:
6570:
6561:
6552:
6543:
6520:
6497:
6491:
6488:
6485:
6482:
6479:
6473:
6470:
6461:
6447:
6427:
6404:
6398:
6378:
6372:
6369:
6360:
6348:
6328:
6319:
6318:
6305:
6302:
6277:
6271:
6251:
6248:
6228:
6221:(even though
6208:
6188:
6185:
6165:
6146:
6133:
6128:
6099:
6078:
6070:
6067:, which in a
6066:
6050:
6045:
6016:
6009:can define a
5996:
5989:
5988:stopping time
5964:
5956:
5952:
5937:
5934:
5926:
5910:
5890:
5870:
5862:
5848:
5845:
5822:
5819:
5816:
5810:
5807:
5804:
5801:
5787:
5784:
5768:
5765:
5757:
5753:
5739:
5736:
5728:
5712:
5709:
5701:
5700:
5699:
5685:
5671:
5669:
5665:
5660:
5656:
5652:
5648:
5643:
5629:
5621:
5617:
5601:
5593:
5557:
5550:
5527:
5524:
5521:
5518:
5495:
5485:
5479:
5476:
5470:
5467:
5464:
5458:
5451:
5447:
5443:
5403:
5395:
5381:
5371:
5357:
5336:
5322:
5308:
5305:
5269:
5266:
5236:
5233:
5222:
5218:
5214:
5205:
5203:
5199:
5195:
5191:
5187:
5183:
5167:
5164:
5156:
5154:
5129:
5105:
5102:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5026:
5023:
5012:
4998:
4995:
4969:
4966:
4963:
4960:
4957:
4954:
4943:
4942:
4941:
4921:
4918:
4895:
4875:
4861:
4848:
4844:
4838:
4821:
4818:
4814:
4809:
4805:
4802:
4786:
4765:
4759:
4742:
4739:
4735:
4730:
4726:
4723:
4708:
4679:
4676:
4672:
4666:
4649:
4646:
4642:
4637:
4633:
4626:This implies
4613:
4603:
4598:
4581:
4578:
4574:
4553:
4543:
4538:
4513:
4510:
4507:
4504:
4496:
4481:
4477:
4473:
4470:
4467:
4461:
4451:
4446:
4442:
4438:
4431:
4427:
4418:
4413:
4409:
4405:
4400:
4396:
4390:
4385:
4382:
4379:
4375:
4370:
4366:
4339:
4335:
4329:
4312:
4309:
4305:
4300:
4296:
4293:
4288:
4271:
4268:
4264:
4255:
4250:
4249:
4245:
4229:
4204:
4199:
4172:
4145:
4137:
4132:
4107:
4085:
4072:
4068:
4055:
4050:
4033:
4030:
4026:
4016:
4015:
4011:
3998:
3995:
3973:
3962:
3959:
3956:
3951:
3942:
3926:
3913:
3903:
3879:
3841:
3816:
3813:
3805:
3786:
3780:
3760:
3735:
3712:
3709:
3703:
3697:
3693:
3689:
3683:
3680:
3677:
3661:
3645:
3636:
3623:
3617:
3611:
3588:
3582:
3562:
3542:
3522:
3519:
3499:
3491:
3473:
3467:
3447:
3444:
3424:
3404:
3392:
3376:
3356:
3336:
3328:
3327:Dynkin system
3324:
3309:
3289:
3282:
3278:
3277:
3276:
3274:
3268:
3258:
3256:
3255:Dynkin system
3252:
3247:
3234:
3225:
3222:
3211:
3207:
3203:
3199:
3195:
3191:
3187:
3171:
3168:
3141:is a set and
3128:
3108:
3099:
3096:
3085:
3080:
3067:
3061:
3055:
3035:
3015:
3012:
2986:
2983:
2972:
2950:
2947:
2924:
2913:
2877:
2870:
2854:
2831:
2824:
2819:
2817:
2814:(by applying
2813:
2812:intersections
2794:
2791:
2788:
2783:
2779:
2775:
2770:
2766:
2762:
2757:
2753:
2749:
2746:
2726:
2703:
2700:
2695:
2691:
2687:
2682:
2678:
2674:
2669:
2665:
2656:
2633:
2619:
2616:
2610:
2602:
2586:
2563:
2555:
2532:
2497:
2490:
2489:
2488:
2486:
2467:
2461:
2458:
2447:
2428:
2422:
2402:
2383:
2338:
2321:
2318:
2315:
2310:
2298:
2293:
2281:
2276:
2258:
2245:
2237:
2229:
2226:
2223:
2217:
2214:
2211:
2198:
2195:
2192:
2186:
2183:
2177:
2172:
2143:
2122:
2113:
2100:
2094:
2091:
2088:
2085:
2079:
2076:
2073:
2067:
2062:
2058:
2054:
2048:
2045:
2040:
2036:
2032:
2027:
2023:
2019:
2014:
2010:
2000:
1987:
1984:
1981:
1975:
1953:
1950:
1930:
1922:
1906:
1886:
1877:
1874:
1864:
1851:
1846:
1842:
1830:
1822:
1817:
1813:
1801:
1793:
1788:
1784:
1772:
1745:
1741:
1729:
1704:
1699:
1695:
1683:
1672:
1664:
1660:
1648:
1620:
1617:
1614:
1609:
1606:
1603:
1599:
1595:
1590:
1586:
1582:
1579:
1559:
1556:
1536:
1533:
1528:
1525:
1522:
1518:
1514:
1509:
1505:
1479:
1476:
1454:
1450:
1438:
1430:
1427:
1417:
1403:
1400:
1397:
1392:
1389:
1386:
1382:
1378:
1373:
1369:
1358:
1355:
1352:
1348:
1344:
1339:
1335:
1324:
1321:
1318:
1314:
1303:
1300:
1297:
1293:
1289:
1284:
1280:
1268:
1243:
1223:
1220:
1215:
1211:
1207:
1202:
1198:
1194:
1189:
1185:
1174:
1171:
1170:limit infimum
1168:
1166:
1152:
1149:
1146:
1139:
1135:
1130:
1126:
1119:
1115:
1110:
1106:
1103:
1083:
1080:
1060:
1057:
1052:
1048:
1044:
1039:
1035:
1014:
1011:
1004:
1000:
995:
991:
984:
980:
975:
967:
949:
945:
933:
925:
922:
914:
910:
895:
875:
872:
869:
864:
861:
858:
854:
850:
845:
841:
830:
827:
824:
820:
816:
811:
807:
796:
793:
790:
786:
775:
772:
769:
765:
761:
756:
752:
740:
715:
695:
692:
687:
683:
679:
674:
670:
666:
661:
657:
646:
643:
640:
638:
637:
636:
634:
630:
620:
605:
602:
594:
590:
574:
571:
561:
558:
556:
555:disjoint sets
539:
536:
528:
524:
508:
500:
490:
482:
480:
476:
472:
468:
464:
459:
445:
437:
421:
415:
412:
407:
403:
399:
394:
390:
386:
381:
377:
364:
362:
346:
326:
317:
314:
311:
308:
305:
302:
299:
293:
287:
284:
281:
275:
269:
266:
263:
257:
248:
225:
202:
199:
196:
193:
190:
187:
184:
178:
175:
166:
164:
160:
156:
151:
149:
145:
141:
137:
133:
128:
126:
122:
117:
115:
93:
90:
79:
78:intersections
75:
71:
67:
64:
60:
56:
52:
48:
44:
40:
33:
19:
18:Sigma algebra
14195:
14170:
14147:
14138:
14111:
14107:
14097:
14084:
13947:
13941:
13924:
13905:
13899:
13876:
13867:
13848:
13839:
13831:
13824:. Retrieved
13819:
13810:
13615:
13542:
13540:
13531:Intersection
12685:
11316:Sample space
11107:
11101:
11100:is called a
10866:
10708:
10676:
10675:is called a
10538:
10526:
10054:
9969:
9627:
9375:
9284:
9276:
8963:
8862:
8716:
8571:
8487:
8460:
7883:
7568:
7471:
7313:
7188:of the sets
7092:
6948:
6940:
6462:
6320:
6294:
6157:
5985:
5782:
5726:
5698:be any set.
5677:
5668:quotient set
5647:pseudometric
5644:
5446:metric space
5373:
5349:
5347:
5217:calligraphic
5212:
5211:
5201:
5193:
5189:
5185:
5181:
5152:
5127:
5125:
4867:
4497:By the case
4494:
4251:
4247:
4246:
4120:is in every
4070:
4069:
4017:
4013:
4012:
3932:
3902:sample space
3637:
3396:
3390:
3270:
3248:
3185:
3184:is called a
3081:
2970:
2911:
2868:
2820:
2809:
2654:
2553:
2484:
2483:is called a
2394:
2259:
2114:
1921:sample space
1899:) or Tails (
1878:
1870:
1635:
1549:all contain
626:
559:
496:
488:
473:through all
460:
365:
167:
152:
129:
124:
118:
112:is called a
72:, countable
62:
54:
50:
46:
36:
13881:McGraw-Hill
13652:of sets in
13618:semialgebra
13586:of sets in
13431:complements
13424:complements
11961:(Semifield)
11958:Semialgebra
8558:integration
8475:closed sets
7334:from a set
7314:A function
7117:denoted by
6768:instead of
5781:called the
4217:Therefore,
3829:defined on
3267:π-λ theorem
3200:, with the
2739:then so is
1175:inner limit
966:subsequence
647:outer limit
593:Vitali sets
527:real number
121:set algebra
53:) on a set
14219:Categories
13802:References
12765:Dual ideal
12690:(𝜎-Field)
12687:𝜎-Algebra
11328:Sigma-ring
10369:for which
8479:Vitali set
8205:such that
7542:Borel sets
7376:measurable
6391:Otherwise
5925:singletons
5590:being the
5570:(and with
5440:that is a
5374:separable
5350:separable
5200:but not a
5150:is just a
3265:See also:
2601:complement
2391:Definition
1497:such that
1418:eventually
631:, involve
564:subset of
485:Motivation
155:statistics
70:complement
14202:EMS Press
14121:1112.1603
14085:separable
14071:μ
14051:μ
14031:μ
14008:μ
13956:μ
13779:∅
13776:≠
13725:…
13633:∖
13630:Ω
13567:∖
13511:∅
13503:contains
13485:Ω
13477:contains
13468:countable
13459:countable
13441:Ω
13345::
13286:∩
13194:∪
13097:∅
13002:∅
12939:Prefilter
12904:∅
12272:they are
12258:↗
12209:⊆
12166:𝜆-system
12127:↗
12095:↘
11767:∈
11764:∅
11734:∈
11731:Ω
11708:⋯
11705:∪
11692:∪
11659:⋯
11656:∩
11643:∩
11607:∖
11604:Ω
11578:∖
11549:∪
11520:∩
11493:⊇
11432::
11373:Ω
11339:Families
11262:∈
11239:−
11222:σ
11189:σ
11186:∈
11163:−
11138:σ
11026:σ
10991:⊆
10985:→
10982:Ω
10884:Σ
10878:Ω
10813:∈
10790:−
10765:σ
10601:→
10598:Ω
10556:Σ
10550:Ω
10496:≤
10490:≤
10463:∈
10429:…
10390:σ
10378:Σ
10346:≤
10340:≤
10321:∈
10302:∈
10294:…
10256:…
10211:∩
10201:∞
10191:×
10178:×
10175:⋯
10172:×
10133:…
9941:…
9924:Σ
9915:≤
9901:∈
9887:⋃
9881:∞
9866:⋃
9809:…
9792:Σ
9762:≤
9756:≤
9729:∈
9695:…
9659:…
9603:≤
9597:≤
9578:∈
9550:∈
9520:…
9486:…
9354:∈
9337:∈
9296:⊆
9246:∈
9184:×
9181:⋯
9178:×
9133:σ
9113:∈
9081:∞
9078:−
9072:×
9069:⋯
9066:×
9047:∞
9044:−
9029:σ
8940:Σ
8936:∈
8914:Σ
8910:∈
8884:×
8831:Σ
8827:∈
8805:Σ
8801:∈
8775:×
8753:σ
8741:Σ
8737:×
8728:Σ
8693:×
8649:Σ
8599:Σ
8471:open sets
8465:over any
8439:∞
8368:Σ
8304:σ
8301:⊆
8289:σ
8182:Σ
8139:Σ
8073:Σ
8030:Σ
7967:Σ
7924:Σ
7853:∈
7788:×
7785:⋯
7782:×
7738:−
7721:σ
7706:σ
7646:σ
7455:Σ
7426:σ
7386:Σ
7354:to a set
7289:∈
7264:−
7239:σ
7231:That is,
7162:−
7125:σ
7080:σ
7072:then the
7037:σ
6989:to a set
6917:…
6879:σ
6854:…
6820:σ
6776:σ
6741:σ
6703:σ
6674:σ
6568:∅
6544:σ
6399:σ
6367:∅
6355:∅
6349:σ
6272:σ
6129:τ
6100:τ
6079:τ
6046:τ
6029:-algebra
6017:σ
5997:τ
5955:partition
5814:∖
5799:∅
5756:power set
5651:separable
5630:σ
5616:countable
5602:σ
5578:△
5558:μ
5528:∈
5491:△
5480:μ
5459:ρ
5416:-algebra
5404:σ
5382:σ
5358:σ
5240:Σ
5138:Σ
5130:-algebra
5080:Λ
5077:∈
5071:∩
5059:⊆
5030:Λ
4973:Σ
4970:∈
4958:∩
4925:Σ
4839:α
4835:Σ
4822:∈
4819:α
4815:⋃
4806:σ
4803:⊆
4787:σ
4760:α
4756:Σ
4743:∈
4740:α
4736:⋃
4727:σ
4724:⊆
4707:generated
4680:σ
4677:⊆
4667:α
4663:Σ
4650:∈
4647:α
4643:⋃
4634:σ
4604:⊆
4599:α
4595:Σ
4582:∈
4579:α
4575:⋃
4544:⊂
4539:α
4535:Σ
4471:≥
4452:∈
4443:α
4428:α
4423:Σ
4419:∈
4376:⋂
4330:α
4326:Σ
4313:∈
4310:α
4306:⋃
4297:σ
4289:α
4285:Σ
4272:∈
4269:α
4265:⋁
4254:generates
4230:∗
4226:Σ
4200:∗
4196:Σ
4173:α
4169:Σ
4146:∗
4142:Σ
4133:α
4129:Σ
4086:∗
4082:Σ
4051:α
4047:Σ
4034:∈
4031:α
4027:⋀
3963:∈
3960:α
3952:α
3948:Σ
3911:Ω
3888:Σ
3694:∫
3681:∈
3658:with the
3612:σ
3583:σ
3490:generated
3468:σ
3229:∞
3206:morphisms
3149:Σ
3103:Σ
2990:∅
2954:∅
2922:Σ
2898:Σ
2867:since by
2852:Σ
2832:∅
2823:empty set
2792:⋯
2789:∪
2776:∪
2763:∪
2724:Σ
2704:…
2641:Σ
2614:∖
2584:Σ
2540:Σ
2518:Σ
2485:σ-algebra
2459:⊆
2456:Σ
2446:power set
2368:∞
2334:∞
2322:⊆
2319:⋯
2316:⊆
2299:⊆
2282:⊆
2218:⊆
2207:∞
2187:×
2092:≥
2068:∈
2049:…
1996:∞
1973:Ω
1837:∞
1834:→
1808:∞
1805:→
1779:∞
1776:→
1736:∞
1733:→
1690:∞
1687:→
1673:⊆
1655:∞
1652:→
1618:⋯
1615:∩
1596:∩
1583:∈
1537:…
1480:∈
1445:∞
1442:→
1431:∈
1401:⋯
1398:∩
1379:∩
1364:∞
1349:⋃
1330:∞
1315:⋂
1309:∞
1294:⋃
1275:∞
1272:→
1224:…
1150:⋯
1147:∩
1127:∩
1107:∈
1061:⋯
1015:…
940:∞
937:→
926:∈
911:cofinally
873:⋯
870:∪
851:∪
836:∞
821:⋂
802:∞
787:⋃
781:∞
766:⋂
747:∞
744:→
696:…
463:real line
436:partition
416:…
361:empty set
347:∅
255:∅
246:Σ
142:, and in
97:Σ
47:σ-algebra
14152:Springer
14146:(2001).
13933:(1995).
13875:(1987).
13847:(2012).
13826:30 March
13545:semiring
13533:Property
13422:relative
13397:downward
13395:directed
13100:∉
13005:∉
12907:∉
12609:Algebra
12274:disjoint
12240:only if
12109:only if
12077:only if
11882:Semiring
11482:Directed
11304:See also
10867:Suppose
10539:Suppose
9285:Suppose
8258:for all
6871:to mean
5370:-algebra
5013:Suppose
4908:and let
4868:Suppose
3933:Suppose
3728:for all
3393:subsets.
3391:disjoint
3281:π-system
3251:π-system
3210:Measures
3198:category
3194:preimage
523:function
132:measures
125:finitely
14204:, 2001
13946:: 262.
13554:-system
13362:or, is
12611:(Field)
12534:𝜎-Ring
12197:only if
11803:-system
11449:or, is
11363:of sets
8488:On the
5549:measure
5396:) is a
5223:. Thus
3802:is the
3192:if the
2716:are in
1027:(where
499:measure
493:Measure
359:is the
59:subsets
51:σ-field
41:and in
14158:
13912:
13887:
13855:
13529:Finite
13470:unions
13415:unions
13413:finite
13404:finite
13323:Never
13224:Never
12844:Filter
12758:Never
12680:Never
12602:Never
12527:Never
12459:δ-Ring
12452:Never
12292:Never
12030:Never
11950:Never
11791:F.I.P.
9348:
7514:metric
6640:By an
5450:metric
5394:-field
4465:
3854:while
3773:where
3253:and a
3121:where
2890:is in
2844:is in
2576:is in
2510:is in
2351:where
1676:
1670:
339:where
74:unions
68:under
66:closed
14116:arXiv
13938:(PDF)
13549:is a
13066:Never
13063:Never
12971:Never
12968:Never
12873:Never
12870:Never
12381:Ring
12299:Ring
11365:over
11080:then
10907:is a
10707:) or
10655:then
10583:. If
10579:is a
9628:Each
8391:is a
8281:then
8119:from
7638:then
7512:is a
7029:is a
5729:over
5653:as a
5448:with
5155:-ring
3876:is a
2657:: If
562:every
521:is a
14156:ISBN
13910:ISBN
13885:ISBN
13853:ISBN
13828:2016
10911:and
10722:>
9972:for
8622:and
8572:Let
7990:and
7520:and
7009:and
6536:is
6158:Let
5754:The
5678:Let
5620:sets
5511:for
5372:(or
5294:or
5198:ring
4248:Join
4073:Let
4014:Meet
3806:for
2910:and
2395:Let
1167:The
1058:<
1045:<
913:many
639:The
45:, a
14126:doi
14083:is
13948:If
13433:in
11484:by
11130:is
11106:or
10967:If
10757:is
9999:of
9440:of
8481:or
8328:If
8162:to
8053:to
7947:to
7609:to
7569:If
7544:on
7516:or
7374:is
7208:in
6949:If
6321:If
6112:is
5957:of
5927:of
5758:of
4566:so
3492:by
3204:as
3048:is
2971:(3)
2912:(2)
2869:(1)
2818:).
2653:is
2552:is
1943:or
1769:lim
1726:lim
1256:is
1173:or
728:is
645:or
501:on
481:).
438:of
366:If
238:is
168:If
153:In
61:of
37:In
14221::
14200:,
14194:,
14124:.
14112:83
14110:.
14106:.
13940:.
13883:.
13879:.
13830:.
13818:.
13614:A
12270:or
9436:A
8564:.
8485:.
5986:A
5348:A
5180:A
5126:A
3325:A
3279:A
3208:.
2603:,
1794::=
557:.
497:A
150:.
116:.
14164:.
14132:.
14128::
14118::
14011:)
14005:,
14002:X
13999:(
13979:,
13976:X
13918:.
13893:.
13861:.
13782:.
13771:F
13747:F
13722:,
13717:2
13713:A
13709:,
13704:1
13700:A
13696:,
13693:B
13690:,
13687:A
13667:.
13662:F
13636:A
13601:.
13596:F
13570:A
13564:B
13552:π
13372:F
13340:F
13298:)
13206:)
13105:F
13010:F
12912:F
12253:i
12249:A
12212:B
12206:A
12122:i
12118:A
12090:i
12086:A
11801:π
11772:F
11739:F
11700:2
11696:A
11687:1
11683:A
11651:2
11647:A
11638:1
11634:A
11610:A
11581:A
11575:B
11552:B
11546:A
11523:B
11517:A
11459:F
11427:F
11405:e
11398:t
11391:v
11349:F
11288:,
11284:)
11280:}
11274:X
11268:F
11259:A
11256::
11253:)
11250:A
11247:(
11242:1
11235:Y
11230:{
11226:(
11219:=
11215:}
11210:)
11205:X
11199:F
11193:(
11183:A
11180::
11177:)
11174:A
11171:(
11166:1
11159:Y
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11150:=
11147:)
11144:Y
11141:(
11118:Y
11088:Y
11068:X
11047:)
11042:X
11036:F
11030:(
11002:T
10996:R
10988:X
10979::
10976:Y
10955:.
10951:T
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10895:)
10891:P
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10881:,
10875:(
10848:.
10844:}
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10834:n
10829:R
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10818:B
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10804:)
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10226:(
10221:{
10217:=
10214:X
10207:)
10196:R
10186:n
10182:B
10167:1
10163:B
10158:(
10154:=
10150:)
10144:n
10140:B
10136:,
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10125:1
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10116:(
10110:n
10106:C
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10041:.
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9870:n
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