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σ-algebra

From Wikipedia, the free encyclopedia
Algebraic structure of set algebra
For an algebraic structure admitting a given signature Σ of operations, seeUniversal algebra.

Inmathematical analysis and inprobability theory, aσ-algebra ("sigma algebra") is part of the formalism for definingsets that can be measured. Incalculus andanalysis, for example, σ-algebras are used to define the concept of sets witharea orvolume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion ofsize.

In formal terms, a σ-algebra (alsoσ-field, where the σ comes from theGerman "Summe",[1] meaning "sum") on a setX is a nonempty collection Σ ofsubsets ofXclosed undercomplement, countableunions, and countableintersections. The ordered pair(X,Σ){\displaystyle (X,\Sigma )} is called ameasurable space.

The setX is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die {1,2,3,4,5,6}), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to capture our intuitive ideas about how sizes combine: if there is a well-defined probability that an event occurs, there should be a well-defined probability that it does not occur (closure under complements); if several sets have a well-defined size, so should their combination (countable unions); if several events have a well-defined probability of occurring, so should the event where they all occur simultaneously (countable intersections).

The definition of σ-algebra resembles other mathematical structures such as atopology (which is required to be closed under all unions but only finite intersections, and which doesn't necessarily contain all complements of its sets) or aset algebra (which is closed only underfinite unions and intersections).

Examples of σ-algebras

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IfX={a,b,c,d}{\displaystyle X=\{a,b,c,d\}} one possible σ-algebra onX{\displaystyle X} isΣ={∅,{a,b},{c,d},{a,b,c,d}},{\displaystyle \Sigma =\{\varnothing ,\{a,b\},\{c,d\},\{a,b,c,d\}\},} where∅{\displaystyle \varnothing } is theempty set. In general, a finite algebra is always a σ-algebra.

If{A1,A2,A3,…},{\displaystyle \{A_{1},A_{2},A_{3},\ldots \},} is a countablepartition ofX{\displaystyle X} then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.

A more useful example is the set of subsets of thereal line formed by starting with allopen intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (bytransfinite iteration through allcountable ordinals) until the relevant closure properties are achieved (a construction known as theBorel hierarchy).

Motivation

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There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.

Measure

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Ameasure onX{\displaystyle X} is afunction that assigns a non-negativereal number to subsets ofX;{\displaystyle X;} 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 ofdisjoint sets.

One would like to assign a size toevery subset ofX,{\displaystyle X,} but in many natural settings, this is not possible. For example, theaxiom of choice 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, theVitali sets. For this reason, one considers instead a smaller collection of privileged subsets ofX.{\displaystyle X.} 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 properties are called σ-algebras.

Limits of sets

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Many uses of measure, such as the probability concept ofalmost sure convergence, involvelimits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.

The inner limit is always a subset of the outer limit:lim infn→∞An ⊆ lim supn→∞An.{\displaystyle \liminf _{n\to \infty }A_{n}~\subseteq ~\limsup _{n\to \infty }A_{n}.} If these two sets are equal then their limitlimn→∞An{\displaystyle \lim _{n\to \infty }A_{n}} exists and is equal to this common set:limn→∞An:=lim infn→∞An=lim supn→∞An.{\displaystyle \lim _{n\to \infty }A_{n}:=\liminf _{n\to \infty }A_{n}=\limsup _{n\to \infty }A_{n}.}

Sub σ-algebras

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In much of probability, especially whenconditional expectation 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 determined only by the partial information. Formally, ifΣ,Σ′{\displaystyle \Sigma ,\Sigma '} are σ-algebras onX{\displaystyle X}, thenΣ′{\displaystyle \Sigma '} is a sub σ-algebra ofΣ{\displaystyle \Sigma } ifΣ′⊆Σ{\displaystyle \Sigma '\subseteq \Sigma }.

TheBernoulli process provides a simple example. This consists of a sequence of random coin flips, coming up Heads (H{\displaystyle H}) or Tails (T{\displaystyle T}), of unbounded length. Thesample space Ω consists of all possible infinite sequences ofH{\displaystyle H} orT:{\displaystyle T:}Ω={H,T}∞={(x1,x2,x3,…):xi∈{H,T},i≥1}.{\displaystyle \Omega =\{H,T\}^{\infty }=\{(x_{1},x_{2},x_{3},\dots ):x_{i}\in \{H,T\},i\geq 1\}.}

The full sigma algebra can be generated from an ascending sequence of subalgebras, by considering the information that might be obtained after observing some or all of the firstn{\displaystyle n} coin flips. This sequence of subalgebras is given byGn={A×{Ω}:A⊆{H,T}n}{\displaystyle {\mathcal {G}}_{n}=\{A\times \{\Omega \}:A\subseteq \{H,T\}^{n}\}}Each of these is finer than the last, and so can be ordered as afiltration

G0⊆G1⊆G2⊆⋯⊆G∞{\displaystyle {\mathcal {G}}_{0}\subseteq {\mathcal {G}}_{1}\subseteq {\mathcal {G}}_{2}\subseteq \cdots \subseteq {\mathcal {G}}_{\infty }}

The first subalgebraG0={∅,Ω}{\displaystyle {\mathcal {G}}_{0}=\{\varnothing ,\Omega \}} is the trivial algebra: it has only two elements in it, the empty set and the total space. The second subalgebraG1{\displaystyle {\mathcal {G}}_{1}} has four elements: the two inG0{\displaystyle {\mathcal {G}}_{0}} plus two more: sequences that start withH{\displaystyle H} and sequences that start withT{\displaystyle T}. Each subalgebra is finer than the last. Then{\displaystyle n}'th subalgebra contains2n+1{\displaystyle 2^{n+1}} elements: it divides the total spaceΩ{\displaystyle \Omega } into all of the possible sequences that might have been observed aftern{\displaystyle n} flips, including the possible non-observation of some of the flips.

The limiting algebraG∞{\displaystyle {\mathcal {G}}_{\infty }} is the smallest σ-algebra containing all the others. It is the algebra generated by theproduct topology orweak topology on the product space{H,T}∞.{\displaystyle \{H,T\}^{\infty }.}

Definition and properties

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Definition

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LetX{\displaystyle X} be some set, and letP(X){\displaystyle P(X)} represent itspower set, the set of all subsets ofX{\displaystyle X}. Then a subsetΣ⊆P(X){\displaystyle \Sigma \subseteq P(X)} is called aσ-algebra if and only if it satisfies the following three properties:[2]

  1. X{\displaystyle X} is inΣ{\displaystyle \Sigma }.
  2. Σ{\displaystyle \Sigma } isclosed under complementation: If some setA{\displaystyle A} is inΣ,{\displaystyle \Sigma ,} then so is itscomplement,X∖A.{\displaystyle X\setminus A.}
  3. Σ{\displaystyle \Sigma } isclosed under countable unions: IfA1,A2,A3,…{\displaystyle A_{1},A_{2},A_{3},\ldots } are inΣ,{\displaystyle \Sigma ,} then so isA=A1∪A2∪A3∪⋯.{\displaystyle A=A_{1}\cup A_{2}\cup A_{3}\cup \cdots .}

From these properties, it follows that the σ-algebra is also closed under countableintersections (by applyingDe Morgan's laws).

It also follows that theempty set∅{\displaystyle \varnothing } is inΣ,{\displaystyle \Sigma ,} since by(1)X{\displaystyle X} is inΣ{\displaystyle \Sigma } and(2) asserts that its complement, the empty set, is also inΣ.{\displaystyle \Sigma .} Moreover, since{X,∅}{\displaystyle \{X,\varnothing \}} satisfies all 3 conditions, it follows that{X,∅}{\displaystyle \{X,\varnothing \}} is the smallest possible σ-algebra onX.{\displaystyle X.} The largest possible σ-algebra onX{\displaystyle X} isP(X).{\displaystyle P(X).}

Elements of the σ-algebra are calledmeasurable sets. An ordered pair(X,Σ),{\displaystyle (X,\Sigma ),} whereX{\displaystyle X} is a set andΣ{\displaystyle \Sigma } is a σ-algebra overX,{\displaystyle X,} is called ameasurable space. A function between two measurable spaces is called ameasurable function if thepreimage of every measurable set is measurable. The collection of measurable spaces forms acategory, with themeasurable functions asmorphisms.Measures are defined as certain types of functions from a σ-algebra to[0,∞].{\displaystyle [0,\infty ].}


A σ-algebra is both aπ-system and aDynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below).

Dynkin's π-λ theorem

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See also:π-λ theorem

This theorem (or the relatedmonotone class theorem) 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.

Dynkin's π-λ theorem says, ifP{\displaystyle P} is a π-system andD{\displaystyle D} is a Dynkin system that containsP,{\displaystyle P,} then the σ-algebraσ(P){\displaystyle \sigma (P)}generated byP{\displaystyle P} is contained inD.{\displaystyle D.} Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets inP{\displaystyle P} enjoy the property under consideration while, on the other hand, showing that the collectionD{\displaystyle D} of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets inσ(P){\displaystyle \sigma (P)} enjoy the property, avoiding the task of checking it for an arbitrary set inσ(P).{\displaystyle \sigma (P).}

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 variableX{\displaystyle X} with theLebesgue-Stieltjes integral typically associated with computing the probability:P(X∈A)=∫AF(dx){\displaystyle \mathbb {P} (X\in A)=\int _{A}\,F(dx)} for allA{\displaystyle A} in the Borel σ-algebra onR,{\displaystyle \mathbb {R} ,}whereF(x){\displaystyle F(x)} is thecumulative distribution function forX,{\displaystyle X,} defined onR,{\displaystyle \mathbb {R} ,} whileP{\displaystyle \mathbb {P} } is aprobability measure, defined on a σ-algebraΣ{\displaystyle \Sigma } of subsets of somesample spaceΩ.{\displaystyle \Omega .}

Combining σ-algebras

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Suppose{Σα:α∈A}{\displaystyle \textstyle \left\{\Sigma _{\alpha }:\alpha \in {\mathcal {A}}\right\}} is a collection of σ-algebras on a spaceX.{\displaystyle X.}

Meet

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:⋀α∈AΣα.{\displaystyle \bigwedge _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }.}

Sketch of Proof: LetΣ∗{\displaystyle \Sigma ^{*}} denote the intersection. SinceX{\displaystyle X} is in everyΣα,Σ∗{\displaystyle \Sigma _{\alpha },\Sigma ^{*}} is not empty. Closure under complement and countable unions for everyΣα{\displaystyle \Sigma _{\alpha }} implies the same must be true forΣ∗.{\displaystyle \Sigma ^{*}.} Therefore,Σ∗{\displaystyle \Sigma ^{*}} is a σ-algebra.

Join

The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but itgenerates a σ-algebra known as the join which typically is denoted⋁α∈AΣα=σ(⋃α∈AΣα).{\displaystyle \bigvee _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }=\sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}A π-system that generates the join isP={⋂i=1nAi:Ai∈Σαi,αi∈A, n≥1}.{\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\}.}Sketch of Proof: By the casen=1,{\displaystyle n=1,} it is seen that eachΣα⊂P,{\displaystyle \Sigma _{\alpha }\subset {\mathcal {P}},} so⋃α∈AΣα⊆P.{\displaystyle \bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\subseteq {\mathcal {P}}.}This impliesσ(⋃α∈AΣα)⊆σ(P){\displaystyle \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)\subseteq \sigma ({\mathcal {P}})}by the definition of a σ-algebragenerated by a collection of subsets. On the other hand,P⊆σ(⋃α∈AΣα){\displaystyle {\mathcal {P}}\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)}which, by Dynkin's π-λ theorem, impliesσ(P)⊆σ(⋃α∈AΣα).{\displaystyle \sigma ({\mathcal {P}})\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}

σ-algebras for subspaces

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SupposeY{\displaystyle Y} is a subset ofX{\displaystyle X} and let(X,Σ){\displaystyle (X,\Sigma )} be a measurable space.

Relation to σ-ring

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Aσ-algebraΣ{\displaystyle \Sigma } is just aσ-ring that contains the universal setX.{\displaystyle X.}[3] Aσ-ring need not be aσ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are aσ-ring, but not aσ-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 aring but not aσ-ring, since the real line can be obtained by their countable union yet its measure is not finite.

Typographic note

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σ-algebras are sometimes denoted usingcalligraphic capital letters, or theFraktur typeface. Thus(X,Σ){\displaystyle (X,\Sigma )} may be denoted as(X,F){\displaystyle \scriptstyle (X,\,{\mathcal {F}})} or(X,F).{\displaystyle \scriptstyle (X,\,{\mathfrak {F}}).}

Particular cases and examples

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Separable σ-algebras

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Aseparableσ{\displaystyle \sigma }-algebra (orseparableσ{\displaystyle \sigma }-field) is aσ{\displaystyle \sigma }-algebraF{\displaystyle {\mathcal {F}}} that is aseparable space when considered as ametric space withmetricρ(A,B)=μ(A△B){\displaystyle \rho (A,B)=\mu (A{\mathbin {\triangle }}B)} forA,B∈F{\displaystyle A,B\in {\mathcal {F}}} and a given finitemeasureμ{\displaystyle \mu } (and with△{\displaystyle \triangle } being thesymmetric difference operator).[4] Anyσ{\displaystyle \sigma }-algebra generated by acountable collection ofsets is separable, but the converse need not hold. For example, the Lebesgueσ{\displaystyle \sigma }-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).

A separable measure space has a naturalpseudometric that renders itseparable as apseudometric space. The distance between two sets is defined as the measure of thesymmetric difference 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 singleequivalence class, the resultingquotient set 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.

Simple set-based examples

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LetX{\displaystyle X} be any set.

Stopping time sigma-algebras

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Astopping timeτ{\displaystyle \tau } can define aσ{\displaystyle \sigma }-algebraFτ,{\displaystyle {\mathcal {F}}_{\tau },} theso-calledstopping time sigma-algebra, which in afiltered probability space describes the information up to the random timeτ{\displaystyle \tau } 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 timeτ{\displaystyle \tau } isFτ.{\displaystyle {\mathcal {F}}_{\tau }.}[5]

σ-algebras generated by families of sets

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σ-algebra generated by an arbitrary family

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LetF{\displaystyle F} be an arbitrary family of subsets ofX.{\displaystyle X.} Then there exists a unique smallest σ-algebra which contains every set inF{\displaystyle F} (even thoughF{\displaystyle F} may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF.{\displaystyle F.} (See intersections of σ-algebras above.) This σ-algebra is denotedσ(F){\displaystyle \sigma (F)} and is calledthe σ-algebra generated byF.{\displaystyle F.}

IfF{\displaystyle F} is empty, thenσ(∅)={∅,X}.{\displaystyle \sigma (\varnothing )=\{\varnothing ,X\}.} Otherwiseσ(F){\displaystyle \sigma (F)} consists of all the subsets ofX{\displaystyle X} that can be made from elements ofF{\displaystyle F} by a countable number of complement, union and intersection operations.

For a simple example, consider the setX={1,2,3}.{\displaystyle X=\{1,2,3\}.} Then the σ-algebra generated by the single subset{1}{\displaystyle \{1\}} isσ({1})={∅,{1},{2,3},{1,2,3}}.{\displaystyle \sigma (\{1\})=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\}\}.} By anabuse of notation, when a collection of subsets contains only one element,A,{\displaystyle A,}σ(A){\displaystyle \sigma (A)} may be written instead ofσ({A});{\displaystyle \sigma (\{A\});} in the prior exampleσ({1}){\displaystyle \sigma (\{1\})} instead ofσ({{1}}).{\displaystyle \sigma (\{\{1\}\}).} Indeed, usingσ(A1,A2,…){\displaystyle \sigma \left(A_{1},A_{2},\ldots \right)} to meanσ({A1,A2,…}){\displaystyle \sigma \left(\left\{A_{1},A_{2},\ldots \right\}\right)} is also quite common.

There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

σ-algebra generated by a function

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Iff{\displaystyle f} is a function from a setX{\displaystyle X} to a setY{\displaystyle Y} andB{\displaystyle B} is aσ{\displaystyle \sigma }-algebra of subsets ofY,{\displaystyle Y,} then theσ{\displaystyle \sigma }-algebra generated by the functionf,{\displaystyle f,} denoted byσ(f),{\displaystyle \sigma (f),} is the collection of all inverse imagesf−1(S){\displaystyle f^{-1}(S)} of the setsS{\displaystyle S} inB.{\displaystyle B.} That is,σ(f)={f−1(S):S∈B}.{\displaystyle \sigma (f)=\left\{f^{-1}(S)\,:\,S\in B\right\}.}

A functionf{\displaystyle f} from a setX{\displaystyle X} to a setY{\displaystyle Y} ismeasurable with respect to a σ-algebraΣ{\displaystyle \Sigma } of subsets ofX{\displaystyle X} if and only ifσ(f){\displaystyle \sigma (f)} is a subset ofΣ.{\displaystyle \Sigma .}

One common situation, and understood by default ifB{\displaystyle B} is not specified explicitly, is whenY{\displaystyle Y} is ametric ortopological space andB{\displaystyle B} is the collection ofBorel sets onY.{\displaystyle Y.}

Iff{\displaystyle f} is a function fromX{\displaystyle X} toRn{\displaystyle \mathbb {R} ^{n}} thenσ(f){\displaystyle \sigma (f)} is generated by the family of subsets which are inverse images of intervals/rectangles inRn:{\displaystyle \mathbb {R} ^{n}:}σ(f)=σ({f−1([a1,b1]×⋯×[an,bn]):ai,bi∈R}).{\displaystyle \sigma (f)=\sigma \left(\left\{f^{-1}(\left[a_{1},b_{1}\right]\times \cdots \times \left[a_{n},b_{n}\right]):a_{i},b_{i}\in \mathbb {R} \right\}\right).}

A useful property is the following. Assumef{\displaystyle f} is a measurable map from(X,ΣX){\displaystyle \left(X,\Sigma _{X}\right)} to(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)} andg{\displaystyle g} is a measurable map from(X,ΣX){\displaystyle \left(X,\Sigma _{X}\right)} to(T,ΣT).{\displaystyle \left(T,\Sigma _{T}\right).} If there exists a measurable maph{\displaystyle h} from(T,ΣT){\displaystyle \left(T,\Sigma _{T}\right)} to(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)} such thatf(x)=h(g(x)){\displaystyle f(x)=h(g(x))} for allx,{\displaystyle x,} thenσ(f)⊆σ(g).{\displaystyle \sigma (f)\subseteq \sigma (g).} IfS{\displaystyle S} is finite or countably infinite or, more generally,(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)} is astandard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true.[6] Examples of standard Borel spaces includeRn{\displaystyle \mathbb {R} ^{n}} with its Borel sets andR∞{\displaystyle \mathbb {R} ^{\infty }} with the cylinder σ-algebra described below.

Borel and Lebesgue σ-algebras

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An important example is theBorel algebra over anytopological space: the σ-algebra generated by theopen sets (or, equivalently, by theclosed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see theVitali set orNon-Borel sets.

On theEuclidean spaceRn,{\displaystyle \mathbb {R} ^{n},} another σ-algebra is of importance: that of allLebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra onRn{\displaystyle \mathbb {R} ^{n}} and is preferred inintegration theory, as it gives acomplete measure space.

Product σ-algebra

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Let(X1,Σ1){\displaystyle \left(X_{1},\Sigma _{1}\right)} and(X2,Σ2){\displaystyle \left(X_{2},\Sigma _{2}\right)} be two measurable spaces. The σ-algebra for the correspondingproduct spaceX1×X2{\displaystyle X_{1}\times X_{2}} is called theproduct σ-algebra and is defined byΣ1×Σ2=σ({B1×B2:B1∈Σ1,B2∈Σ2}).{\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).}

Observe that{B1×B2:B1∈Σ1,B2∈Σ2}{\displaystyle \{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\}} is a π-system.

The Borel σ-algebra forRn{\displaystyle \mathbb {R} ^{n}} is generated by half-infinite rectangles and by finite rectangles. For example,B(Rn)=σ({(−∞,b1]×⋯×(−∞,bn]:bi∈R})=σ({(a1,b1]×⋯×(an,bn]:ai,bi∈R}).{\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).}

For each of these two examples, the generating family is a π-system.

σ-algebra generated by cylinder sets

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Main article:Cylinder σ-algebra

SupposeX⊆RT={f:f(t)∈R, t∈T}{\displaystyle X\subseteq \mathbb {R} ^{\mathbb {T} }=\{f:f(t)\in \mathbb {R} ,\ t\in \mathbb {T} \}}

is a set of real-valued functions. LetB(R){\displaystyle {\mathcal {B}}(\mathbb {R} )} denote the Borel subsets ofR.{\displaystyle \mathbb {R} .} Acylinder subset ofX{\displaystyle X} is a finitely restricted set defined asCt1,…,tn(B1,…,Bn)={f∈X:f(ti)∈Bi,1≤i≤n}.{\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\}.}

Each{Ct1,…,tn(B1,…,Bn):Bi∈B(R),1≤i≤n}{\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\}}is a π-system that generates a σ-algebraΣt1,…,tn.{\displaystyle \textstyle \Sigma _{t_{1},\dots ,t_{n}}.} Then the family of subsetsFX=⋃n=1∞⋃ti∈T,i≤nΣt1,…,tn{\displaystyle {\mathcal {F}}_{X}=\bigcup _{n=1}^{\infty }\bigcup _{t_{i}\in \mathbb {T} ,i\leq n}\Sigma _{t_{1},\dots ,t_{n}}}is an algebra that generates thecylinder σ-algebra forX.{\displaystyle X.} This σ-algebra is a subalgebra of the Borel σ-algebra determined by theproduct topology ofRT{\displaystyle \mathbb {R} ^{\mathbb {T} }} restricted toX.{\displaystyle X.}

An important special case is whenT{\displaystyle \mathbb {T} } is the set of natural numbers andX{\displaystyle X} is a set of real-valued sequences. In this case, it suffices to consider the cylinder setsCn(B1,…,Bn)=(B1×⋯×Bn×R∞)∩X={(x1,x2,…,xn,xn+1,…)∈X:xi∈Bi,1≤i≤n},{\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\},}for whichΣn=σ({Cn(B1,…,Bn):Bi∈B(R),1≤i≤n}){\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)}is a non-decreasing sequence of σ-algebras.

Ball σ-algebra

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The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than theBorel σ-algebra. 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.[7]

σ-algebra generated by random variable or vector

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Suppose(Ω,Σ,P){\displaystyle (\Omega ,\Sigma ,\mathbb {P} )} is aprobability space. IfY:Ω→Rn{\displaystyle \textstyle Y:\Omega \to \mathbb {R} ^{n}} is measurable with respect to the Borel σ-algebra onRn{\displaystyle \mathbb {R} ^{n}} thenY{\displaystyle Y} is called arandom variable (n=1{\displaystyle n=1}) orrandom vector (n>1{\displaystyle n>1}). The σ-algebra generated byY{\displaystyle Y} isσ(Y)={Y−1(A):A∈B(Rn)}.{\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in {\mathcal {B}}\left(\mathbb {R} ^{n}\right)\right\}.}

σ-algebra generated by a stochastic process

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Suppose(Ω,Σ,P){\displaystyle (\Omega ,\Sigma ,\mathbb {P} )} is aprobability space andRT{\displaystyle \mathbb {R} ^{\mathbb {T} }} is the set of real-valued functions onT.{\displaystyle \mathbb {T} .} IfY:Ω→X⊆RT{\displaystyle \textstyle Y:\Omega \to X\subseteq \mathbb {R} ^{\mathbb {T} }} is measurable with respect to the cylinder σ-algebraσ(FX){\displaystyle \sigma \left({\mathcal {F}}_{X}\right)} (see above) forX{\displaystyle X} thenY{\displaystyle Y} is called astochastic process orrandom process. The σ-algebra generated byY{\displaystyle Y} isσ(Y)={Y−1(A):A∈σ(FX)}=σ({Y−1(A):A∈FX}),{\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),}the σ-algebra generated by the inverse images of cylinder sets.

See also

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FamiliesF{\displaystyle {\mathcal {F}}} of sets overΩ{\displaystyle \Omega }
Is necessarily true ofF:{\displaystyle {\mathcal {F}}\colon }
or, isF{\displaystyle {\mathcal {F}}}closed under:		Directed
by⊇{\displaystyle \,\supseteq }		A∩B{\displaystyle A\cap B}A∪B{\displaystyle A\cup B}B∖A{\displaystyle B\setminus A}Ω∖A{\displaystyle \Omega \setminus A}A1∩A2∩⋯{\displaystyle A_{1}\cap A_{2}\cap \cdots }A1∪A2∪⋯{\displaystyle A_{1}\cup A_{2}\cup \cdots }Ω∈F{\displaystyle \Omega \in {\mathcal {F}}}∅∈F{\displaystyle \varnothing \in {\mathcal {F}}}F.I.P.
π-systemYesYesNoNoNoNoNoNoNoNo
SemiringYesYesNoNoNoNoNoNoYesNever
Semialgebra(Semifield)YesYesNoNoNoNoNoNoYesNever
Monotone classNoNoNoNoNoonly ifAi↘{\displaystyle A_{i}\searrow }only ifAi↗{\displaystyle A_{i}\nearrow }NoNoNo
𝜆-system(Dynkin System)YesNoNoonly if
A⊆B{\displaystyle A\subseteq B}		YesNoonly ifAi↗{\displaystyle A_{i}\nearrow } or
they aredisjoint		YesYesNever
Ring(Order theory)YesYesYesNoNoNoNoNoNoNo
Ring(Measure theory)YesYesYesYesNoNoNoNoYesNever
δ-RingYesYesYesYesNoYesNoNoYesNever
𝜎-RingYesYesYesYesNoYesYesNoYesNever
Algebra(Field)YesYesYesYesYesNoNoYesYesNever
𝜎-Algebra(𝜎-Field)YesYesYesYesYesYesYesYesYesNever
Dual idealYesYesYesNoNoNoYesYesNoNo
FilterYesYesYesNeverNeverNoYesYes∅∉F{\displaystyle \varnothing \not \in {\mathcal {F}}}Yes
Prefilter(Filter base)YesNoNoNeverNeverNoNoNo∅∉F{\displaystyle \varnothing \not \in {\mathcal {F}}}Yes
Filter subbaseNoNoNoNeverNeverNoNoNo∅∉F{\displaystyle \varnothing \not \in {\mathcal {F}}}Yes
Open TopologyYesYesYesNoNoNo
(even arbitrary∪{\displaystyle \cup })		YesYesNever
Closed TopologyYesYesYesNoNo
(even arbitrary∩{\displaystyle \cap })		NoYesYesNever
Is necessarily true ofF:{\displaystyle {\mathcal {F}}\colon }
or, isF{\displaystyle {\mathcal {F}}}closed under:		directed
downward		finite
intersections		finite
unions		relative
complements		complements
inΩ{\displaystyle \Omega }		countable
intersections		countable
unions		containsΩ{\displaystyle \Omega }contains∅{\displaystyle \varnothing }Finite
Intersection
Property

Additionally, asemiring is aπ-system where every complementB∖A{\displaystyle B\setminus A} is equal to a finitedisjoint union of sets inF.{\displaystyle {\mathcal {F}}.}
Asemialgebra is a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A} is equal to a finitedisjoint union of sets inF.{\displaystyle {\mathcal {F}}.}
A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements ofF{\displaystyle {\mathcal {F}}} and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}

References

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  1. ^Elstrodt, J. (2018). Maß- Und Integrationstheorie. Springer Spektrum Berlin, Heidelberg.https://doi.org/10.1007/978-3-662-57939-8
  2. ^Rudin, Walter (1987).Real & Complex Analysis.McGraw-Hill.ISBN 0-07-054234-1.
  3. ^Vestrup, Eric M. (2009).The Theory of Measures and Integration. John Wiley & Sons. p. 12.ISBN 978-0-470-31795-2.
  4. ^Džamonja, Mirna;Kunen, Kenneth (1995)."Properties of the class of measure separable compact spaces"(PDF).Fundamenta Mathematicae: 262.Ifμ{\displaystyle \mu } is a Borel measure onX,{\displaystyle X,} the measure algebra of(X,μ){\displaystyle (X,\mu )} is the Boolean algebra of all Borel sets moduloμ{\displaystyle \mu }-null sets. Ifμ{\displaystyle \mu } 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 thatμ{\displaystyle \mu } isseparableif and only if this metric space is separable as a topological space.
  5. ^Fischer, Tom (2013)."On simple representations of stopping times and stopping time sigma-algebras".Statistics and Probability Letters.83 (1):345–349.arXiv:1112.1603.doi:10.1016/j.spl.2012.09.024.
  6. ^Kallenberg, Olav (2001).Foundations of Modern Probability (2nd ed.).Springer. p. 7.ISBN 0-387-95313-2.
  7. ^van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York.https://doi.org/10.1007/978-1-4757-2545-2

External links

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