σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space.
The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]
The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, 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 conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If one possible σ-algebra on is where is the empty set. In general, a finite algebra is always a σ-algebra.
If is a countable partition of 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 the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).
Motivation edit
There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
Measure edit
A measure on is a function that assigns a non-negative real number to subsets of 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 disjoint sets.
One would like to assign a size to every subset of but in many natural settings, this is not possible. For example, the axiom 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, the Vitali sets. For this reason, one considers instead a smaller collection of privileged subsets of 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 edit
Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.
- The limit supremum or outer limit of a sequence of subsets of is
- The limit infimum or inner limit of a sequence of subsets of is
The inner limit is always a subset of the outer limit:
Sub σ-algebras edit
In much of probability, especially when conditional 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. A simple example suffices to illustrate this idea.
Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads ( ) or Tails ( ). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the sample space Ω must consist of all possible infinite sequences of or
However, after 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 2n possibilities for the first flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra
Observe that then
Definition and properties edit
Definition edit
Let be some set, and let represent its power set. Then a subset is called a σ-algebra if and only if it satisfies the following three properties:[3]
- is in and is considered to be the universal set in the following context.
- is closed under complementation: If some set is in then so is its complement,
- is closed under countable unions: If are in then so is
From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).
It also follows that the empty set is in since by (1) is in and (2) asserts that its complement, the empty set, is also in Moreover, since satisfies condition (3) as well, it follows that is the smallest possible σ-algebra on The largest possible σ-algebra on is
Elements of the σ-algebra are called measurable sets. An ordered pair where is a set and is a σ-algebra over is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to
A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below).
Dynkin's π-λ theorem edit
This theorem (or the related monotone 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.
- A π-system is a collection of subsets of that is closed under finitely many intersections, and
- A Dynkin system (or λ-system) is a collection of subsets of that contains and is closed under complement and under countable unions of disjoint subsets.
Dynkin's π-λ theorem says, if is a π-system and is a Dynkin system that contains then the σ-algebra generated by is contained in Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in enjoy the property under consideration while, on the other hand, showing that the collection 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 enjoy the property, avoiding the task of checking it for an arbitrary set in
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 with the Lebesgue-Stieltjes integral typically associated with computing the probability:
Combining σ-algebras edit
Suppose is a collection of σ-algebras on a space
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:
Sketch of Proof: Let denote the intersection. Since is in every is not empty. Closure under complement and countable unions for every implies the same must be true for Therefore, is a σ-algebra.
Join
The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates a σ-algebra known as the join which typically is denoted
σ-algebras for subspaces edit
Suppose is a subset of and let be a measurable space.
- The collection is a σ-algebra of subsets of
- Suppose is a measurable space. The collection is a σ-algebra of subsets of
Relation to σ-ring edit
A σ-algebra is just a σ-ring that contains the universal set [4] A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-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 ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite.
Typographic note edit
σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus may be denoted as or
Particular cases and examples edit
Separable σ-algebras edit
A separable -algebra (or separable -field) is a -algebra that is a separable space when considered as a metric space with metric for and a given finite measure (and with being the symmetric difference operator).[5] Any -algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue -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 natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric 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 single equivalence class, the resulting quotient 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 edit
Let be any set.
- The family consisting only of the empty set and the set called the minimal or trivial σ-algebra over
- The power set of called the discrete σ-algebra.
- The collection is a simple σ-algebra generated by the subset
- The collection of subsets of which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of if and only if is uncountable). This is the σ-algebra generated by the singletons of Note: "countable" includes finite or empty.
- The collection of all unions of sets in a countable partition of is a σ-algebra.
Stopping time sigma-algebras edit
A stopping time can define a -algebra the so-called stopping time sigma-algebra, which in a filtered probability space describes the information up to the random time 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 is [6]
σ-algebras generated by families of sets edit
σ-algebra generated by an arbitrary family edit
Let be an arbitrary family of subsets of Then there exists a unique smallest σ-algebra which contains every set in (even though may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing (See intersections of σ-algebras above.) This σ-algebra is denoted and is called the σ-algebra generated by
If is empty, then Otherwise consists of all the subsets of that can be made from elements of by a countable number of complement, union and intersection operations.
For a simple example, consider the set Then the σ-algebra generated by the single subset is By an abuse of notation, when a collection of subsets contains only one element, may be written instead of in the prior example instead of Indeed, using to mean 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 edit
If is a function from a set to a set and is a -algebra of subsets of then the -algebra generated by the function denoted by is the collection of all inverse images of the sets in That is,
A function from a set to a set is measurable with respect to a σ-algebra of subsets of if and only if is a subset of
One common situation, and understood by default if is not specified explicitly, is when is a metric or topological space and is the collection of Borel sets on
If is a function from to then is generated by the family of subsets which are inverse images of intervals/rectangles in
A useful property is the following. Assume is a measurable map from to and is a measurable map from to If there exists a measurable map from to such that for all then If is finite or countably infinite or, more generally, is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true.[7] Examples of standard Borel spaces include with its Borel sets and with the cylinder σ-algebra described below.
Borel and Lebesgue σ-algebras edit
An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets.
On the Euclidean space another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on and is preferred in integration theory, as it gives a complete measure space.
Product σ-algebra edit
Let and be two measurable spaces. The σ-algebra for the corresponding product space is called the product σ-algebra and is defined by
Observe that is a π-system.
The Borel σ-algebra for is generated by half-infinite rectangles and by finite rectangles. For example,
For each of these two examples, the generating family is a π-system.
σ-algebra generated by cylinder sets edit
Suppose
is a set of real-valued functions. Let denote the Borel subsets of A cylinder subset of is a finitely restricted set defined as
Each
An important special case is when is the set of natural numbers and is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets
Ball σ-algebra edit
The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than the Borel 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.[8]
σ-algebra generated by random variable or vector edit
Suppose is a probability space. If is measurable with respect to the Borel σ-algebra on then is called a random variable ( ) or random vector ( ). The σ-algebra generated by is
σ-algebra generated by a stochastic process edit
Suppose is a probability space and is the set of real-valued functions on If is measurable with respect to the cylinder σ-algebra (see above) for then is called a stochastic process or random process. The σ-algebra generated by is
See also edit
- Measurable function – Function for which the preimage of a measurable set is measurable
- Sample space – Set of all possible outcomes or results of a statistical trial or experiment
- Sigma-additive set function – Mapping function
- Sigma-ring – Ring closed under countable unions
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: | Directed by | F.I.P. | ||||||||
π-system | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
𝜆-system (Dynkin System) | only if | only if or they are disjoint | Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
δ-Ring | Never | |||||||||
𝜎-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
𝜎-Algebra (𝜎-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
Open Topology | (even arbitrary ) | Never | ||||||||
Closed Topology | algebra, algebraic, structure, admitting, given, signature, operations, universal, algebra, mathematical, analysis, probability, theory, also, field, nonempty, collection, subsets, closed, under, complement, countable, unions, countable, intersections, ordered. For an algebraic structure admitting a given signature S of operations see Universal algebra In mathematical analysis and in probability theory a s algebra also s field on a set X is a nonempty collection S of subsets of X closed under complement countable unions and countable intersections The ordered pair X S displaystyle X Sigma is called a measurable space The s algebras are a subset of the set algebras elements of the latter only need to be closed under the union or intersection of finitely many subsets which is a weaker condition 1 The main use of s algebras is in the definition of measures specifically the collection of those subsets for which a given measure is defined is necessarily a s algebra This concept is important in mathematical analysis as the foundation for Lebesgue integration and in probability theory where it is interpreted as the collection of events which can be assigned probabilities Also in probability s algebras are pivotal in the definition of conditional expectation In statistics sub s algebras are needed for the formal mathematical definition of a sufficient statistic 2 particularly when the statistic is a function or a random process and the notion of conditional density is not applicable If X a b c d displaystyle X a b c d one possible s algebra on X displaystyle X is S a b c d a b c d displaystyle Sigma varnothing a b c d a b c d where displaystyle varnothing is the empty set In general a finite algebra is always a s algebra If A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots is a countable partition of X displaystyle X then the collection of all unions of sets in the partition including the empty set is a s algebra A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions countable intersections and relative complements and continuing this process by transfinite iteration through all countable ordinals until the relevant closure properties are achieved a construction known as the Borel hierarchy Contents 1 Motivation 1 1 Measure 1 2 Limits of sets 1 3 Sub s algebras 2 Definition and properties 2 1 Definition 2 2 Dynkin s p l theorem 2 3 Combining s algebras 2 4 s algebras for subspaces 2 5 Relation to s ring 2 6 Typographic note 3 Particular cases and examples 3 1 Separable s algebras 3 2 Simple set based examples 3 3 Stopping time sigma algebras 4 s algebras generated by families of sets 4 1 s algebra generated by an arbitrary family 4 2 s algebra generated by a function 4 3 Borel and Lebesgue s algebras 4 4 Product s algebra 4 5 s algebra generated by cylinder sets 4 6 Ball s algebra 4 7 s algebra generated by random variable or vector 4 8 s algebra generated by a stochastic process 5 See also 6 References 7 External linksMotivation editThere are at least three key motivators for s algebras defining measures manipulating limits of sets and managing partial information characterized by sets Measure edit A measure on X displaystyle X nbsp is a function that assigns a non negative real number to subsets of X displaystyle X nbsp 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 disjoint sets One would like to assign a size to every subset of X displaystyle X nbsp but in many natural settings this is not possible For example the axiom 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 the Vitali sets For this reason one considers instead a smaller collection of privileged subsets of X displaystyle X nbsp 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 s algebras Limits of sets edit Many uses of measure such as the probability concept of almost sure convergence involve limits of sequences of sets For this closure under countable unions and intersections is paramount Set limits are defined as follows on s algebras The limit supremum or outer limit of a sequence A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp of subsets of X displaystyle X nbsp is lim sup n A n n 1 m n A m n 1 A n A n 1 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 nbsp It consists of all points x displaystyle x nbsp that are in infinitely many of these sets or equivalently that are in cofinally many of them That is x lim sup n A n displaystyle x in limsup n to infty A n nbsp if and only if there exists an infinite subsequence A n 1 A n 2 displaystyle A n 1 A n 2 ldots nbsp where n 1 lt n 2 lt displaystyle n 1 lt n 2 lt cdots nbsp of sets that all contain x displaystyle x nbsp that is such that x A n 1 A n 2 displaystyle x in A n 1 cap A n 2 cap cdots nbsp The limit infimum or inner limit of a sequence A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp of subsets of X displaystyle X nbsp is lim inf n A n n 1 m n A m n 1 A n A n 1 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 nbsp It consists of all points that are in all but finitely many of these sets or equivalently that are eventually in all of them That is x lim inf n A n displaystyle x in liminf n to infty A n nbsp if and only if there exists an index N N displaystyle N in mathbb N nbsp such that A N A N 1 displaystyle A N A N 1 ldots nbsp all contain x displaystyle x nbsp that is such that x A N A N 1 displaystyle x in A N cap A N 1 cap cdots nbsp The inner limit is always a subset of the outer limit lim inf n A n lim sup n A n displaystyle liminf n to infty A n subseteq limsup n to infty A n nbsp If these two sets are equal then their limit lim n A n displaystyle lim n to infty A n nbsp exists and is equal to this common set lim n A n lim inf n A n lim sup n A n displaystyle lim n to infty A n liminf n to infty A n limsup n to infty A n nbsp Sub s algebras edit In much of probability especially when conditional 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 s algebra which is a subset of the principal s algebra it consists of the collection of subsets relevant only to and determined only by the partial information A simple example suffices to illustrate this idea Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads H displaystyle H nbsp or Tails T displaystyle T nbsp Since you and your opponent are each infinitely wealthy there is no limit to how long the game can last This means the sample space W must consist of all possible infinite sequences of H displaystyle H nbsp or T displaystyle T nbsp W H T x 1 x 2 x 3 x i H T i 1 displaystyle Omega H T infty x 1 x 2 x 3 dots x i in H T i geq 1 nbsp However after n displaystyle n nbsp 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 2n possibilities for the first n displaystyle n nbsp flips Formally since you need to use subsets of W this is codified as the s algebraG n A H T A H T n displaystyle mathcal G n A times H T infty A subseteq H T n nbsp Observe that thenG 1 G 2 G 3 G displaystyle mathcal G 1 subseteq mathcal G 2 subseteq mathcal G 3 subseteq cdots subseteq mathcal G infty nbsp where G displaystyle mathcal G infty nbsp is the smallest s algebra containing all the others Definition and properties editDefinition edit Let X displaystyle X nbsp be some set and let P X displaystyle P X nbsp represent its power set Then a subset S P X displaystyle Sigma subseteq P X nbsp is called a s algebra if and only if it satisfies the following three properties 3 X displaystyle X nbsp is in S displaystyle Sigma nbsp and X displaystyle X nbsp is considered to be the universal set in the following context S displaystyle Sigma nbsp is closed under complementation If some set A displaystyle A nbsp is in S displaystyle Sigma nbsp then so is its complement X A displaystyle X setminus A nbsp S displaystyle Sigma nbsp is closed under countable unions If A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp are in S displaystyle Sigma nbsp then so is A A 1 A 2 A 3 displaystyle A A 1 cup A 2 cup A 3 cup cdots nbsp From these properties it follows that the s algebra is also closed under countable intersections by applying De Morgan s laws It also follows that the empty set displaystyle varnothing nbsp is in S displaystyle Sigma nbsp since by 1 X displaystyle X nbsp is in S displaystyle Sigma nbsp and 2 asserts that its complement the empty set is also in S displaystyle Sigma nbsp Moreover since X displaystyle X varnothing nbsp satisfies condition 3 as well it follows that X displaystyle X varnothing nbsp is the smallest possible s algebra on X displaystyle X nbsp The largest possible s algebra on X displaystyle X nbsp is P X displaystyle P X nbsp Elements of the s algebra are called measurable sets An ordered pair X S displaystyle X Sigma nbsp where X displaystyle X nbsp is a set and S displaystyle Sigma nbsp is a s algebra over X displaystyle X nbsp is called a measurable space A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable The collection of measurable spaces forms a category with the measurable functions as morphisms Measures are defined as certain types of functions from a s algebra to 0 displaystyle 0 infty nbsp A s algebra is both a p system and a Dynkin system l system The converse is true as well by Dynkin s theorem see below Dynkin s p l theorem edit See also p l theorem This theorem or the related monotone class theorem is an essential tool for proving many results about properties of specific s algebras It capitalizes on the nature of two simpler classes of sets namely the following A p system P displaystyle P nbsp is a collection of subsets of X displaystyle X nbsp that is closed under finitely many intersections and A Dynkin system or l system D displaystyle D nbsp is a collection of subsets of X displaystyle X nbsp that contains X displaystyle X nbsp and is closed under complement and under countable unions of disjoint subsets Dynkin s p l theorem says if P displaystyle P nbsp is a p system and D displaystyle D nbsp is a Dynkin system that contains P displaystyle P nbsp then the s algebra s P displaystyle sigma P nbsp generated by P displaystyle P nbsp is contained in D displaystyle D nbsp Since certain p systems are relatively simple classes it may not be hard to verify that all sets in P displaystyle P nbsp enjoy the property under consideration while on the other hand showing that the collection D displaystyle D nbsp of all subsets with the property is a Dynkin system can also be straightforward Dynkin s p l Theorem then implies that all sets in s P displaystyle sigma P nbsp enjoy the property avoiding the task of checking it for an arbitrary set in s P displaystyle sigma P nbsp One of the most fundamental uses of the p l theorem is to show equivalence of separately defined measures or integrals For example it is used to equate a probability for a random variable X displaystyle X nbsp with the Lebesgue Stieltjes integral typically associated with computing the probability P X A A F d x displaystyle mathbb P X in A int A F dx nbsp for all A displaystyle A nbsp in the Borel s algebra on R displaystyle mathbb R nbsp where F x displaystyle F x nbsp is the cumulative distribution function for X displaystyle X nbsp defined on R displaystyle mathbb R nbsp while P displaystyle mathbb P nbsp is a probability measure defined on a s algebra S displaystyle Sigma nbsp of subsets of some sample space W displaystyle Omega nbsp Combining s algebras edit Suppose S a a A displaystyle textstyle left Sigma alpha alpha in mathcal A right nbsp is a collection of s algebras on a space X displaystyle X nbsp MeetThe intersection of a collection of s algebras is a s algebra To emphasize its character as a s algebra it often is denoted by a A S a displaystyle bigwedge alpha in mathcal A Sigma alpha nbsp Sketch of Proof Let S displaystyle Sigma nbsp denote the intersection Since X displaystyle X nbsp is in every S a S displaystyle Sigma alpha Sigma nbsp is not empty Closure under complement and countable unions for every S a displaystyle Sigma alpha nbsp implies the same must be true for S displaystyle Sigma nbsp Therefore S displaystyle Sigma nbsp is a s algebra JoinThe union of a collection of s algebras is not generally a s algebra or even an algebra but it generates a s algebra known as the join which typically is denoted a A S a s a A S a displaystyle bigvee alpha in mathcal A Sigma alpha sigma left bigcup alpha in mathcal A Sigma alpha right nbsp A p system that generates the join is P i 1 n A i A i S a i a i 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 nbsp Sketch of Proof By the case n 1 displaystyle n 1 nbsp it is seen that each S a P displaystyle Sigma alpha subset mathcal P nbsp so a A S a P displaystyle bigcup alpha in mathcal A Sigma alpha subseteq mathcal P nbsp This implies s a A S a s P displaystyle sigma left bigcup alpha in mathcal A Sigma alpha right subseteq sigma mathcal P nbsp by the definition of a s algebra generated by a collection of subsets On the other hand P s a A S a displaystyle mathcal P subseteq sigma left bigcup alpha in mathcal A Sigma alpha right nbsp which by Dynkin s p l theorem implies s P s a A S a displaystyle sigma mathcal P subseteq sigma left bigcup alpha in mathcal A Sigma alpha right nbsp s algebras for subspaces edit Suppose Y displaystyle Y nbsp is a subset of X displaystyle X nbsp and let X S displaystyle X Sigma nbsp be a measurable space The collection Y B B S displaystyle Y cap B B in Sigma nbsp is a s algebra of subsets of Y displaystyle Y nbsp Suppose Y L displaystyle Y Lambda nbsp is a measurable space The collection A X A Y L displaystyle A subseteq X A cap Y in Lambda nbsp is a s algebra of subsets of X displaystyle X nbsp Relation to s ring edit A s algebra S displaystyle Sigma nbsp is just a s ring that contains the universal set X displaystyle X nbsp 4 A s ring need not be a s algebra as for example measurable subsets of zero Lebesgue measure in the real line are a s ring but not a s 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 ring but not a s ring since the real line can be obtained by their countable union yet its measure is not finite Typographic note edit s algebras are sometimes denoted using calligraphic capital letters or the Fraktur typeface Thus X S displaystyle X Sigma nbsp may be denoted as X F displaystyle scriptstyle X mathcal F nbsp or X F displaystyle scriptstyle X mathfrak F nbsp Particular cases and examples editSeparable s algebras edit A separable s displaystyle sigma nbsp algebra or separable s displaystyle sigma nbsp field is a s displaystyle sigma nbsp algebra F displaystyle mathcal F nbsp that is a separable space when considered as a metric space with metric r A B m A B displaystyle rho A B mu A mathbin triangle B nbsp for A B F displaystyle A B in mathcal F nbsp and a given finite measure m displaystyle mu nbsp and with displaystyle triangle nbsp being the symmetric difference operator 5 Any s displaystyle sigma nbsp algebra generated by a countable collection of sets is separable but the converse need not hold For example the Lebesgue s displaystyle sigma nbsp 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 natural pseudometric that renders it separable as a pseudometric space The distance between two sets is defined as the measure of the symmetric 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 single equivalence class the resulting quotient 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 edit Let X displaystyle X nbsp be any set The family consisting only of the empty set and the set X displaystyle X nbsp called the minimal or trivial s algebra over X displaystyle X nbsp The power set of X displaystyle X nbsp called the discrete s algebra The collection A X A X displaystyle varnothing A X setminus A X nbsp is a simple s algebra generated by the subset A displaystyle A nbsp The collection of subsets of X displaystyle X nbsp which are countable or whose complements are countable is a s algebra which is distinct from the power set of X displaystyle X nbsp if and only if X displaystyle X nbsp is uncountable This is the s algebra generated by the singletons of X displaystyle X nbsp Note countable includes finite or empty The collection of all unions of sets in a countable partition of X displaystyle X nbsp is a s algebra Stopping time sigma algebras edit A stopping time t displaystyle tau nbsp can define a s displaystyle sigma nbsp algebra F t displaystyle mathcal F tau nbsp the so called stopping time sigma algebra which in a filtered probability space describes the information up to the random time t displaystyle tau nbsp 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 t displaystyle tau nbsp is F t displaystyle mathcal F tau nbsp 6 s algebras generated by families of sets edits algebra generated by an arbitrary family edit Let F displaystyle F nbsp be an arbitrary family of subsets of X displaystyle X nbsp Then there exists a unique smallest s algebra which contains every set in F displaystyle F nbsp even though F displaystyle F nbsp may or may not itself be a s algebra It is in fact the intersection of all s algebras containing F displaystyle F nbsp See intersections of s algebras above This s algebra is denoted s F displaystyle sigma F nbsp and is called the s algebra generated by F displaystyle F nbsp If F displaystyle F nbsp is empty then s X displaystyle sigma varnothing varnothing X nbsp Otherwise s F displaystyle sigma F nbsp consists of all the subsets of X displaystyle X nbsp that can be made from elements of F displaystyle F nbsp by a countable number of complement union and intersection operations For a simple example consider the set X 1 2 3 displaystyle X 1 2 3 nbsp Then the s algebra generated by the single subset 1 displaystyle 1 nbsp is s 1 1 2 3 1 2 3 displaystyle sigma 1 varnothing 1 2 3 1 2 3 nbsp By an abuse of notation when a collection of subsets contains only one element A displaystyle A nbsp s A displaystyle sigma A nbsp may be written instead of s A displaystyle sigma A nbsp in the prior example s 1 displaystyle sigma 1 nbsp instead of s 1 displaystyle sigma 1 nbsp Indeed using s A 1 A 2 displaystyle sigma left A 1 A 2 ldots right nbsp to mean s A 1 A 2 displaystyle sigma left left A 1 A 2 ldots right right nbsp is also quite common There are many families of subsets that generate useful s algebras Some of these are presented here s algebra generated by a function edit If f displaystyle f nbsp is a function from a set X displaystyle X nbsp to a set Y displaystyle Y nbsp and B displaystyle B nbsp is a s displaystyle sigma nbsp algebra of subsets of Y displaystyle Y nbsp then the s displaystyle sigma nbsp algebra generated by the function f displaystyle f nbsp denoted by s f displaystyle sigma f nbsp is the collection of all inverse images f 1 S displaystyle f 1 S nbsp of the sets S displaystyle S nbsp in B displaystyle B nbsp That is s f f 1 S S B displaystyle sigma f left f 1 S S in B right nbsp A function f displaystyle f nbsp from a set X displaystyle X nbsp to a set Y displaystyle Y nbsp is measurable with respect to a s algebra S displaystyle Sigma nbsp of subsets of X displaystyle X nbsp if and only if s f displaystyle sigma f nbsp is a subset of S displaystyle Sigma nbsp One common situation and understood by default if B displaystyle B nbsp is not specified explicitly is when Y displaystyle Y nbsp is a metric or topological space and B displaystyle B nbsp is the collection of Borel sets on Y displaystyle Y nbsp If f displaystyle f nbsp is a function from X displaystyle X nbsp to R n displaystyle mathbb R n nbsp then s f displaystyle sigma f nbsp is generated by the family of subsets which are inverse images of intervals rectangles in R n displaystyle mathbb R n nbsp s f s f 1 a 1 b 1 a n b n a i b i 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 nbsp A useful property is the following Assume f displaystyle f nbsp is a measurable map from X S X displaystyle left X Sigma X right nbsp to S S S displaystyle left S Sigma S right nbsp and g displaystyle g nbsp is a measurable map from X S X displaystyle left X Sigma X right nbsp to T S T displaystyle left T Sigma T right nbsp If there exists a measurable map h displaystyle h nbsp from T S T displaystyle left T Sigma T right nbsp to S S S displaystyle left S Sigma S right nbsp such that f x h g x displaystyle f x h g x nbsp for all x displaystyle x nbsp then s f s g displaystyle sigma f subseteq sigma g nbsp If S displaystyle S nbsp is finite or countably infinite or more generally S S S displaystyle left S Sigma S right nbsp is a standard Borel space for example a separable complete metric space with its associated Borel sets then the converse is also true 7 Examples of standard Borel spaces include R n displaystyle mathbb R n nbsp with its Borel sets and R displaystyle mathbb R infty nbsp with the cylinder s algebra described below Borel and Lebesgue s algebras edit An important example is the Borel algebra over any topological space the s algebra generated by the open sets or equivalently by the closed sets This s algebra is not in general the whole power set For a non trivial example that is not a Borel set see the Vitali set or Non Borel sets On the Euclidean space R n displaystyle mathbb R n nbsp another s algebra is of importance that of all Lebesgue measurable sets This s algebra contains more sets than the Borel s algebra on R n displaystyle mathbb R n nbsp and is preferred in integration theory as it gives a complete measure space Product s algebra edit Let X 1 S 1 displaystyle left X 1 Sigma 1 right nbsp and X 2 S 2 displaystyle left X 2 Sigma 2 right nbsp be two measurable spaces The s algebra for the corresponding product space X 1 X 2 displaystyle X 1 times X 2 nbsp is called the product s algebra and is defined byS 1 S 2 s B 1 B 2 B 1 S 1 B 2 S 2 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 nbsp Observe that B 1 B 2 B 1 S 1 B 2 S 2 displaystyle B 1 times B 2 B 1 in Sigma 1 B 2 in Sigma 2 nbsp is a p system The Borel s algebra for R n displaystyle mathbb R n nbsp is generated by half infinite rectangles and by finite rectangles For example B R n s b 1 b n b i R s a 1 b 1 a n b n a i b i 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 nbsp For each of these two examples the generating family is a p system s algebra generated by cylinder sets edit SupposeX R T f f t R t T displaystyle X subseteq mathbb R mathbb T f f t in mathbb R t in mathbb T nbsp is a set of real valued functions Let B R displaystyle mathcal B mathbb R nbsp denote the Borel subsets of R displaystyle mathbb R nbsp A cylinder subset of X displaystyle X nbsp is a finitely restricted set defined asC t 1 t n B 1 B n f X f t i B i 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 nbsp Each C t 1 t n B 1 B n B i 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 nbsp is a p system that generates a s algebra S t 1 t n displaystyle textstyle Sigma t 1 dots t n nbsp Then the family of subsets F X n 1 t i T i n S t 1 t n displaystyle mathcal F X bigcup n 1 infty bigcup t i in mathbb T i leq n Sigma t 1 dots t n nbsp is an algebra that generates the cylinder s algebra for X displaystyle X nbsp This s algebra is a subalgebra of the Borel s algebra determined by the product topology of R T displaystyle mathbb R mathbb T nbsp restricted to X displaystyle X nbsp An important special case is when T displaystyle mathbb T nbsp is the set of natural numbers and X displaystyle X nbsp is a set of real valued sequences In this case it suffices to consider the cylinder setsC n B 1 B n B 1 B n R X x 1 x 2 x n x n 1 X x i B i 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 nbsp for which S n s C n B 1 B n B i 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 nbsp is a non decreasing sequence of s algebras Ball s algebra edit The ball s algebra is the smallest s algebra containing all the open and or closed balls This is never larger than the Borel s algebra Note that the two s 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 s algebra useful in the analysis of such maps 8 s algebra generated by random variable or vector edit Suppose W S P displaystyle Omega Sigma mathbb P nbsp is a probability space If Y W R n displaystyle textstyle Y Omega to mathbb R n nbsp is measurable with respect to the Borel s algebra on R n displaystyle mathbb R n nbsp then Y displaystyle Y nbsp is called a random variable n 1 displaystyle n 1 nbsp or random vector n gt 1 displaystyle n gt 1 nbsp The s algebra generated by Y displaystyle Y nbsp iss Y Y 1 A A B R n displaystyle sigma Y left Y 1 A A in mathcal B left mathbb R n right right nbsp s algebra generated by a stochastic process edit Suppose W S P displaystyle Omega Sigma mathbb P nbsp is a probability space and R T displaystyle mathbb R mathbb T nbsp is the set of real valued functions on T displaystyle mathbb T nbsp If Y W X R T displaystyle textstyle Y Omega to X subseteq mathbb R mathbb T nbsp is measurable with respect to the cylinder s algebra s F X displaystyle sigma left mathcal F X right nbsp see above for X displaystyle X nbsp then Y displaystyle Y nbsp is called a stochastic process or random process The s algebra generated by Y displaystyle Y nbsp iss Y Y 1 A A s F X s Y 1 A A F X 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 nbsp the s algebra generated by the inverse images of cylinder sets See also editMeasurable function Function for which the preimage of a measurable set is measurable Sample space Set of all possible outcomes or results of a statistical trial or experiment Sigma additive set function Mapping function Sigma ring Ring closed under countable unionsFamilies F displaystyle mathcal F nbsp of sets over W displaystyle Omega nbsp vteIs necessarily true of F displaystyle mathcal F colon nbsp or is F displaystyle mathcal F nbsp closed under Directedby displaystyle supseteq nbsp A B displaystyle A cap B nbsp A B displaystyle A cup B nbsp B A displaystyle B setminus A nbsp W A displaystyle Omega setminus A nbsp A 1 A 2 displaystyle A 1 cap A 2 cap cdots nbsp A 1 A 2 displaystyle A 1 cup A 2 cup cdots nbsp W F displaystyle Omega in mathcal F nbsp F displaystyle varnothing in mathcal F nbsp F I P p system nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Semiring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverSemialgebra Semifield nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverMonotone class nbsp nbsp nbsp nbsp nbsp only if A i displaystyle A i searrow nbsp only if A i displaystyle A i nearrow nbsp nbsp nbsp nbsp 𝜆 system Dynkin System nbsp nbsp nbsp only ifA B displaystyle A subseteq B nbsp nbsp nbsp only if A i displaystyle A i nearrow nbsp orthey are disjoint nbsp nbsp NeverRing Order theory nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Ring Measure theory nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Neverd Ring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Never𝜎 Ring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverAlgebra Field nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Never𝜎 Algebra 𝜎 Field nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverDual ideal nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Filter nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Prefilter Filter base nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Filter subbase nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Open Topology nbsp nbsp nbsp nbsp nbsp nbsp nbsp even arbitrary displaystyle cup nbsp nbsp nbsp NeverClosed Topology nbsp nbsp nbsp span, wikipedia, wiki, book, books, library, article, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games. |