σ-algebra

A σ-algebra , also called σ-set algebra , closed set system , sigma field or Borel set field , is a set system in measure theory , i.e. a set of sets. A σ-algebra is characterized by its closure with regard to certain set-theoretic operations. σ-algebras play a central role in modern stochastics and integration theory , since they appear there as domains of definition for measures and contain all quantities to which an abstract volume or probability is assigned.

σ-algebras are used in many areas of mathematics. For example, they enable the temporal availability of information to be modeled through filtering or the compression of data through sufficient σ-algebra .

definition

Let be a non-empty set and be the power set of this set. ${\ displaystyle \ Omega}$${\ displaystyle {\ mathcal {P}} (\ Omega)}$

A system of sets , i.e. a set of subsets of , is called σ-algebra (up or over ) if it fulfills the following three conditions: ${\ displaystyle {\ mathcal {A}} \ subseteq {\ mathcal {P}} (\ Omega)}$${\ displaystyle \ Omega}$${\ displaystyle \ Omega}$

1. ${\ displaystyle {\ mathcal {A}}}$contains the basic amount. So it applies${\ displaystyle \ Omega \ in {\ mathcal {A}}.}$
2. ${\ displaystyle {\ mathcal {A}}}$is stable in terms of complement formation . If so , then also in included.${\ displaystyle A \ in {\ mathcal {A}}}$${\ displaystyle A ^ {\ mathsf {c}} = \ Omega \ setminus A}$${\ displaystyle {\ mathcal {A}}}$
3. ${\ displaystyle {\ mathcal {A}}}$is stable with respect to countable associations . So are sets
${\ displaystyle A_ {1}, A_ {2}, A_ {3}, \ dots}$in included, so also in included.${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ bigcup _ {n \ in \ mathbb {N}} A_ {n}}$${\ displaystyle {\ mathcal {A}}}$

motivation

If you want to mathematically specify the intuitive concept of volume in or in other spaces, you usually require the following properties: ${\ displaystyle \ mathbb {R} ^ {3}}$

1. Any amount has a volume .${\ displaystyle M \ subseteq \ mathbb {R} ^ {3}}$${\ displaystyle \ operatorname {Vol} (M) \ in [0, \ infty]}$
2. ${\ displaystyle \ operatorname {Vol}}$should be shift-invariant, because the position of a set intuitively has no influence on its volume. For and therefore applies . Likewise, the volume should be invariant under rotations. Congruent quantities should therefore have identical volumes.${\ displaystyle M \ subseteq \ mathbb {R} ^ {3}}$${\ displaystyle a \ in \ mathbb {R} ^ {3}}$${\ displaystyle \ operatorname {Vol} (M + a) = \ operatorname {Vol} (M)}$
3. The volume is normalized. For example, the unit cube should have the volume 1.${\ displaystyle [0,1] ^ {3}}$
4. The union of countably many disjoint sets has as a volume exactly the sum of the volumes of the individual sets. This property is called σ-additivity and is important for the later consideration of limit values.

With this implicit definition of a volume concept, the question arises whether such a function even exists. This question is called the dimension problem . According to Vitali's theorem, however , the dimensional problem is unsolvable, so there is no mapping with the required properties.

An attempt is now being made, by sensibly weakening the above requirements, to define a concept of volume, which on the one hand still largely corresponds to our intuitive concept, but on the other hand is also mathematically well-defined and provides a fruitful theory of measure. To do this, one weakens the first of the above requirements and accepts that one cannot assign a volume to all quantities. One then restricts oneself to a system of sets which have a volume that corresponds to the following practical considerations:

• The basic set should have a (not necessarily finite) volume and therefore be contained in the set system.
• If the set has a volume, one also wants to know the volume of the complement. So for every set there should also be its complement in the set system.${\ displaystyle M}$
• The fourth condition in the list above implies that if countably many sets have a volume, then the union of these sets also has a volume and is therefore contained in the set system.

Direct consequences from this are that the empty set and countable sections of sets with volume also have a volume again.

These requirements are precisely the defining properties of a σ-algebra. Thus, σ-algebras are the set systems on which one sensibly defines volume terms and measures in order to avoid contradictions like those caused by Vitali's theorem.

properties

Stability against set operations

From the conditions 1 and 2 of the definition it follows directly that always contains the complement of , i.e. the empty set . ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ Omega}$ ${\ displaystyle \ emptyset}$

Furthermore, identity follows from De Morgan's laws

${\ displaystyle \ bigcap _ {n \ in \ mathbb {N}} A_ {n} = {\ biggl (} \ bigcup _ {n \ in \ mathbb {N}} A_ {n} ^ {\ mathsf {c} } {\ biggr)} ^ {\! \! {\ mathsf {c}}}}$

It follows from points 2 and 3 of the definition that σ-algebras are also closed with respect to countable averages .

The stability with regard to finitely many cuts or unions also follows directly from the stability with regard to countable infinite intersections and unions. In the case of the union one sets for everyone at a fixed one , then is ${\ displaystyle A_ {j} = \ emptyset}$${\ displaystyle j> m}$${\ displaystyle m}$

${\ displaystyle A_ {1} \ cup A_ {2} \ cup \ dotsb \ cup A_ {m} = \ bigcup _ {i \ in \ mathbb {N}} A_ {i}}$

The procedure is the same for cuts, you then bet for everyone . ${\ displaystyle A_ {j} = \ Omega}$${\ displaystyle j> m}$

Thus σ-algebras are closed under set difference , because it is

${\ displaystyle A \ setminus B = A \ cap B ^ {\ mathsf {c}}}$.

Mightiness

If a finite σ-algebra, there is always a positive integer with , that is: the power of is a power of two . ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle n}$${\ displaystyle | {\ mathcal {A}} | = 2 ^ {n}}$ ${\ displaystyle | {\ mathcal {A}} |}$${\ displaystyle {\ mathcal {A}}}$

Examples

For any amount is ${\ displaystyle \ Omega}$

${\ displaystyle {\ mathcal {A}} _ {1}: = \ {\ emptyset, \ Omega \}}$

the smallest possible σ-algebra. It is also called the trivial σ-algebra . The power set

${\ displaystyle {\ mathcal {A}} _ {2}: = {\ mathcal {P}} (\ Omega)}$

is the largest possible σ-algebra with as a basic set. ${\ displaystyle \ Omega}$

For any set and a subset is ${\ displaystyle \ Omega}$${\ displaystyle A \ subseteq \ Omega}$

${\ displaystyle {\ mathcal {A}} _ {3} = \ {\ emptyset, A, A ^ {\ mathsf {c}}, \ Omega \}}$

a σ-algebra. It is the smallest σ-algebra that contains. ${\ displaystyle A}$

The set system is above a basic set${\ displaystyle \ Omega}$

${\ displaystyle {\ mathcal {A}} _ {4} = \ {A \ subset \ Omega \ mid A \ \ mathrm {abz {\ ddot {a}} hlbar \ or} \ A ^ {\ mathsf {c} } \ \ mathrm {abz {\ ddot {a}} hlbar} \}}$

a σ-algebra. Here, countable means that finite or countable is infinite. ${\ displaystyle A}$

Are and any two sets, a σ-algebra in and a map. Then ${\ displaystyle \ Omega}$${\ displaystyle \ Omega '}$${\ displaystyle {\ mathcal {A}} '}$${\ displaystyle \ Omega '}$${\ displaystyle T \ colon \ Omega \ rightarrow \ Omega '}$

${\ displaystyle {\ mathcal {A}} _ {5}: = T ^ {- 1} ({\ mathcal {A}} ') = \ lbrace T ^ {- 1} (A'): A '\ in {\ mathcal {A}} '\ rbrace}$

a σ-algebra in . This follows directly from the stability of the archetype with regard to the set operations . It is a simple example of an initial σ-algebra , a common method for constructing σ-algebras. ${\ displaystyle \ Omega}$

The most important example in the application is the Borel σ-algebra , which can be assigned to any topological space . It is by definition the smallest σ-algebra that contains all open subsets, but it can only very rarely be fully described.

meaning

σ-algebras form the starting point for the definition of the measure space and the probability space . The Banach-Tarski paradox demonstrates that on uncountable sets the σ-algebra formed by the power set can be too large as a basis for determining the volume and that considering other σ-algebras is mathematically necessary. In the theory of stochastic processes , especially in stochastic financial mathematics , the information that can in principle be observed up to a point in time is described by a σ-algebra, which leads to the concept of filtration , i.e. a family of σ-algebras that increase over time. Filtration is essential to the general theory of stochastic integration ; Integrands (i.e. financial mathematical trading strategies) may only depend on the information up to (exclusively) at a time ; in particular, they must not “look into the future”. ${\ displaystyle t}$${\ displaystyle t}$

Operations

Sections of σ-algebras

Intersections of two σ-algebras and , that is, the system of sets ${\ displaystyle {\ mathcal {A}} _ {1}}$${\ displaystyle {\ mathcal {A}} _ {2}}$

${\ displaystyle {\ mathcal {A}} _ {1} \ cap {\ mathcal {A}} _ {2} = \ {A \ subseteq \ Omega \; | \; A \ in {\ mathcal {A}} _ {1} {\ text {and}} A \ in {\ mathcal {A}} _ {2} \}}$,

are always σ-algebras again. Because is exemplary , so is ${\ displaystyle A \ in {\ mathcal {A}} _ {1} \ cap {\ mathcal {A}} _ {2}}$

• ${\ displaystyle \ Omega \ setminus A}$in , there is also in .${\ displaystyle {\ mathcal {A}} _ {1}}$${\ displaystyle A}$${\ displaystyle {\ mathcal {A}} _ {1}}$
• ${\ displaystyle \ Omega \ setminus A}$in , there is also in .${\ displaystyle {\ mathcal {A}} _ {2}}$${\ displaystyle A}$${\ displaystyle {\ mathcal {A}} _ {2}}$

So is also in , the cut is complementary stable. The stability with regard to the other set operations follows analogously. ${\ displaystyle \ Omega \ setminus A}$${\ displaystyle {\ mathcal {A}} _ {1} \ cap {\ mathcal {A}} _ {2}}$

The statement also applies to the intersection of any number of σ-algebras, since the above argument can then be extended to all of these σ-algebras. This property forms the basis for the σ operator, cf. below.

Unions of σ-algebras

The union of two σ-algebras and , that is, the system of sets ${\ displaystyle {\ mathcal {A}} _ {1}}$${\ displaystyle {\ mathcal {A}} _ {2}}$

${\ displaystyle {\ mathcal {A}} _ {1} \ cup {\ mathcal {A}} _ {2} = \ {A \ subseteq \ Omega \; | \; A \ in {\ mathcal {A}} _ {1} {\ text {or}} A \ in {\ mathcal {A}} _ {2} \}}$

is generally no longer a σ-algebra. For example, consider the two σ-algebras

${\ displaystyle {\ mathcal {A}} _ {1} = \ {\ emptyset, \ {1,2,3 \}, \ {1 \}, \ {2,3 \} \}}$

such as

${\ displaystyle {\ mathcal {A}} _ {2} = \ {\ emptyset, \ {1,2,3 \}, \ {3 \}, \ {1,2 \} \}}$,

so is

${\ displaystyle {\ mathcal {A}} _ {1} \ cup {\ mathcal {A}} _ {2} = \ {\ emptyset, \ {1,2,3 \}, \ {1,2 \} , \ {2,3 \}, \ {1 \}, \ {3 \} \}}$.

This system of quantities is neither union-stable, since it does not contain, nor is it cut-stable, since it does not contain. ${\ displaystyle \ {1 \} \ cup \ {3 \} = \ {1,3 \}}$${\ displaystyle \ {2 \} = \ {1,2 \} \ cap \ {2,3 \}}$

Products of σ-algebras

Are and systems of sets on and and is the product of and defined as ${\ displaystyle {\ mathcal {M}} _ {1}}$${\ displaystyle {\ mathcal {M}} _ {2}}$${\ displaystyle \ Omega _ {1}}$${\ displaystyle \ Omega _ {2}}$${\ displaystyle {\ mathcal {M}} _ {1}}$${\ displaystyle {\ mathcal {M}} _ {2}}$

${\ displaystyle {\ mathcal {M}} _ {1} \ times {\ mathcal {M}} _ {2}: = \ {A \ times B \ subseteq \ Omega _ {1} \ times \ Omega _ {2 } \; | \; A \ in {\ mathcal {M}} _ {1}, \; B \ in {\ mathcal {M}} _ {2} \}}$,

so the product of two σ-algebras is generally no longer a σ-algebra, but just a half ring . Because if you look at it

${\ displaystyle {\ mathcal {B}} = \ {\ emptyset, \ {1 \}, \ {2 \}, \ {1,2 \} \}}$,

so the system of sets contains both sets ${\ displaystyle {\ mathcal {B}} \ times {\ mathcal {B}}}$

${\ displaystyle M_ {1} = \ {1.2 \} \ times \ {1.2 \} = \ {(1.1), (1.2), (2.1), (2.2) \}}$as well .${\ displaystyle M_ {2} = \ {2 \} \ times \ {2 \} = \ {(2.2) \}}$

The amount

${\ displaystyle M_ {1} \ setminus M_ {2} = M_ {2} ^ {\ mathsf {c}} = \ {(1,1), (1,2), (2,1) \}}$

is not included, however, because it cannot be represented as a Cartesian product of two elements . Thus, the product is not complementary stable and consequently cannot be a σ-algebra. ${\ displaystyle {\ mathcal {B}}}$

The product of σ-algebras is therefore not defined as the Cartesian product of the individual σ-algebras, but via the product σ-algebra . This uses the set systems of the Cartesian products as the generator of a σ-algebra. In the case of the product of finitely many σ-algebras, this means that the product σ-algebra is the smallest σ-algebra that contains all Cartesian products of elements of the individual σ-algebras.

σ operator

For any subset of the power set , the operator is defined as ${\ displaystyle {\ mathcal {M}}}$${\ displaystyle {\ mathcal {P}} (\ Omega)}$${\ displaystyle \ sigma}$

${\ displaystyle \ sigma ({\ mathcal {M}}) = \ bigcap _ {{\ mathcal {A}} \ in {\ mathcal {F}} ({\ mathcal {M}})} \! \! { \ mathcal {A}},}$

in which

${\ displaystyle {\ mathcal {F}} ({\ mathcal {M}}) = \ {{\ mathcal {A}} \ subseteq {\ mathcal {P}} (\ Omega) \ mid {\ mathcal {M} } \ subseteq {\ mathcal {A}}, {\ mathcal {A}} \ \ sigma {\ text {-Algebra}} \}.}$

Since the intersection of a family of σ-algebras (over the same basic set ) is again a σ-algebra, it is thus the smallest σ-algebra that includes. ${\ displaystyle \ Omega}$${\ displaystyle \ sigma ({\ mathcal {M}})}$${\ displaystyle {\ mathcal {M}}}$

The operator fulfills the fundamental properties of an envelope operator : ${\ displaystyle \ sigma}$

• ${\ displaystyle {\ mathcal {M}} \ subseteq \ sigma ({\ mathcal {M}})}$, so the -operator is extensive .${\ displaystyle \ sigma}$
• If it is , it is also ( monotony or isotony ).${\ displaystyle {\ mathcal {M}} \ subseteq {\ mathcal {N}}}$${\ displaystyle \ sigma ({\ mathcal {M}}) \ subseteq \ sigma ({\ mathcal {N}})}$
• It is ( idempotence ).${\ displaystyle \ sigma (\ sigma ({\ mathcal {M}})) = \ sigma ({\ mathcal {M}})}$

${\ displaystyle \ sigma ({\ mathcal {M}})}$is called the σ-algebra generated by, is called the producer of this σ-algebra. The designation as generated σ-algebra is not clear, however, since the initial σ-algebra is also referred to as the (by the functions ) generated σ-algebra. ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle {\ mathcal {M}}}$${\ displaystyle f_ {i}}$

In many cases the elements of cannot be specified explicitly (see e.g. Borel hierarchy ). A frequently used method of proving statements that hold for all elements of is the principle of good sets . The Dynkinsche π-λ set makes statements about when a generated σ-algebra and a generated Dynkin system match. ${\ displaystyle \ sigma ({\ mathcal {M}})}$${\ displaystyle \ sigma ({\ mathcal {M}})}$

Special σ-algebras

Trace σ algebras

For , the system of sets is called the trace from in or trace-σ-algebra from above . One can show that the trace of in is again a σ-algebra (but with the basic set ), which justifies the name “trace σ-algebra”. Analogously, the trace σ algebra can also be understood as an initial σ algebra with regard to natural embedding . If is a producer of , then . The trace of the creator thus creates the trace σ-algebra. ${\ displaystyle E \ subseteq \ Omega}$${\ displaystyle {\ mathcal {A}} | _ {E} = \ {A \ cap E \, | \, A \ in {\ mathcal {A}} \}}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle E}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle E}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle E}$${\ displaystyle E}$ ${\ displaystyle i \ colon E \ mapsto \ Omega, \, i (e) = e}$${\ displaystyle {\ mathcal {E}}}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {A}} | _ {E} = \ sigma ({\ mathcal {E}} | _ {E})}$

Sub-σ-algebras

If a σ-algebra is valid for a system of sets that is and is also a σ-algebra, then a sub-σ-algebra , part-σ-algebra or sub-σ-algebra of . ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {M}}}$${\ displaystyle {\ mathcal {M}} \ subseteq {\ mathcal {A}}}$${\ displaystyle {\ mathcal {M}}}$${\ displaystyle {\ mathcal {M}}}$${\ displaystyle {\ mathcal {A}}}$

Borel σ-algebra

Borel's σ-algebra is the most important σ-algebra in use. This is due to the fact that it is naturally compatible with the corresponding underlying topological space and contains many important sets such as the open and closed sets. Furthermore, large classes of measurable functions can be given for Borel's σ-algebra. In particular, all continuous functions are always measurable with respect to Borel's σ-algebra.

Initial σ algebras and final σ algebra

The initial σ-algebra is a σ-algebra that is defined by means of mappings on a base set on which no σ-algebra per se exists. It is then even the smallest σ-algebra with respect to which the functions used in the construction can be measured . The counterpart is the final σ-algebra , it is the largest σ-algebra, so that a given set of functions can be measured. This construction thus forms an analogue to the initial topology and the final topology in the topology . Product σ-algebras and trace σ-algebras can both be understood as special cases of initial σ-algebras.

Product σ algebras

Product σ algebras play a role when dimensions are to be defined on the product of two measuring spaces. Since the product of two σ-algebras is generally not a σ-algebra, one is interested in extending the products of the σ-algebras to the product space. This extension is then the product σ-algebra. It plays an important role in the definition of product dimensions , which in turn are the basis for Fubini's theorem , the modeling of multi-level experiments in stochastics, and serve as the theoretical basis for stochastic processes .

Separable σ-algebras

A σ-algebra that has a countable generator is called separable. An example of this would be Borel's σ-algebra , which can be generated from cuboids with rational corner points. ${\ displaystyle \ mathbb {R} ^ {n}}$

σ-algebras in sub-areas of mathematics

A variety of σ-algebras still exist within the sub-areas of mathematics. The list below provides a rough overview.

Probability theory

In probability theory , σ-algebras are sometimes called event systems, as they contain events according to the stochastic nomenclature .

Another important σ-algebra in probability theory is the terminal σ-algebra that occurs when examining limit values . For a sequence of σ-algebras it says which sets are independent of all finite initial parts of the sequence.

Theory of stochastic processes

The most important use of σ-algebras in the theory of stochastic processes is the filtering . These are nested families of σ-algebras that model how much information is available to a stochastic process at a certain point in time. When modeling games of chance, they ensure that the participating players have no information about the upcoming game.

Other important σ-algebras are the predictable σ-algebra for the formulation of predictable processes in continuous time and the σ-algebra of the τ-past , which results from a combination with a stopping time .

Furthermore, there is the exchangeable σ-algebra , which only contains sets that are exchangeable in the sense that they are invariant to permutations of finitely many successor members of the stochastic process.

Ergodic theory

In the ergodic important σ-algebras are the σ-algebra of invariant events and P-trivial σ-algebra . P-trivial σ-algebras are those that only contain sets with probability 0 or 1. Both σ-algebras are used, for example, to define ergodic transformations or related basic concepts of ergodic theory.

Mathematical Statistics

In mathematical statistics several different σ-algebras occur. One of them is the sufficient σ-algebra . It contains all sets that contain information relating to a given distribution class. Thus, all sets that are not included in the σ-algebra can be omitted without any loss of information. A tightening is the minimally sufficient σ-algebra, it is the smallest sufficient σ-algebra (except for zero sets). In addition, there is the related, strongly sufficient σ-algebra , which under certain circumstances corresponds to the sufficient σ-algebra. The counterpart to the sufficient σ-algebra is the distribution-free σ-algebra , it carries no information and is therefore maximally uninformative. Furthermore, there is, for example, the complete σ-algebra .

Related set systems

Dynkin systems

Every σ-algebra is always a Dynkin system . Conversely, every average stable Dynkin system is also a σ-algebra. An example of a Dynkin system that is not a σ-algebra is

${\ displaystyle {\ mathcal {M}} = \ {\ emptyset, \ {1,2 \}, \ {3,4 \}, \ {1,4 \}, \ {2,3 \}, \ { 1,2,3,4 \} \}}$

on the basic set . The set system is a Dynkin system, but not an algebra (since it is not stable to the average) and therefore also not a σ-algebra. ${\ displaystyle \ Omega = \ {1,2,3,4 \}}$

In addition , Dynkin's π-λ theorem applies : If the system of sets is stable on average, then the σ-algebra generated by and the Dynkin system generated by coincide. ${\ displaystyle {\ mathcal {E}}}$${\ displaystyle {\ mathcal {E}}}$${\ displaystyle {\ mathcal {E}}}$

Algebras

Every σ-algebra is always a set algebra . Conversely, not every set algebra is a σ-algebra. Example of this would be

${\ displaystyle {\ mathcal {A}} = \ {A \ subseteq \ Omega \ mid | A | {\ text {or}} | A ^ {C} | {\ text {is finite}} \}}$

with an infinite basic amount . ${\ displaystyle \ Omega}$

σ-rings

Every σ-algebra is by definition a σ-ring that contains the basic set. Not every σ-ring is a σ-algebra.

Individual evidence

1. Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 92 , doi : 10.1007 / 978-3-642-45387-8 .
2. Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 4 , doi : 10.1007 / 978-3-642-36018-3 .