Sigma-algebra
Encyclopedia
In mathematics
, a σ-algebra (also sigma-algebra, σ-field, sigma-field) is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures
; specifically, the collection of sets over which a measure is defined is 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.
The definition is that a σ-algebra over a set X is a nonempty collection Σ of subset
s of X (including X itself) that is closed
under complement
ation and countable union
s of its members. It is an algebra of sets
, completed
to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space.
Thus, if X = {a, b, c, d}, one possible sigma algebra on X is
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 (a construction known as the Borel set
).
on X is a function
which assigns a real number
to subsets of 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 of disjoint sets.
One would like to assign a size to every subset of X, 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 set
s. For this reason, one considers instead a smaller collection of privileged subsets of 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.
The collection of subsets of X that form the σ-algebra is usually denoted by Σ, the capital Greek letter sigma
. The pair (X, Σ) is an algebra of sets
and also a field of sets, called a measurable space. If the subsets of X in Σ correspond to numbers in elementary algebra
, then the two set operations union (symbol ∪)
and intersection (∩)
correspond to addition and multiplication. The collection of sets Σ is completed
to include countably infinite operations.
Σ is closed under complementation: If A is in Σ, then so is its complement
, . Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1 ∪ A2 ∪ A3 ∪ … .
From these axioms, it follows that the σ-algebra is also closed under countable intersections
(by applying De Morgan's laws).
It also follows that the X itself and the empty set
are both in Σ, because since by (1) Σ is non-empty, you can pick some A ⊂ X, and by (2) you know that X \ A is also in Σ. By (3) A ∪ (X \ A) = X is in Σ. And finally, since X is in Σ, you know by (2) that its complement, the empty set, is also in Σ.
In fact, this is precisely the difference between a σ-algebra and a σ-ring: a σ-algebra Σ is just a σ-ring that contains the universal set X. 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.
Elements of the σ-algebra are called measurable sets. An ordered pair , where X is a set and Σ is a σ-algebra over X, 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 function
s as morphism
s. Measures
are defined as certain types of functions from a σ-algebra to [0, ∞].
σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface
. Thus may be denoted as
or 
. This is handy to avoid situations where the letter Σ may be confused for the summation
operator.
To see that such a σ-algebra always exists, let
. The σ-algebra generated by F will therefore be the smallest element in Φ. Indeed, such a smallest element exists: First, Φ is not empty because the power set 2X is in Φ. Consequently, let σ* denote the (nonempty!) intersection of all elements in Φ. Because each element in Φ contains F, the intersection σ* will also contain F. Moreover, because each element in Φ is a σ-algebra, the intersection σ* will also be a σ algebra (observe that if every element in Φ has the three properties of a σ-algebra, then the intersection of Φ will as well). Hence, because σ* is a σ-algebra which contains F, σ* is in Φ, and because it is the intersection of all sets in Φ, σ* is indeed the smallest set in Φ by definition, which in turn implies that , the σ-algebra generated by F.
For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the single subset {1} is σ({1}) = {∅, {1}, {2,3}, {1,2,3}}. By an abuse of notation
, when a collection of subsets contains only one member, A, one may write σ(A) instead of σ({A}); in the prior example σ(1) instead of σ({1}).
over any topological space
: the σ-algebra generated by the open set
s (or, equivalently, by the closed set
s). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example, see the Vitali set
.
On the Euclidean space
Rn, another σ-algebra is of importance: that of all Lebesgue measurable
sets. This σ-algebra contains more sets than the Borel σ-algebra on Rn and is preferred in integration
theory, as it gives a complete measure space
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a σ-algebra (also sigma-algebra, σ-field, sigma-field) is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
; specifically, the collection of sets over which a measure is defined is a σ-algebra. This concept is important in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
as the foundation for Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
, and in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, where it is interpreted as the collection of events which can be assigned probabilities.
The definition is that a σ-algebra over a set X is a nonempty collection Σ of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X (including X itself) that is closed
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
ation and countable union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
s of its members. It is an algebra of sets
Algebra of sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...
, completed
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space.
Thus, if X = {a, b, c, d}, one possible sigma algebra on X is
A more useful example is the set of subsets of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements (a construction known as the Borel set
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
).
Motivation
A measureMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
on X is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which assigns a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
to subsets of 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 of disjoint sets.
One would like to assign a size to every subset of X, 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 set
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
s. For this reason, one considers instead a smaller collection of privileged subsets of 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.
The collection of subsets of X that form the σ-algebra is usually denoted by Σ, the capital Greek letter sigma
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...
. The pair (X, Σ) is an algebra of sets
Algebra of sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...
and also a field of sets, called a measurable space. If the subsets of X in Σ correspond to numbers in elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
, then the two set operations union (symbol ∪)
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
and intersection (∩)
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
correspond to addition and multiplication. The collection of sets Σ is completed
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
to include countably infinite operations.
Definition and properties
Let X be some set, and 2X symbolically represent its power set. Then a subset is called a σ-algebra if it satisfies the following three properties:- Σ is non-empty: There is at least one
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
, .
From these axioms, it follows that the σ-algebra is also closed under countable intersections
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
(by applying De Morgan's laws).
It also follows that the X itself and the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
are both in Σ, because since by (1) Σ is non-empty, you can pick some A ⊂ X, and by (2) you know that X \ A is also in Σ. By (3) A ∪ (X \ A) = X is in Σ. And finally, since X is in Σ, you know by (2) that its complement, the empty set, is also in Σ.
In fact, this is precisely the difference between a σ-algebra and a σ-ring: a σ-algebra Σ is just a σ-ring that contains the universal set X. 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.
Elements of the σ-algebra are called measurable sets. An ordered pair , where X is a set and Σ is a σ-algebra over X, is called a measurable space. A function between two measurable spaces is called a measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, with the measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s as morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s. Measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
are defined as certain types of functions from a σ-algebra to [0, ∞].
σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface
Fraktur (typeface)
Fraktur is a calligraphic hand and any of several blackletter typefaces derived from this hand. The word derives from the past participle fractus of Latin frangere...
. Thus may be denoted as
or . This is handy to avoid situations where the letter Σ may be confused for the summationSummation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
operator.
Generated
σ-algebra Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every set in F (even though F may or may not itself be a σ-algebra). This σ-algebra is denoted σ(F) and called the σ-algebra generated by F.To see that such a σ-algebra always exists, let
. The σ-algebra generated by F will therefore be the smallest element in Φ. Indeed, such a smallest element exists: First, Φ is not empty because the power set 2X is in Φ. Consequently, let σ* denote the (nonempty!) intersection of all elements in Φ. Because each element in Φ contains F, the intersection σ* will also contain F. Moreover, because each element in Φ is a σ-algebra, the intersection σ* will also be a σ algebra (observe that if every element in Φ has the three properties of a σ-algebra, then the intersection of Φ will as well). Hence, because σ* is a σ-algebra which contains F, σ* is in Φ, and because it is the intersection of all sets in Φ, σ* is indeed the smallest set in Φ by definition, which in turn implies that , the σ-algebra generated by F.For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the single subset {1} is σ({1}) = {∅, {1}, {2,3}, {1,2,3}}. By an abuse of notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
, when a collection of subsets contains only one member, A, one may write σ(A) instead of σ({A}); in the prior example σ(1) instead of σ({1}).
Examples
Let X be any set, then the following are σ-algebras over X:- The family consisting only of the empty set and the set X, called the minimal or trivial σ-algebra over X.
- The power set of X.
- The collection of subsets of X which are countable or whose complements are countable (which is distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by the singletons of X.
- If {Σλ} is a family of σ-algebras over X indexed by λ then the intersection of all Σλ's is a σ-algebra over X.
Examples for generated algebras
An important example is the Borel algebraBorel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
over any topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
: the σ-algebra generated by the open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s (or, equivalently, by the closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example, see the Vitali set
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
.
On the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn, another σ-algebra is of importance: that of all Lebesgue measurable
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
sets. This σ-algebra contains more sets than the Borel σ-algebra on Rn and is preferred in integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
theory, as it gives a complete measure space
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
.
See also
- Measurable functionMeasurable functionIn mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
- Sample space
- Separable sigma algebraSeparable sigma algebraIn mathematics, σ-algebras are usually studied in the context of measure theory. A separable σ-algebra is a sigma algebra that can be generated by a countable collection of sets...
- Sigma ring
- Sigma additivitySigma additivityIn mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...