Sigma-algebra
From Free net encyclopedia
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations, meaning that the union or the intersection of countably many members of the algebra is also a member. The main use of σ-algebras is in the definition of measures on X. The concept is important in mathematical analysis and probability theory.
Formally, a subset Σ of the powerset of a set X is a σ-algebra if and only if it has the following properties:
- The empty set is in Σ.
- If E is in Σ then so is the complement X\E of E.
- The union of countably many sets in Σ is also in Σ.
From 1 and 2 it follows that X is in Σ; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections (via De Morgan's laws).
A measure 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. One might like to assign such a size to every subset of X, but the axiom of choice implies that when the size under consideration is standard length for subsets of the real line, then there exist sets known as Vitali sets for which no size exists. For this reason, one considers instead a smaller collection of privileged subsets of X whose measure is defined; these sets constitute the σ-algebra.
Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), 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 measurable 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 [0,∞].
Notation
σ-algebras are sometimes denoted using capital letters of the Fraktur typeface. Thus, <math>\mathfrak{F}</math> may be used to denote (X,Σ). Another common convention is to use calligraphic capital letters in place of Σ, thus <math>(X,\mathcal{A})\,\;</math> is often used in place of (X,Σ). This is handy to avoid situations where Σ might be confused for the summation operator.
Examples
If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another σ-algebra over X is given by the full power set of X. The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X iff X is uncountable.
If {Σa} is a family of σ-algebras over X, then the intersection of all Σa is also a σ-algebra over X.
If U is an arbitrary family of subsets of X then we can form a special σ-algebra from U, called the σ-algebra generated by U. We denote it by σ(U) and define it as follows. First note that there is a σ-algebra over X that contains U, namely the power set of X. Let Φ be the family of all σ-algebras over X that contain U (that is, a σ-algebra Σ over X is in Φ if and only if U is a subset of Σ.) Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra over X that contains U. For a simple example, consider the set X={1,2,3}. Then the σ-algebra generated by the subset {1} is σ({1}) = { ∅, {1}, {2,3}, X}. Note that by an abuse of notation, when my collection of subsets C is a singleton containing only A, one may write σ(A) instead of σ(C).
This leads to the most important example: the Borel algebra over any topological space is the σ-algebra generated by the open sets (or, equivalently, by the closed sets). 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 algebra on Rn and is preferred in integration theory.
See also
es:Sigma-álgebra fr:Tribu (mathématiques) is:Sigma-algebra he:סיגמא-אלגברה nl:Sigma-algebra ja:完全加法族 no:Sigma-algebra pl:Przestrzeń mierzalna pt:Sigma-álgebra ru:Сигма-алгебра sk:Sigma algebra su:Aljabar sigma fi:Sigma-algebra th:พีชคณิตซิกมา uk:Сигма-алгебра