Sigma-algebra
From Freepedia
In mathematics, a σ-algebra (or σ-field) [pronounced sigma-algebra] Σ over a set X is a family of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory.
Formally, Σ 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.
- If E1, E2, E3, ... is a (countable) sequence in Σ then their (countable) union 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).
An ordered pair (X, Σ), where X is a set and Σ is a σ-algebra over X, is called a measurable space. The collection of measurable spaces form a category with the measurable functions as morphisms.
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.
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.
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.
