Sigma-algebra

In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory.

Formally, X is a σ-algebra if and only if it has the following properties:

The empty set is in X,
If E is in X then so is the complement of E.
If E₁, E₂, E₃, ... is a sequence in X then their (countable) union is also in X.

X is an algebra of sets or as some say, a field of sets when 3 is:

   If E and F are in X, then E \/ F, ie E union F, is in X

From 1 and 2 it follows that S is in X; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections (via De Morgan's laws).

An ordered pair (S, X), where S is a set and X is a σ-algebra over S, is called a measurable space.

Examples

If S is any set, then the family consisting only of the empty set and S is a σ-algebra over S, the so-called trivial σ-algebra. Another σ-algebra over S is given by the full power set of S.

If {X_a} is a family of σ-algebras over S, then the intersection of all X_a is also a σ-algebra over S.

If U is an arbitrary family of subsets of S 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 S that contains U, namely the power set of S. Let Φ be the family of all σ-algebras over S that contain U (that is, a σ-algebra X over S is in Φ if and only if U is a subset of X.) Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra over S 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.