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A σ-algebra X over a set E is a family of subsets of E which is closed under countable set operations; σ-algebras are mainly used in order to define measures on E. The concept is important in analysis and probability and statistics.

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

1. The empty set is in X,
2. If S is in X then so is the complement of S.
3. If S₁, S₂, S₃, ... is a sequence in X then their (countable) union is also in X.

From 1 and 2 it follows that E is in X; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections.

Examples

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

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

If U is an arbitrary family of subsets of E 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 E that contains U, namely the power set of E. Let Φ be the family of all σ-algebras over E that contain U (that is, a σ-algebra X over E 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 E that contains U; its elements are all sets that can be gotten from sets in U by applying a countable sequence of the set operations union, intersection and complement.

This leads to the most important example: the Borel σ-algebra over any topological space is the σ-algebra generated by the open sets (or, equivalently, by the closed sets). It is important to 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 R, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra.

Measurable functions

If X is a σ-algebra over E and Y is a σ-algebra over F, then a function f : E -> F is called measurable if the preimage of every set in Y is in X. A function f : E -> R is called measurable if it is measurable with respect to the Borel σ-algebra on R.