# Online Encyclopedia

# 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
*E*_{1},*E*_{2},*E*_{3}, ... is a sequence in*X*then their (countable) union is also 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 **R**^{n}, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel algebra on **R**^{n} and is preferred in integration theory.

See also measurable function.