# Sigma algebra  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In mathematics, a sigma algebra is a formal mathematical structure intended among other things to provide a rigid basis for measure theory and axiomatic probability theory. In essence it is a collection of subsets of an arbitrary set $\Omega$ that contains $\Omega$ itself and which is closed under the taking of complements (with respect to $\Omega$ ) and countable unions. It is found to be just the right structure that allows construction of non-trivial and useful measures on which a rich theory of (Lebesgue) integration can be developed which is much more general than Riemann integration.

## Formal definition

Given a set $\Omega$ , let $P\,=\,2^{\Omega }$ be its power set, i.e. set of all subsets of $\Omega$ . Then a subset FP (i.e., F is a collection of subset of $\Omega$ ) is a sigma algebra if it satisfies all the following conditions or axioms:

1. $\Omega \,\in \,F.$ 2. If $A\,\in \,F$ then the complement $A^{c}\in F$ 3. If $G_{i}\,\in \,F$ for $i\,=\,1,2,3,\dots$ then $\bigcup _{i=1}^{\infty }G_{i}\in F$ ## Examples

• For any set S, the power set 2S itself is a σ algebra.
• The set of all Borel subsets of the real line is a sigma-algebra.
• Given the set $\Omega$ = {Red, Yellow, Green}, the subset F = {{}, {Green}, {Red, Yellow}, {Red, Yellow, Green}} of $2^{\Omega }$ is a σ algebra.