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# Sigma algebra

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 that contains itself and which is closed under the taking of complements (with respect to ) 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 , let be its power set, i.e. set of all subsets of .
Then a subset *F* ⊆ *P* (i.e., *F* is a collection of subset of ) is a sigma algebra if it satisfies all the following conditions or axioms:

- If then the complement
- If for then

## Examples

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

## See also

## External links

- Tutorial on sigma algebra at probability.net