# Sigma algebra

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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 FP (i.e., F is a collection of subset of ) is a sigma algebra if it satisfies all the following conditions or axioms:

1. 
2. If  then the complement 
3. If  for  then 

## 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  = {Red, Yellow, Green}, the subset F = {{}, {Green}, {Red, Yellow}, {Red, Yellow, Green}} of  is a σ algebra.