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 ${\displaystyle \scriptstyle \Omega }$ that contains ${\displaystyle \scriptstyle \Omega }$ itself and which is closed under the taking of complements (with respect to ${\displaystyle \scriptstyle \Omega }$) 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 ${\displaystyle \scriptstyle \Omega }$, let ${\displaystyle \scriptstyle P\,=\,2^{\Omega }}$ be its power set, i.e. set of all subsets of ${\displaystyle \Omega }$. Then a subset FP (i.e., F is a collection of subset of ${\displaystyle \scriptstyle \Omega }$) is a sigma algebra if it satisfies all the following conditions or axioms:

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

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 ${\displaystyle \scriptstyle \Omega }$ = {Red, Yellow, Green}, the subset F = {{}, {Green}, {Red, Yellow}, {Red, Yellow, Green}} of ${\displaystyle \scriptstyle 2^{\Omega }}$ is a σ algebra.