Sigma algebra

In mathematics, a sigma algebra is a formal mathematical structure intended among other things to provide a rigid basis for axiomatic probability theory.

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

1. $\varnothing \in \Omega .$
2. If $A\in F$ then $A^{c}\in F$
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 2^S 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.

See also

Set

Set theory

Borel set

Measure theory

measure

External links

Tutorial on sigma algebra at probability.net