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 $\scriptstyle \Omega$ that contains $\scriptstyle \Omega$ itself and which is closed under the taking of complements (with respect to $\scriptstyle \Omega$) 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 $\scriptstyle \Omega$, let $\scriptstyle P\,=\,2^{\Omega }$ 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 $\scriptstyle \Omega$) is a sigma algebra if it satisfies all the following conditions or axioms:

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

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

See also

Set

Set theory

Borel set

Measure theory

Measure

External links

• Tutorial on sigma algebra at probability.net