# Definition:Sigma-Algebra

## Definition

### Definition 1

Let $X$ be a set.

Let $\Sigma$ be a system of subsets of $X$.

$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:

 $(\text {SA} 1)$ $:$ Unit: $\ds X \in \Sigma$ $(\text {SA} 2)$ $:$ Closure under Complement: $\ds \forall A \in \Sigma:$ $\ds \relcomp X A \in \Sigma$ $(\text {SA} 3)$ $:$ Closure under Countable Unions: $\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$ $\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$

### Definition 2

Let $X$ be a set.

Let $\Sigma$ be a system of subsets of $X$.

$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:

 $(\text {SA} 1')$ $:$ Unit: $\ds X \in \Sigma$ $(\text {SA} 2')$ $:$ Closure under Set Difference: $\ds \forall A, B \in \Sigma:$ $\ds A \setminus B \in \Sigma$ $(\text {SA} 3')$ $:$ Closure under Countable Disjoint Unions: $\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$ $\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma$

### Definition 3

A $\sigma$-algebra $\Sigma$ is a $\sigma$-ring with a unit.

### Definition 4

Let $X$ be a set.

A $\sigma$-algebra $\Sigma$ over $X$ is an algebra of sets which is closed under countable unions.

## Examples

### Trivial $\sigma$-Algebra

Let $X$ be a set.

The trivial $\sigma$-algebra on $X$ is the $\sigma$-algebra defined as:

$\set {\O, X}$

## Also see

• Results about $\sigma$-algebras can be found here.

## Linguistic Note

The $\sigma$ in $\sigma$-algebra is the Greek letter sigma which equates to the letter s.

$\sigma$ stands for for somme, which is French for union, and also summe, which is German for union.