# Definition:Sigma-Algebra

## Definition

### Definition 1

Let $X$ be a set.

A $\sigma$-algebra $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

 $(SA \, 1)$ $:$ Unit: $\displaystyle X \in \mathcal R$ $(SA \, 2)$ $:$ Closure under Complement: $\displaystyle \forall A \in \mathcal R:$ $\displaystyle \complement_X \left({A}\right) \in \mathcal R$ $(SA \, 3)$ $:$ Closure under Countable Unions: $\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:$ $\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

### Definition 2

Let $X$ be a set.

A $\sigma$-algebra $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

 $(SA \, 1')$ $:$ Unit: $\displaystyle X \in \mathcal R$ $(SA \, 2')$ $:$ Closure under Complement: $\displaystyle \forall A \in \mathcal R:$ $\displaystyle \complement_X \left({A}\right) \in \mathcal R$ $(SA \, 3')$ $:$ Closure under Countable Disjoint Unions: $\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:$ $\displaystyle \bigsqcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

### Definition 3

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

### Definition 4

Let $X$ be a set.

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

## Also known as

This is also seen as $\sigma$-algebra, from $\sigma$ being the Greek letter sigma.

## 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.