# Definition:Sigma-Algebra

## Contents

## 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.

- Equivalence of Definitions of Sigma-Algebra
- $\sigma$-Algebra as Magma of Sets, proving that
**$\sigma$-algebras**instantiate the general concept of a magma of sets.

## 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**.