# Definition:Sigma-Algebra/Definition 1

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## Definition

Let $X$ be a set.

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

\((\text {SA} 1)\) | $:$ | 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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |

## Also see

## 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**.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.1$