Definition:Sigma-Algebra/Definition 1

Definition

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 $

The term sigma-algebra can also be seen without the hyphen: sigma algebra.

Some sources refer to a sigma-algebra as a sigma-field

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.

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