Definition:Sigma-Algebra/Definition 1

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

Also known as

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

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

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.