Definition:Sigma-Algebra/Definition 1

From ProofWiki
Jump to navigation Jump to search


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.