Category:Definitions/Sigma-Algebras

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This category contains definitions related to Sigma-Algebras.
Related results can be found in Category:Sigma-Algebras.


Let $X$ be a set.

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

\((\text {SA} 1)\)   $:$   Unit:    \(\displaystyle X \in \RR \)             
\((\text {SA} 2)\)   $:$   Closure under Complement:      \(\displaystyle \forall A \in \RR:\) \(\displaystyle \relcomp X A \in \RR \)             
\((\text {SA} 3)\)   $:$   Closure under Countable Unions:      \(\displaystyle \forall A_n \in \RR: n = 1, 2, \ldots:\) \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \RR \)