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 $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

\((SA \, 1)\)   $:$   Unit:    \(\displaystyle X \in \mathcal R \)             
\((SA \, 2)\)   $:$   Closure under Complement:      \(\displaystyle \forall A \in \mathcal R:\) \(\displaystyle \complement_X \left({A}\right) \in \mathcal R \)             
\((SA \, 3)\)   $:$   Closure under Countable Unions:      \(\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:\) \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R \)