Category:Axioms/Sigma-Algebra Axioms

This category contains axioms related to Sigma-Algebra Axioms.

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 $

Pages in category "Axioms/Sigma-Algebra Axioms"

The following 3 pages are in this category, out of 3 total.

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