Axiom:Sigma-Algebra Axioms

Definition

Let $X$ be a set.

Let $\Sigma$ be a system of subsets over $X$.

$\Sigma$ is said to satisfy the sigma-algebra axioms if and only if:

$(\text {SA 1})$	$:$	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 $

$\Sigma$ is said to satisfy the sigma-algebra axioms if and only if:

$(\text {SA 1}')$	$:$	Unit:		$\ds X \in \Sigma $
$(\text {SA 2}')$	$:$	Closure under Set Difference:	$\ds \forall A, B \in \Sigma:$	$\ds A \setminus B \in \Sigma $
$(\text {SA 3}')$	$:$	Closure under Countable Disjoint Unions:	$\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$	$\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma $

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

\((\text {SA 1}')\)	$:$	Unit:		\(\ds X \in \Sigma \)
\((\text {SA 2}')\)	$:$	Closure under Set Difference:	\(\ds \forall A, B \in \Sigma:\)	\(\ds A \setminus B \in \Sigma \)
\((\text {SA 3}')\)	$:$	Closure under Countable Disjoint Unions:	\(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\)	\(\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \)