Axiom:Sigma-Ring Axioms

Definition

$\Sigma$ satisfies the $\sigma$-ring axioms if and only if:

$(\text {SR} 1)$	$:$	Empty Set:		$\ds \O \in \Sigma $
$(\text {SR} 2)$	$:$	Closure under Set Difference:	$\ds \forall A, B \in \Sigma:$	$\ds A \setminus B \in \Sigma $
$(\text {SR} 3)$	$:$	Closure under Countable Unions:	$\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$	$\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma $

$\Sigma$ satisfies the $\sigma$-ring axioms if and only if:

$(\text {SR} 1')$	$:$	Empty Set:		$\ds \O \in \Sigma $
$(\text {SR} 2')$	$:$	Closure under Set Difference:	$\ds \forall A, B \in \Sigma:$	$\ds A \setminus B \in \Sigma $
$(\text {SR} 3')$	$:$	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 $

\((\text {SR} 1)\)	$:$	Empty Set:		\(\ds \O \in \Sigma \)
\((\text {SR} 2)\)	$:$	Closure under Set Difference:	\(\ds \forall A, B \in \Sigma:\)	\(\ds A \setminus B \in \Sigma \)
\((\text {SR} 3)\)	$:$	Closure under Countable Unions:	\(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\)	\(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \)

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