Axiom:Sigma-Ring Axioms/Formulation 1

Definition

$\Sigma$ is a $\sigma$-ring if and only if the following axioms are satisfied:

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

These criteria are called the $\sigma$-ring axioms.

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