Definition:Sigma-Ring

Definition

A $\sigma$-ring is a ring of sets which is closed under countable unions.

That is, a ring of sets $\Sigma$ is a $\sigma$-ring if and only if:

$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$

$\Sigma$ is a $\sigma$-ring if and only if $\Sigma$ satisfies 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 $

Let $\Sigma$ be a system of sets.

$\Sigma$ is a $\sigma$-ring if and only if $\Sigma$ satisfies 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 Disjoint Unions:	$\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$	$\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma $

The $\sigma$ in $\sigma$-ring is the Greek letter sigma which equates to the letter s.

$\sigma$ stands for for somme, which is French for union.

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