Definition:Sigma-Ring

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Definition

Definition 1

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:

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


Definition 2

A $\sigma$-ring $\Sigma$ is a system of sets with the following properties:

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


Definition 3

A $\sigma$-ring $\Sigma$ is a system of sets with the following properties:

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


Also see


Linguistic Note

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.