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 $\mathcal R$ is a $\sigma$-ring if and only if:

$\displaystyle A_1, A_2, \ldots \in \mathcal R \implies \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$


Definition 2

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

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


Definition 3

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

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


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.