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
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 Unions: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 3
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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Also see
- Equivalence of Definitions of Sigma-Ring
- Results about $\sigma$-rings can be found here.
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.