Axiom:Sigma-Ring Axioms
Jump to navigation
Jump to search
Definition
Let $\Sigma$ be a system of sets.
Formulation 1
$\Sigma$ satisfies the $\sigma$-ring axioms if and only if:
\((\text {SR} 1)\) | $:$ | 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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Formulation 2
$\Sigma$ satisfies the $\sigma$-ring axioms if and only if:
\((\text {SR} 1')\) | $:$ | 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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |