Axiom:Sigma-Ring Axioms/Formulation 2

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Let $\Sigma$ be a system of sets.

$\Sigma$ is a $\sigma$-ring if and only if the following axioms are satisfied:

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

These criteria are called the $\sigma$-ring axioms.

Also see