Axiom:Sigma-Algebra Axioms/Formulation 2

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Let $X$ be a set.

Let $\Sigma$ be a system of subsets over $X$.

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

\((\text {SA} 1')\)   $:$   Unit:    \(\ds X \in \Sigma \)      
\((\text {SA} 2')\)   $:$   Closure under Set Difference:      \(\ds \forall A, B \in \Sigma:\) \(\ds A \setminus B \in \Sigma \)      
\((\text {SA} 3')\)   $:$   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 $\sigma$-algebra axioms.

Also see