Axiom:Sigma-Algebra Axioms/Formulation 1

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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 Complement:      \(\ds \forall A \in \Sigma:\) \(\ds \relcomp X A \in \Sigma \)      
\((\text {SA 3})\)   $:$   Closure under Countable Unions:      \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \)      

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

Also see