Axiom:Sigma-Algebra Axioms/Formulation 2

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

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