Sigma-Algebra is Monotone Class

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Theorem

Let $\Sigma$ be a $\sigma$-algebra on a set $X$.


Then $\Sigma$ is also a monotone class.


Proof

By definition, $\Sigma$, being a $\sigma$-algebra, is closed under countable unions.

From Sigma-Algebra Closed under Countable Intersection, it is also closed under countable intersections.


Thence, by definition, $\Sigma$ is a monotone class.

$\blacksquare$