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.
$\blacksquare$