Sigma-Algebra Closed under Union/Corollary

Theorem
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $A_1, \ldots, A_n \in \Sigma$.

Then $\ds \bigcup_{k \mathop = 1}^n A_k \in \Sigma$.

Proof
Define for $k \in \N, k > n: A_k = \O$.

Then by Sigma-Algebra Contains Empty Set, axiom $(3)$ of a $\sigma$-algebra applies.

Hence:
 * $\ds \bigcup_{k \mathop \in \N} A_k = \bigcup_{k \mathop = 1}^n A_k \in \Sigma$