Sigma-Algebra Closed under Finite Intersection

Theorem

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

Let $A_1, \ldots, A_n \in \Sigma$.

Then $\displaystyle \bigcap_{k \mathop = 1}^n A_k \in \Sigma$.

Proof

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

By axiom $(1)$ of a $\sigma$-algebra, it follows that $\forall k \in \N, k > n: A_k \in \Sigma$.

From Sigma-Algebra Closed under Countable Intersection, it follows that $\displaystyle \bigcap_{k \mathop \in \N} A_k = \bigcap_{k \mathop = 1}^n A_k \in \Sigma$.

$\blacksquare$