(Intersection of algebras). Let $\left(\mathcal{B}_\alpha\right)_{\alpha \in I}$ be a family of Boolean algebras on a set $X$, indexed by a (possibly infinite or uncountable) label set $I$. Show that the intersection $\Lambda_{\alpha \in I} \mathcal{B}_\alpha:= \bigcap_{\alpha \in I} \mathcal{B}_\alpha$ of these algebras is still a Boolean algebra, and is the finest Boolean algebra that is coarser than all of the $\mathcal{B}_\alpha$. (If $I$ is empty, we adopt the convention that $\Lambda_{\alpha \in I} \mathcal{B}_\alpha$ is the discrete algebra.)