> [!NOTE] Lemma > Let be an [[Event Space|event space]] on $\Omega.$ Then is closed under [[Finite Set|finite]] [[Set Intersection|intersection]] (a union of a finite set of sets). **Proof**: Let $A_{1},A_{2},\dots,A_{n} \in \mathcal{F}$. By [[De Morgan's Laws for Intersection]], $\bigcap_{j=1}^{n} A_{j} = \left(\bigcup_{j=1}^{n} A_{j}^{c} \right)^{c} $Since $\mathcal{F}$ is closed under compliments, $A_{j}^{c}\in \mathcal{F}$, for every $j = 1,\dots,n$. Since [[Event Spaces are Closed Under Finite Unions]], $\bigcup_{j=1}^{n} A_{j}^{c} \in \mathcal{F}.$ By taking compliments again $\cap_{j=1}^{n} A_{j} \in \mathcal{F}$.