> [!NOTE] Lemma > Let be an [[Event Space|event space]] on $\Omega.$ Then is closed under [[Finite Set|finite]] [[Union of sets|unions]] (a union of a finite set of sets). **Proof**: Let $A_{1}, A_{2}, \dots A_{n} \in \mathcal{F}$. Set $A_{j} = \emptyset \in \mathcal{F} \quad \forall j > n$. Thus $\forall n \in \mathbb{N}^{+}: \; A_{n} \in \mathcal{F}.$ Since the empty sets will not contribute to the union: $\bigcup_{j=1}^{n} = \bigcup_{j=1}^{\infty} A_{j} \in \mathcal{F},$since, by definition, $\mathcal{F}$ is closed under countable unions.