Definition (Monotone class)

Let $\mathcal{G}$ be a collection of sets. Suppose that if $(A_{j})_{j\ge 1},(B_{j})_{j\ge 1}\subseteq \mathcal{G}$ are such that $$A_{j}\subseteq A_{j+1}\text{ and }B_{j}\supseteq B_{j+1}\quad\forall j\ge 1$$then upon setting $A=\bigcup_{j=1}^{\infty}A_{j}$ and $B=\bigcap_{j=1}^{\infty}B_{j}$, we have that $A,B\in\mathcal{G}$. In this case, we call $\mathcal{G}$ a monotone class.