Let $(X, \mathscr{A}, \mathbb{P})$ be a probability space. Recently I learned about the definition of the terminal $\sigma$-Algebra for a series of sub-$\sigma$-algebras $(\mathcal{A}_n)_{n \in \mathbb{N}}$.
$$\mathscr{A}_\infty := \bigcap_{n \in \mathbb{N}} \sigma(\bigcup_{m \geq n} \mathcal{A_m})$$
But I can't seem to get an intuition for this terminal $\sigma$-Algebra. What kind of events are contained?
If $A_n \in \mathcal A_n$ for each $n$ then $\lim \sup A_n \equiv \cap_n \cup _{m\geq n} A_m$ and $ \lim \inf A_n \equiv \cup_n \cap_{m\geq n} A_m$ belong to $\mathcal A_{\infty}$.