Let $\{X_n\}$ be a sequence of random variables, X a random variable, and let $\{X_n\} \rightarrow X$ a.e.
Let $A_m(\epsilon)$ be the event:
$A_m(\epsilon) = \bigcap_{n=m}^{\infty} \{|X_n-X| <\epsilon\}$
I see this type of notation alot, and I am a bit confused. Can we think of $\{|X_n-X| <\epsilon\}$ as the sequence $\{|X_n-X|: |X_n-X| <\epsilon\}$, which is the collection of random variables $|X_n-X|$ such that $|X_n-X|<\epsilon $ for all $\omega \in \Omega$.
Let me know. Help with advanced probability theory notation is greatly appreciated.