Infinitely often and almost always

Let $A_{1},A_{2},A_{3},\ldots$ be a sequence of events in some sample space $\Omega$.

perspective	infinitely often	almost always
unions and intersections	$\omega \in \bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }A_{n}$	$\omega \in \bigcup _{N=1}^{\infty }\bigcap _{n=N}^{\infty }A_{n}$
first-order quantifiers	$\forall N\geq 1\ \exists n\geq N\colon \omega \in A_{n}$	$\exists N\leq 1\ \forall n\geq N\colon \omega \in A_{n}$
verbal expression	$\omega \in A_{n}$ for infinitely many $n\geq 1$	$\omega \in A_{n}$ for almost all $n\geq 1$, i.e. $\omega \in A_{n}$ for all but finitely many $n\geq 1$, i.e. $\omega \notin A_{n}$ for finitely many $n\geq 1$
lim sup/lim inf	$\omega \in \limsup _{n\to \infty }A_{n}$	$\omega \in \liminf _{n\to \infty }A_{n}$
limit of sup/inf	$\omega \in \lim _{N\to \infty }\bigcup _{n=N}^{\infty }A_{n}$	$\omega \in \lim _{N\to \infty }\bigcap _{n=N}^{\infty }A_{n}$