Let $(X_n)$ be a sequence of real random variables on some probability space $(\Omega,\mathcal{F},P)$ and let $a\in\mathbb{R}$.
Note that
\begin{align} \limsup_n\,[X_n\geq a] &= \bigcap_{n}\bigcup_{k\geq n}[X_k\geq a]\\ &= \{\omega\in\Omega\colon \forall_n\exists_{k\geq n}X_k(\omega)\geq a\}\\ &= \{\omega\in\Omega\colon \forall_n\sup_{k\geq n}X_k(\omega)\geq a\}\\ &= \{\omega\in\Omega\colon \inf_n\sup_{k\geq n}X_k(\omega)\geq a\}\\ &= [\limsup_n X_n\geq a]\\ \end{align}
(Is this even correct?)
When can we change the $\limsup$ "into the set"? It seems to work for functions on the left side of $\geq$ as then $\inf_n$ translates to "and" and $\sup$ translates to "or".
I really like the analogies $\bigcap\leftrightarrow\forall$ and $\bigcup\leftrightarrow\exists$. Is there a more general framework for extending these to $$\bigcap\leftrightarrow\inf\leftrightarrow\forall\quad\text{and}\quad \bigcup\leftrightarrow\sup\leftrightarrow\exists?$$