I was looking for connections between the limit superior and inferior of sets and the limit superior and inferior of sequences. Let $(X_k)_{k\in\mathbb{N}}$ be a stochastic process. Then is $\limsup\limits_k \{\omega : X_k(\omega)>a\}=\{\omega : \limsup\limits_k X_k(\omega)>a\}$ and $\liminf\limits_k \{\omega : X_k(\omega)<a\}=\{\omega : \liminf\limits_k X_k(\omega)<a\}$.
Is the following reasonning correct?
$\limsup\limits_k \{\omega : X_k(\omega)>a\}=\{X_k(\omega)>a$ for infinitely many $k\}\subset \{\omega : \limsup\limits_k X_k(\omega)>a\} $ since $X_k(\omega)>a$ for infinitely many $k\implies \limsup\limits_k X_k(\omega)>a$. The converse seems to be true also.
If $\limsup\limits_k X_k(\omega)>a$ then $X_k(\omega)>a$ for infinitely many $k$ and so $\{\limsup\limits_k X_k>a\}\subset\{X_k>a$ for infinitely many $k\}= \limsup\limits_k\{X_k>a\}$