Suppose $X_n$ are iid random variables with $\mathbb{P}(X_n\le x)=1-e^{-x}$. By using the Borel Cantelli Lemmas it's fairly easy to show that $\mathbb{P}(\lim\sup X_n/\log n=1)=1$. My lecture notes go on to claim that hence $\lim\sup X_n/\log n=1$ almost surely and hence $\sup X_n=\infty$ almost surely. I don't understand this at all. As far as I know $\{\lim\sup X_n/\log n=1\}=\cap_n\cup_{m\geq n}\{\omega \in \Omega:X_m(\omega)=\log n\}$
I have no definition for what $\lim\sup X_n$ is, however. How does one define the $\lim\sup$ of a sequence of measurable functions? And moreover surely it is nontrivial to show that this precisely 'factors out' of the event in the required way? I'm clearly missing some critical background information here, so a clear explanation would be greatly appreciated! Many thanks.