Trying to get a better understanding of limsup/liminf in terms of probability.
An intuitive explanation of the limsup of a sequence of sets $(A_n)_{n\geq 1}$, in the context of probability, is that $\limsup_{\substack{n \to \infty}}A_n$ is "the event that infinitely many events occur".
Events are subsets of the sample space. Take the simple example of rolling a die, so $\Omega = \{1,2,3,4,5,6\}$.
Consider the sequence of events $A_n$ defined by (for $i\in\mathbb{N}$)
$$ \begin{align*} A_{3i-2}&=\{1,2,3\}\\ A_{3i-1}&=\{1\}\\ A_{3i}&=\{1,2,3,6\} \end{align*} $$
Then, $$ \begin{align*} \limsup_{n\rightarrow\infty}A_n&=\bigcap_{n=1}^{\infty}\bigcup_{j=n}^{\infty}A_j\\ &=\{1,2,3,6\}\cap \{1,2,3,6\}\cap\{1,2,3,6\}\cap\dots\\ &=\{1,2,3,6\} \end{align*} $$
Also, $$ \begin{align*} \liminf_{n\rightarrow\infty}A_n&=\bigcup_{n=1}^{\infty}\bigcap_{j=n}^{\infty}A_j\\ &=\{1\}\cup \{1\}\cup\{1\}\cup\dots\\ &=\{1\} \end{align*} $$
Firstly, just want to check my understanding is correct with this trivial example. Limsup contains $\{1,2,3,6\}$ because limsup is "the set of all $x\in\Omega$ that occur infinitely many times". The liminf set is smaller because $1$ is the only element that occurs "all but finitely many times". Sound correct?
Now, when we say that limsup is "the event that infinitely many events occur", does the "the" imply we take all possible values of $x\in\Omega$ contained infinitely many times within the $A_n$-s. I guess my point is that the set $A=\{1,2\}$ is an event where infinitely many of the $A_n$ occur... but we don't say it is "the" event?
Yes, this is correct. In fact, you should stick to the definitions you refer to here, that $\lim \sup$ is the set of all elements $x \in \Omega$ that appear in infinitely many $A_n$, and $\lim \inf$ is the set of elements that appear in all but finitely many $A_n$. Try to prove these statements are equivalent to the union/intersection definitions if you can.
I am unfamiliar with the definition of $\lim \sup$ as "the event that infinitely many events occur", and don't know how to interpret such a statement much less prove it. Can you point to where this definition comes from? What is the sample space in this context? What does it mean for an event to occur in this context?
All we are given is a sequence of events. We can, for example, consider the set of events that occur infinitely many times, but this "set of sets" $A$ isn't an event in our context as its elements are not elements of $\Omega$, but sets of elements of $\Omega.$