I am having trouble trying to understand the topic of my question. For reference please use Virtual Laboratory of Probability and Statistics.
Let's start with limit superior: $\limsup_{n \to \infty} A_n = \bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i$ It works for arbitrary sequences of events $A_n$, right? Let's assume we have independent replication of the same basic event $A$ whose probability is strictly positive $\mathbb{P}(A) = p \in (0, 1]$, so that each $A_n$ is just $A$. And since set union and set intersection are both idempotent operations, we get: $\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i = \bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A = A$ so that $\limsup_{n \to \infty} A_n = A$ and $\mathbb{P} \left( \limsup_{n \to \infty} A_n \right) = \mathbb{P} \left( A \right) = p$ So far, so good. But from the second Borel-Cantelli lemma, since $p > 0$, we get $\mathbb{P} \left( \limsup_{n \to \infty} A_n \right) = 1$ For me it's hard to accept, that arbitrarily chosen $p \in (0, 1]$ will always be equal to 1. Where's my mistake?
Anyway, what is that event $\limsup_{n \to \infty} A_n$? Since it is assigned probability 1, it must cover the entire sample space $\Omega$ of our probability space $(\Omega, \mathscr{F }, \mathbb{P})$, minus some countable subset.
The first term in the definition of limit superior is $\bigcap_{n=1}^\infty$, and it specifies a decreasing (non increasing) sequence of events. From this it follows that any of it "tails" $\bigcup_{i=n}^\infty A_i$ must have probability 1 -- $\mathbb{P} \left( \bigcup_{i=n}^\infty A_i \right) = 1$.
Putting it all together, any of the "tails" $\bigcup_{i=n}^\infty A_i$ must include every possible event from the probability space. Is this line of thinking is correct? I mean, do I use the right words to describe my intuition behind limit superior/inferior?
One last question, can limit superior/inferior of arbitrary sequence of events ever take any other probability different from 0 and 1?