What is the point of the definition $\lim_{n \to \infty }\sup A_n$ for events $A_n$?

Question

I'm looking to get to grips with the purpose of this definition. " Let $(\omega,\mathbb{F},P)$ be a probability space and let $A_n$ belong to $\mathbb{F}$ for $n \geq 1$. We define $$\limsup_{n\to \infty}A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$$"

In the notes through which I am working, it says that $$\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$$

is the event that infinitely many of $A_n$ occur. But why is this definition necessary if we can say that $$\bigcap_{n=1}^{\infty}A_n$$ is the event that infinitely many of the A_n occur?

I feel that there's a subtlety that I'm overlooking at the moment...

Answer 1

If one $A_n$ in a hundred is the empty set - or even just the first one - then the intersection of them all is the empty set.
With their formula, the union is everything that will happen in the future. The intersection is those things that will always occur in the future - there will always be another one of them; so they occur infinitely often.

Answer 2

The notion that infinitely many of the event $A_n$ occurs (i.e. $\limsup A_n$) is weaker than the notion that all of the events $A_n$ occur (i.e. $\bigcap_{n=1}^{\infty}A_n$).

We have: $$\bigcap_{n=1}^{\infty}A_n\subseteq\limsup A_n\tag1$$but $\supseteq$ is not necessary here.

If e.g $B$ is a proper subset of $D$ and $A_n=B$ if $n$ is odd and $A_n=D$ otherwise then the LHS of $(1)$ equals $B$ and the RHS equals $D$.

Answer 3

$\bigcap_{n=1}^{\infty}A_n$ is the event that "all the events happen".

$\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$ is the event that "for all $n$, at least one of $A_n,A_{n+1},A_{n+2},...$ happens", which is equivalent to "infinitely many of the $A_n$ occur".