TITLE: Interpretation of limsup-liminf of sets
QUESTION [44 upvotes]: What is an intuitive interpretation of the 'events'
$$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$
and
$$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$
when $A_n$ are subsets of a measured space $(\Omega, F,\mu)$.
Of the first it should be that 'an infinite number of those events is verified', but
I don't see how to explain (or interpret this). Thanks for any help!

REPLY [11 votes]: For the begining note that
$$
s\in\bigcap\limits_{\lambda\in\Lambda} S_\lambda\Longleftrightarrow
\forall\lambda\in\Lambda\quad s\in S_\lambda
$$
$$
s\in\bigcup\limits_{\lambda\in\Lambda} S_\lambda\Longleftrightarrow
\exists \lambda\in\Lambda\quad s\in S_\lambda
$$
Now we can show more formal explanation of what $\limsup$ is.
$$
x\in\limsup\limits_{n\to\infty}A_n
\Longleftrightarrow
x\in\bigcap\limits_{n=0}^\infty\bigcup\limits_{k=n}^\infty A_k
\Longleftrightarrow
\forall n\in\mathbb{N}\quad x\in\bigcup\limits_{k=n}^\infty A_k
\Longleftrightarrow
$$
$$
\forall n\in\mathbb{N}\quad\exists k\geq n\quad x\in A_k
$$
This means exactly that $x$ occurs in the sequence of sets $\{A_n:n\in\mathbb{N}\}$ infinitely many times.
Here is a formal explanation for $\liminf$
$$
x\in\liminf\limits_{n\to\infty}A_n
\Longleftrightarrow
x\in\bigcup\limits_{n=0}^\infty\bigcap\limits_{k=n}^\infty A_k
\Longleftrightarrow
\exists n\in\mathbb{N}\quad x\in\bigcap\limits_{k=n}^\infty A_k
\Longleftrightarrow
$$
$$
\exists n\in\mathbb{N}\quad\forall k\geq n\quad x\in A_k
$$
This means exactly that $x$ occurs in all the sets of sequence $\{A_n:n\in\mathbb{N}\}$ except maybe in the first $n$ sets.

REPLY [10 votes]: Read 'forall' with intersection, 'exists' with union. I prefer to think of events as follows:
$x \in\limsup A_n$ iff $\forall n$ $\exists k \geq n$ such that $x \in A_k$.
$x \in\liminf A_n$ iff $\exists n$ $\forall k \geq n$ we have $x \in A_k$.