Motivation to define $\limsup$ and $\liminf$ of sets Let $\{A_n\}$ a collection of set. We define $$\limsup_{n\to \infty }A_n:=\bigcap_{n\in\mathbb N}\bigcup_{m\geq n}A_m\quad \text{and}\quad \liminf_{n\to \infty }A_n:=\bigcup_{n\in\mathbb N}\bigcap_{m\geq n}A_m.$$
These definition looks a bit coming from nowhere.

*

*Is there any reason to define the limsup and liminf of sets like that?

*Is there any connection with limsup and liminf defined for a real sequence for example ?

 A: If $(x_n)$ is a sequence in a lattice $(P,\leq)$, $$\liminf_{n\to \infty }a_n:=\sup_{n\in\mathbb N} \inf_{k\geq n}x_k\quad \text{and}\quad \limsup_{n\to \infty }x_n=\inf_{n\in\mathbb N}\sup_{k\geq n}x_k.$$
If we use standard notation, we use$$a\wedge b:=\inf\{a,b\}\quad \text{and}\quad a\vee b:=\sup\{a,b\}.$$
So, $$\limsup_{n\to \infty }x_n:=\bigvee_{n\in\mathbb N}\bigwedge_{m\geq n}x_m\quad \text{and}\quad \liminf_{n\to \infty }x_n:=\bigwedge_{n\in\mathbb R}\bigvee_{m\geq n}x_m.$$
Take the lattice $(2^{\mathbb R},\subset )$ where the order is the usual inclusion. Then, $A\wedge B$ is the biggest set contain in $A$ and $B$ and $A\vee B$ is the smallest set where $A$ and $B$ are contained. Therefore,
$$A\wedge B:=A\cap B\quad \text{and}\quad A\vee B:=A\cup B.$$
So, $$\inf_{k\geq n}A_k:=\bigwedge_{k\geq n}A_k=\bigcap_{k\geq n}A_k,$$
and thus $$\sup_{n\in\mathbb N}\inf_{k\geq n}A_k=\bigvee_{n\in\mathbb N}\bigwedge_{k\geq n}A_k=\bigcup_{n\in\mathbb N}\bigcap_{k\geq n}A_k=:\liminf_{n\to \infty }A_k.$$
Same idea with the $\limsup$.
A: First of all there is a typo, the second set should be $ \bigcup $ $ \bigcap $.
With that in mind, see if you can spot an analogy between statements 1 and 2 here:

*

*The lim sup of a sequence $ \{ a_n \} $ is a number $ M $ such that $ M $ is "nearly" larger than every term of the sequence "after a while," while being the smallest such $ M $.


*The lim sup of a sequence $ \{ A_n \} $ is a set $ M $ which contains every $ A_n $ "after a while," and is the smallest such set.
In these statements, "... after a while" can be interpreted to "there exists $ N $ such that for all $ n \geq N $ ... For the real sequences, "a is 'nearly' greater than b" means that, for a given $ \varepsilon > 0 $, $ a > b - \varepsilon $ .
Similarly, spot the analogy between statements 3 and 4:


*the lim inf of a sequence $ \{ a_n \} $ is a number $ m $ such that $ m $ is "nearly" smaller than each $ a_n $ "after a while," while being the largest such $ m $.


*The lim inf of a sequence of sets $ \{ A_n \} $ is a set $ M $ which is contained in every $ A_n $ "after a while," while being the largest such $ M $.
A: For a set $A$, define the characteristic function
$$\chi_A(x)=\begin{cases} 1 \text{ if } x\in A \\ 0 \text{ otherwise}\end{cases}$$
Then the definition of $\liminf$ and $\limsup$ of sets are such that $\liminf \chi_{A_i}=\chi_{\liminf A_i}$ and $\limsup \chi_{A_i}=\chi_{\limsup A_i}$.  Even if you ignore the actual interpretation of the sets, this is enough to make the definitions useful.
