Motivation to define $\limsup$ and $\liminf$ of sets

Question

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 ?

Answer 1

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 $\limsup$.

Answer 2

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:

1. 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 $.

2. 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:

3. 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 $.

4. 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 $.

Answer 3

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.