# Limsup and liminf (sequence of sets)

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Requires further expansion, are limsup and liminf always defined? What does limit of a sequence of sets mean?

## Definition

Given a set [ilmath]X[/ilmath] and a sequence [ilmath](A_n)^\infty_{n=1}[/ilmath] of subsets of [ilmath]X[/ilmath], so [ilmath](A_n)^\infty_{n=1}\subseteq\mathcal{P}(X)[/ilmath], we may define the *superior limit* ([ilmath]\text{Lim sup} [/ilmath]) and *inferior limit* ([ilmath]\text{Lim inf} [/ilmath]) of [ilmath](A_n)[/ilmath] as follows^{[1]}:

### Lim sup

- [math]\mathop{\text{Lim sup} }_{n\rightarrow\infty}(A_n):=\Big\{x\in X\ \Big\vert\vert\{n\in\mathbb{N}\ \vert\ x\in A_n\}\vert=\aleph_0 \Big\}[/math]
^{[1]}- In words: The superior limit of [ilmath](A_n)[/ilmath] is the set that contains [ilmath]x\in X[/ilmath] given that [ilmath]x[/ilmath] is in (countably) infinitely many elements of the sequence.

### Lim inf

- [math]\mathop{\text{Lim inf} }_{n\rightarrow\infty}(A_n):=\Big\{x\in X\ \Big\vert\vert\{n\in\mathbb{N}\ \vert x\notin A_n\}\vert\ne\aleph_0\Big\}[/math]
^{[1]}- In words: The inferior limit of [ilmath](A_n)[/ilmath] is the set that contains [ilmath]x\in X[/ilmath] given that [ilmath]x[/ilmath] is in all
*but*a finite number of elements of [ilmath](A_n)[/ilmath].

- In words: The inferior limit of [ilmath](A_n)[/ilmath] is the set that contains [ilmath]x\in X[/ilmath] given that [ilmath]x[/ilmath] is in all

## Distinction

One may think to "not be in a finite number of elements" is "to be in an infinite number of elements" and conclude wrongly that these definitions are the same. This is because an element can *both* be in *and* not be in an infinite number of elements!

### Example

Let [ilmath]a\in X[/ilmath] be some arbitrary point.

- Consider the sequence [ilmath](a_n)_{n=1}^\infty[/ilmath] with [ilmath]A_n:=\left\{\begin{array}{lr}\{a\} & n\text{ is even}\\\emptyset & n\text{ is odd}\end{array}\right.[/ilmath]
- Now [ilmath]a[/ilmath] is in an infinite number (namely all the even [ilmath]n[/ilmath]s) and
*not in*an infinite number (all the odd [ilmath]n[/ilmath]s) too.- [math]\mathop{\text{Lim sup} }_{n\rightarrow\infty}(A_n)=\{a\}[/math] as the number of elements containing [ilmath]a[/ilmath] is (countably) infinite.
- [math]\mathop{\text{Lim inf} }_{n\rightarrow\infty}(A_n)=\emptyset[/math] as no [ilmath]x\in X[/ilmath] is in an infinite number and only not in a finite number.

- Now [ilmath]a[/ilmath] is in an infinite number (namely all the even [ilmath]n[/ilmath]s) and

## References

- ↑
^{1.0}^{1.1}^{1.2}Measure Theory - Paul R. Halmos