Intuitive explanation of lim sup of sequence of sets

[madhavpr]

Hi,

I can derive a few properties of the limit inferior and limit superior of a sequence of sets but I have trouble in understanding what they actually mean. However, my understand of lim inf and lim sup of a sequence isn't all that bad. Is there a way to understand them intuitively (something like slope of the tangent line ~ derivative) ? Also, is there a connection between lim inf and lim sup of sequences of numbers with sequences of sets?

Thanks,
Madhav

 

[Svein]

I can derive a few properties of the limit inferior and limit superior of a sequence of sets

What kind of sets? If they are subsets of ℝ (the real numbers), ℚ (the rational numbers) or ℕ (the integers), the notion of "limit inferior" etc. has meaning. If you are talking about ℂ (the complex numbers) or sets of fishes in the sea, the limits are per se meaningless.

 

[madhavpr]

Thanks Svein, for the reply. I was solving problems from a real analysis text.

The problem looked like this.

For a sequence of sets, E_n, (n=1,2,3,...), lim sup E_n = { x | x is an element of E_k for infinitely many k }. I don't understand the significance of this definition (and lim inf's definition as well). When and where do they show up? What was the necessity to define such quantities?

 

[Svein]

For a sequence of sets, E_n, (n=1,2,3,...), lim sup E_n = { x | x is an element of E_k for infinitely many k }. I don't understand the significance of this definition (and lim inf's definition as well). When and where do they show up? What was the necessity to define such quantities?

Makes no sense. If E_n = [0, 1] for all n, every number in [0, 1] satisfies that definition. There must be some additional requirement somewhere.

 

[Samy_A]

For sets, $\displaystyle \limsup A_n= \cap_{N=1}^\infty ( \cup_{n\ge N} A_n )$ and $\displaystyle \liminf A_n= \cup_{N=1}^\infty (\cap_{n \ge N} A_n)$.
(All $A_n$ are understood to be subset of some set $X$.)

An element is in limsup if it is an element of infinitely many $A_n$.
An element is in liminf if it is an element of all the $A_n$, except possibly a finite number of them.

The notion is used in measure theory. The one example I remember is the Borel-Cantelli lemma.

 

[Svein]

The notion is used in measure theory. The one example I remember is the Borel-Cantelli lemma.

Must be something beyond what I did in real analysis. My thesis was in complex function algebras.

 

[Samy_A]

Must be something beyond what I did in real analysis. My thesis was in complex function algebras.

I remember these notions being used quite a lot in my 3rd year Measure Theory course.
I tried to search the Internet for examples now, and don't find much.
So maybe it was something specific to my professor, or maybe things are done differently now than they were 40 years ago. (Or maybe my memory is playing tricks with me.)

To the OP:
A connection between liminf and limsup for real numbers and for sets can be seen in the following.
Let $(A_n)_n$ be subsets of some set $X$, $A=\liminf A_n,\ B=\limsup A_n$.
If we denote by $\chi_n$ the characteristic function of $A_n$, and by $\chi_A, \chi_B$ the characteristic functions of $A,\ B$, then:
$\chi_A=\liminf \ \chi_n, \ \chi_B=\limsup \ \chi_n$ (where these liminf and limsup are the usual liminf and limsup for real numbers).