Liminf and limsup of sequence of sets

[jbalthrop]

Homework Statement

I am attempting to learn some measure theory and am starting with liminf and limsup of sequences of sets.

I found an example that is as follows:

A_n={0/n, 1/n, ... , n^2/n} and I am trying to find the limsup and liminf.

Homework Equations

liminf \subset limsup

The Attempt at a Solution

My understanding is that both deal with the tail sequences, and that limsup involves values that appear "infinitely often" and liminf covers values that appear "all but finitely often". Also I understand that lim infAn⊂lim supAn.

For the above example, if I enumerate the first few sets, it is clearly evident that {0} appears i.o. It also seems (to me) that as n--> infinity, all of the positive rational numbers appear. Now, since the sequence is monotone non-decreasing, An⊂An+1, I am having trouble seeing the limits. For example, no matter how large I choose N, there is some n≥N in which all of the rationals appear, right?

Obviously I am confused (this is all self-taught), so any explanation would be greatly appreciated. I seem to be able to make sense of liminf and limsup when the sequence is of a form similar to [0, n/n+1) and other examples, but I'm struggling with this example.

Thank you.

 

[mighty2000]

Thank you for sharing your question and example with us. It is great that you are taking the initiative to learn about measure theory on your own. I will try my best to provide an explanation that will help you understand liminf and limsup in this context.

First, let's define what liminf and limsup are. Liminf, or limit inferior, is the infimum (greatest lower bound) of the set of all values that appear in the tail sequence of a given sequence of sets. In simpler terms, it is the smallest value that appears infinitely often in the sequence. On the other hand, limsup, or limit superior, is the supremum (least upper bound) of the set of all values that appear in the tail sequence. In other words, it is the largest value that appears infinitely often in the sequence.

Now, let's look at your example. A_n={0/n, 1/n, ... , n^2/n}. As you correctly pointed out, liminf is the set of values that appear "all but finitely often" in the sequence. In this case, it would be the set of all positive rational numbers, as they all appear in the sequence for infinitely many values of n. This can be seen by choosing any positive rational number r and finding an n such that r=n/m, where m is a positive integer. For example, if we choose r=1/2, we can find n=2 such that 1/2=2/4 is in the sequence. Similarly, if we choose r=1/3, we can find n=3 such that 1/3=3/9 is in the sequence. This holds true for any positive rational number.

On the other hand, limsup is the set of values that appear "infinitely often" in the sequence. In this case, it would be the set of all positive rational numbers, as they all appear in the sequence for infinitely many values of n. This can be seen by choosing any positive rational number r and finding an n such that r=n/m, where m is a positive integer. For example, if we choose r=1/2, we can find n=2 such that 1/2=2/4 is in the sequence. Similarly, if we choose r=1/3, we can find n=3 such that 1/3=3/9 is in