Fatou's lemma. Examples with limit inferior $\neq$ lim.

Question

I have problems with understanding Fatou's Lemma. What is the reason for using $\liminf$? Can someone please give an example where $\liminf \neq \lim$. When the reason does not depend on one of the two sides of the lemma, it is good to give an example when $\lim f_n \neq \liminf f_n$ and one when $\lim \mu(f_n) \neq \liminf \mu(f_n)$. Then I can see why $\liminf$ is used instead of $\lim$? Thanks in advance.

Answer 1

You can take two examples:

Example 1: Lebesgue measure on $\mathbb{R}$. Take $f_{2n}=\dfrac{1}{n}$ and $f_{2n+1}=1-\dfrac{1}{n}$. Here note that $f=0$, and so it's integral is equal to $0$. Now, $f_{n}$'s all have integral infinity. If we just restrict our attention to $[0,1]$ say, then of course, the RHS from Fatou's lemma is $0$.

Example 2: Counting measure on $\mathbb{N}$. Take $a_{2n}=\dfrac{1}{n^2}$, and $a_{2n+1}=0$.

Here, $f=0$ again, and so is the RHS.

Apply Fatou's lemma and see what conclusions you get. Note that the limit doesn't exist in either case.

Answer 2

If the sequence $a_n$ ain't convergent $\lim a_n$ doesn't exist, $\liminf a_n$ always exists, so proving something using $\liminf$ is actually a stronger result.

If $a_n$ is convergent $\lim a_n = \liminf a_n$.