Thread: A problem with limsup/liminf

1. A problem with limsup/liminf

Hi !

This is a proof my teacher wanted us to do, but when coming to giving the solution, he got confused

The problem is :
$\displaystyle \left(\liminf_{n \to \infty} f_n(x)\right)^-=\limsup_{n \to \infty} \left(f_n(x)\right)^-$

Where $\displaystyle .^-$ is 'the negative part of the function' :
$\displaystyle f(x)=(f(x))^+-(f(x))^-$
$\displaystyle |f(x)|=(f(x))^++(f(x))^-$
etc...

Also, $\displaystyle f_n$ is a sequence of measurable functions, but I'm not sure it intervenes here.
As a sidenote, this was used in a proof of the generalization of Beppo-Levi's lemma.

Thanks in advance !

2. Originally Posted by Moo
The problem is :
$\displaystyle \left(\liminf_{n \to \infty} f_n(x)\right)^-=\limsup_{n \to \infty} \left(f_n(x)\right)^-$

Where $\displaystyle .^-$ is 'the negative part of the function' :
$\displaystyle f(x)=(f(x))^+-(f(x))^-$
It will suffice to prove this for each fixed x. So fix x.

Let $\displaystyle x_n = f_n(x)$ and $\displaystyle y_n = (f_n(x))_-$ (I prefer to use a subscript rather than a superscript to denote the "negative part"). Then $\displaystyle y_n = \begin{cases}0&\text{if }x_n\geqslant0,\\ -x_n&\text{if }x_n<0.\end{cases}$. Also, $\displaystyle \liminf_{n \to \infty} f_n(x) = \liminf_{n \to \infty}x_n := l$, and $\displaystyle \limsup_{n \to \infty} \left(f_n(x)\right)_- = \limsup_{n \to \infty}y_n$.

If $\displaystyle l\geqslant0$ then $\displaystyle x_n\geqslant0$ for all sufficiently large n, hence $\displaystyle y_n = 0$ for all sufficiently large n, and therefore $\displaystyle \limsup_{n \to \infty}y_n = 0$. In this case, $\displaystyle l_- = 0$ also, and so $\displaystyle \limsup_{n \to \infty}y_n = \left(\liminf_{n \to \infty}x_n\right)_-$.

Now suppose that $\displaystyle l<0$. This implies that $\displaystyle x_n<0$ infinitely often. But $\displaystyle y_n=-x_n$ whenever $\displaystyle x_n<0$, and therefore $\displaystyle \limsup_{n \to \infty}y_n = \limsup_{n \to \infty}(-x_n) = -l$. Thus in this case it is again true that $\displaystyle \limsup_{n \to \infty}y_n = \left(\liminf_{n \to \infty}x_n\right)_-$.

I hope that argument is convincing. I found it quite hard to write it out coherently, and I'm sure that I would also have got confused if I tried to present it to a class.

3. I don't understand the idea of fixing x, because maybe we have to consider the whole function if we want to take the negative part or the liminf ? There is no doubt you're correct, but I'm just trying to understand why ><

Can you enlighten this point please ?

I'm sure that I would also have got confused if I tried to present it to a class.
My teacher is less than 40 years old, it's his first year lecturing this class, and this was a serious attack to his image of "perfect teacher". I'm happier now

Thanks a lot, as usual

4. Originally Posted by Moo
I don't understand the idea of fixing x, because maybe we have to consider the whole function if we want to take the negative part or the liminf ?
If you don't like the idea of fixing x, then simply replace $\displaystyle x_n$ by $\displaystyle f_n(x)$ and $\displaystyle y_n$ by $\displaystyle (f_n(x))_-$ throughout the proof. I was just trying to make it easier to read by simplifying the notation. Obviously I failed.

The point is that all these operations (limsup and liminf of a sequence of functions, taking the positive or negative part) are defined pointwise. So you might as well look restrict attention to what happens at a single point x.

Originally Posted by Moo
My teacher is less than 40 years old, it's his first year lecturing this class, and this was a serious attack to his image of "perfect teacher". I'm happier now
It has to happen to everyone sooner or later.

5. Originally Posted by Opalg
If you don't like the idea of fixing x, then simply replace $\displaystyle x_n$ by $\displaystyle f_n(x)$ and $\displaystyle y_n$ by $\displaystyle (f_n(x))_-$ throughout the proof. I was just trying to make it easier to read by simplifying the notation. Obviously I failed.

The point is that all these operations (limsup and liminf of a sequence of functions, taking the positive or negative part) are defined pointwise. So you might as well look restrict attention to what happens at a single point x.
No, it's all clear if we omit the hypothesis of fixing x !
The problem was if f_n is alternatively > or < 0 while x varies.
And ould the liminf depend on x ?

But I'm such a fool... I think there wasn't any 'x', that is he wrote f_n instead of f_n(x)
Anyway, it's a good extension

Also, there is some weird way I just retrieved... Would you be kind enough to tell me if it's a good start ?

Let $\displaystyle f~:~ x \mapsto x_-$
It's a decreasing and continuous function.

So if I'm not mistaking, there is a property letting us say that $\displaystyle f(\liminf u_n)=\limsup f(u_n)$, but I don't know a proof of it :/

Haha sorry, my post is just a mess as ideas come up ><

It has to happen to everyone sooner or later.
What was your first time ?