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Math Help - A problem with limsup/liminf

  1. #1
    Moo
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    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 :
    \left(\liminf_{n \to \infty} f_n(x)\right)^-=\limsup_{n \to \infty} \left(f_n(x)\right)^-

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

    Also, 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 !
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    Quote Originally Posted by Moo View Post
    The problem is :
    \left(\liminf_{n \to \infty} f_n(x)\right)^-=\limsup_{n \to \infty} \left(f_n(x)\right)^-

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

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

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

    Now suppose that l<0. This implies that x_n<0 infinitely often. But y_n=-x_n whenever x_n<0, and therefore \limsup_{n \to \infty}y_n = \limsup_{n \to \infty}(-x_n) = -l. Thus in this case it is again true that \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.
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  3. #3
    Moo
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    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
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    Quote Originally Posted by Moo View Post
    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 x_n by  f_n(x) and y_n by  (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.

    Quote Originally Posted by Moo View Post
    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.
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  5. #5
    Moo
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    Quote Originally Posted by Opalg View Post
    If you don't like the idea of fixing x, then simply replace x_n by  f_n(x) and y_n by  (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 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 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 ><



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