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Math Help - example, sequence of measurable

  1. #1
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    example, sequence of measurable

    Give an example in [0, 1] of a sequence of measurable functions (f_n)_n such that \lim_{n \rightarrow \infty   } || f_n ||_1=0 but (f_n)_n does not converge to zero a.e.


    For this one, I can't think of a sequence that fits this criteria. What would this sequence look like? I can't think of one right now.
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  2. #2
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    Quote Originally Posted by xianghu21 View Post
    Give an example in [0, 1] of a sequence of measurable functions (f_n)_n such that \lim_{n \rightarrow \infty   } || f_n ||_1=0 but (f_n)_n does not converge to zero a.e.
    Have a look at this:

    Choose a sequence (a_n)_n in [0,1] and real numbers r_n\to_n 0 such that every x\in[0,1] belong to infinitely many intervals [a_n,a_n+r_n].

    For instance, a_0=0, a_1=\frac{1}{2}, a_2=0, a_3 =\frac{1}{4}, a_4=\frac{1}{2}, a_5=\frac{3}{4}, a_6=0, etc. (dyadic points) and r_0=r_1=\frac{1}{2}, r_2=r_3=r_4=r_5=\frac{1}{4}, r_6=\cdots = r_{13} = \frac{1}{8},...

    Then let f_n(x)={\bf 1}_{[a_n,a_n+r_n]}(x).

    There may be simpler, that's what came to my mind. Since (f_n)_n can't be monotonic, it must look like my example anyway.
    Last edited by Laurent; December 14th 2009 at 07:48 AM. Reason: Simplification: no need to divide by r_n.
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