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Math Help - Sequence of Lebesgue measurable functions

  1. #1
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    Sequence of Lebesgue measurable functions

    I want to find a sequence f_n(x) \in L^+ such that f_n is defined on (0,1) for all natural numbers n, we have lim_{n\rightarrow\infty}\int f_n(x)dm=0 (m=Lebesgue measure), and lim_{n\rightarrow\infty}f_n(x) does not exist for any x in (0,1).

    My initial idea, which I know does not work, was to define something like this. For n odd, let f_n(x) be \displaystyle\frac{2}{n+1} for irrational x, and 0 otherwise. For n even, let f_n(x) be 0 for irrational x and 1 for rational x. That satisfies the first condition about the limit of the integral, but does not satisfy the second condition since the limit of f_n(x) is 0 on the irrationals.

    Even though I know this doesn't work, I was wondering if someone here would know of a way I could think about tweaking this to work, or am I barking up the wrong tree, so to speak?
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  2. #2
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    Quote Originally Posted by Diamondlance View Post
    I want to find a sequence f_n(x) \in L^+ such that f_n is defined on (0,1) for all natural numbers n, we have \lim_{n\rightarrow\infty}\int f_n(x)dm=0 (m=Lebesgue measure), and lim_{n\rightarrow\infty}f_n(x) does not exist for any x in (0,1).
    The idea is to construct a sequence of functions that are 0 on most of the interval but jump up to 1 on a small subinterval that sweeps across the whole interval.

    Specifically, given any natural number n, let 2^q be the largest power of 2 that is \leqslant n, so that n = 2^q + r, where 0\leqslant r<2^q. Then define f_n(x) = \begin{cases}1&\text{if }r/2^q\leqslant x<(r+1)/2^q,\\ 0&\text{otherwise.}\end{cases}
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