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For a homework problem I'm trying to find a sequence $(f_n)_{n\in \mathbb N}$ of functions such that the functions map from $[0,1]$ to $[0,1]$, the sequence shall not converge pointwise almost everywhere, but $\int_0^1 f_n (x)dx$ shall converge to $0$, as $n$ goes to infinity.

My intuition was to use the functions: $f_n:= \sin(\pi x)^{2n}$. As I believe $\lim_{n\to \infty} \int_0^1f_n(x)dx =0$, but I believe it converges pointwise to zero on the interval except at the fixed zeros. I have no clue how to approach this now and any help would be appreciated.

Ted Shifrin
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Evan Dickinson
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    This https://math.stackexchange.com/questions/1412091/the-typewriter-sequence might inspire you. Also, please edit your math formulas. They are not properly displaying. – Severin Schraven Apr 04 '25 at 21:55
  • Here you find a lot of information on "how to ask a good question". Questions tend to get downvoted quickly, if they are "below" these quality guidelines. To avoid this annoying process you should read it, and adapt your question(s) to it. https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question – Tina Apr 04 '25 at 22:17
  • $\sin(0)=0$. Did you mean "... except $\sin(\pi /2)^{2n}, \sin(3\pi /2)^{2n}$"? – Tina Apr 04 '25 at 22:22
  • You can choose the sequence of indicatorfunctions $\mathbb I_{[0, 1/n]}, \mathbb I_{[1/n,2/n]}, \dots, \mathbb I_{[(n-1)/n, 1]}, \mathbb I_{[0, 1/(n+1)]}, \mathbb I_{[1/(n+1), 2/(n+1)]}, \dots $ – Tina Apr 04 '25 at 22:29

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