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Problem Statement.

Good evening. As the title suggests, I am having a hard time with the following exercise:

Prove that there exists a sequence $f_1,f_2,\ldots$ of functions on $\mathcal{L}^1([0,1])$ such that $\lim_{n\rightarrow \infty}||f_n||_1=0$ but $$\sup\{f_n(x): n\in\mathbf{Z}^+\}=\infty$$ for every $x\in [0,1].$


Notation.

Here $\mathcal{L}^1([0,1])$ means $\mathcal{L}^1(\lambda_{[0,1]})$ where $\lambda_{[0,1]}$ represents Lebesgue measure restricted to the Borel subsets of $\mathbf{R}$ that are contained in $[0,1]$.


Questions.

Ok, so my first question is how do I interpret $\lim_{n\rightarrow \infty}||f_n||_1=0$? I know that by definition $$||f||_1=\int |f|\,d\mu$$ so I think this statement means $$\lim_{n\rightarrow \infty}||f_n||=\lim_{n\rightarrow\infty}\int|f_n|\,d\mu=0.$$ But I haven't been able to get anywhere with this, possibly because of my confusion with the other part of the question, which is even finding a sequence of functions that have the property $\sup\{f_n(x): n\in\mathbf{Z}^+\}=\infty$. I initially tried playing around with functions like $$f_n(x)=\frac{1}{\sqrt{nx}}$$ but this is undefined at $x=0$. So to summarize, my two questions are:

  1. Understanding the limit notation I pointed out.
  2. Finding a sequence of functions with an undefined supremum at every $x$.

Any suggestions towards either of these points of confusion would be wonderful.

Thank you for your time!

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    $\begingroup$ Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow. $\endgroup$ – rubikscube09 Dec 2 '18 at 5:21
  • $\begingroup$ @rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work? $\endgroup$ – Thy Art is Math Dec 2 '18 at 5:24
  • $\begingroup$ @rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion! $\endgroup$ – Thy Art is Math Dec 2 '18 at 5:26
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    $\begingroup$ Do you know about indicator functions? They might be useful for something like this. $\endgroup$ – rubikscube09 Dec 2 '18 at 5:28
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    $\begingroup$ Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions. $\endgroup$ – rubikscube09 Dec 2 '18 at 5:34
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Here is an example of such a function: $$ f_n:=k\times 1_{[j/2^k,(j+1)/2^k]}, $$ where $n=2^k+j$ and $0\le j< 2^k$.


$$ f_1\equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \\ f_4=2\times 1_{[0,1/4]}, f_5=2\times 1_{[1/4,1/2]}, f_6=2\times 1_{[1/2,3/4]}, f_7=2\times 1_{[3/4,1]} \\ \ldots $$

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  • $\begingroup$ Ok, the examples made that way clearer, thank you! $\endgroup$ – Thy Art is Math Dec 2 '18 at 5:54

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