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For all $p,\,q\in[1,\infty)$, $p \not = q$, find a sequence of Lebesgue measurable function $f_n:\mathbb{R}\rightarrow\mathbb{R}$ so that $f_n\in \cap_{r\in[1,\infty)}L^r(\mathbb{R})$, and $(f_n)_{n\in\mathbb{N}}$ is a Cauchy sequence in $L^p(\mathbb{R})$, but not a Cauchy sequence in $L^q(\mathbb{R})$.

I was able to this problem given $p < q$, using result in here, but I cannot find one that true on $p \not = q$.

Awoo
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1 Answers1

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Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.

Robert Israel
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  • I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $p\not = q$. – Awoo Nov 15 '18 at 05:13
  • You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q \ne p$. – Robert Israel Nov 15 '18 at 13:50