This question looks simple at the first glance but ... I have tried to combine the theorems and definitions on $L^p$ spaces to solve this question but I have not been able to do so. I need help to show that, there is a measurable function $g\in L^ { p_{0}}\setminus \Bigg( \bigcup_{p\neq {p_{0}}}L^p \Bigg)$ for every $ 0
$L^p$ spaces in integration measure
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real-analysis
measure-theory
lebesgue-integral
lp-spaces
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1I'm voting to close as duplicate of the question t.b. links to. Technically it asks for slightly more, but the general case is an immediate corollary of the case for a single $p\in(0,\infty)$ (just take the $p/p_0$ power). – 2011-12-17
1 Answers
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The answer for $p\geqslant 1$ is given in this theread. Given a $p>0$, take a function $f$ which is only in in $L^{Np}$ where $N$ is such that $Np\geqslant 1$. Then consider $g:=|f|^{1/N}\in L^p$ but not in $L^q$ if $q\neq p$.