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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<p_{0} < \infty$.

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Try a function of the form $x^a\log^b(x)$ on $(0,1)$ and another function of that form on $(1,\infty)$ – GEdgar Nov 21 '11 at 3:18
The second half of this faq-entry explains how you can vote on answers and accept them. Also, a more descriptive title would be very nice. How about "A function belonging to only one $L^p$-space"? Here's a question dealing with the case $1 \lt p \lt \infty$. – t.b. Nov 21 '11 at 3:34
@ Arturo and t.b, well noted. I new on this site and trying to learn the rules and procedures. I will do my best to do the right thing. Thanks – wright Nov 21 '11 at 3:56
No you don't... – leo Dec 11 '11 at 4:01
I'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). – Jonas Meyer Dec 17 '11 at 5:58

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 $q\neq p$.

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