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I am solving old comprehensive real analysis exams and there are two questions that I can not be sure,

  1. If $f$ is Riemann integrable then $|f|^r$ is Riemann integrable for any $r>0$.( True or False) I feel it is wrong indeed but couldnt find any example)

  2. $1<p<q<r<\infty$, any function $f$ is in $L^q$ but not in $L^p$ and not in $L^r$.

Thank you for any help.

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What have you tried? –  Gautam Shenoy Dec 16 '12 at 5:54
    
It suffices to prove the result for $r \in (0,1)$ because one can use the following result: If f and g are integrable, so is fg. –  Gautam Shenoy Dec 16 '12 at 6:01
1  
I thought the same thing r should be between 1 and zero. I tried to use the theorem saying that if f is riemann integrable then the total measure of the jump points are zero, So I have to increase the measure of jump points. –  BrK Dec 16 '12 at 6:04
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