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I am trying to prove the following relation , If $u \in L^p(\Omega)$ $\Omega \subset R^n $and $0 < p <\infty$ , the the following relation is valid , $$\|u\|_{L^p(\Omega)}^p = p\int_0^\infty t^{p-1} d_u(t)dt$$ where $d_u(t) $ is a distribution function defined by $\mu (L^n(x\in \Omega : |u(x)| >t))$

How do i go about proving the above relation. Can you give me some suggestions. Thanks . here is my solution , but i am not fully satisfied because i cannot argue some of the steps that i have done myself : $$\int_0^\infty t^{p-1}d_u(t) d(t)= \int_0^\infty t^{p-1} L^n\{x\in \Omega : |u(x)| >t\}dt $$ $$=\int_0^\infty t^{p-1 } \int_{\{x:|u(x)| > t \}} 1.dL^n(x) dt$$ $$=\int_0^\infty \int_{\{x:|u(x)| > t \}} t^{p-1}.dL^n(x) dt $$

Now i know here i have to use fubini , but i am not able to argue myself satisfactorily why ?

$$=\int_{\{x:|u(x)| > t \}}\int_0^t t^{p-1}.dt dL^n(x)$$ (am i allowed to do this here ? if yes why if not why not please ) $$=\int_\Omega |u(x)|^p.dL^n(x) dt$$ , in this step also i am not very clear . I am not satisfied much although i kind of got the solution :( Thank you for your explanation. Please do comment and help me .

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There should be a $t$ outside the integral. Hint: Fubini's theorem. –  Davide Giraudo Dec 17 '12 at 19:25
    
@DavideGiraudo : Thanks , i think i got it . will get back if i have some doubts. –  Theorem Dec 17 '12 at 19:28
    
@DavideGiraudo : I have done the problem but i am not satisfied myself with my own solution, because i am not able to argue myself why i did so . Can you see and give me some explanation. Thank you :) –  Theorem Dec 17 '12 at 20:08

1 Answer 1

up vote 1 down vote accepted

Replace the step where i am not able to argue myself satisfactorily why by $$ p\int_0^\infty \int_{\{x:|u(x)| > t \}} t^{p-1}\mathrm dL^n(x) \mathrm dt = \int_X\int_0^{|u(x)|} pt^{p-1}\mathrm dt\mathrm dL^n(x) = \int_X|u(x)|^{p}\mathrm dL^n(x). $$

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I am still not satisfied with my understanding about how u did the first step ie change of limits of integration. –  Theorem Dec 18 '12 at 12:34
    
Tonelli for the integral of the function $(x,t)\mapsto pt^{p-1}\mathbf 1_{|u(x)|\gt t}$ with respect to the measure $L^n\otimes\mathrm{Leb}$ on $X\times(0,+\infty)$. –  Did Dec 18 '12 at 13:00

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