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 .