Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$ [duplicate]
Solution 1:
Let $$ \chi_W (x,t) = \{ (x,t) | 0< t< |f(x)| \}$$
Then
$$\int_X |f|^p d\mu= \int_X \int_{[0,|f(x)| ]} pt^{p-1} dt d\mu $$ $$ = \int_X \int_{\mathbb{R}} pt^{p-1} \chi_W (x,t) dt d\mu $$ $$ =\int_{\mathbb{R}}\int_X pt^{p-1} \chi_W (x,t) d\mu dt $$ $$ = p\int_{[0,\infty]} t^{p-1} \mu ( \{ x| |f(x)| > t \} ) dt $$