in R. Moser's book "partial regularity for harmonic maps and related problems" there is the following equation (p. 47): $$ \int_{\Omega} u^pdx=\int_0^{\infty} \vert S(u^p,s)\vert ds=p\int_0^{\infty}s^{p-1} \vert S(u,s)\vert ds $$ where $u:\Omega\rightarrow [0,\infty)$ on a measurable set $\Omega$, for $s>0$, $S(u,s)=\left\lbrace x\in \Omega: u(x)>s\right\rbrace$ and $p\geq1$. The justification for the first equality is by Fubini's theorem. But i can't see that. Can anyone explain this, please? The second equality follows by the substitution formula.
Best regards.
Assuming that by $\lvert S(u^p, s)\rvert$ you are referring to the measure of $S(u^p, s)$ on $\Omega$, we have; \begin{align*} \int_0^\infty \lvert S(u^p, s)\rvert ds =\int_0^\infty \int_{\Omega} \chi_{\{u^p(x)>s\}}(x) dx ds &=\int_{\Omega} \int_0^\infty \chi_{\{u^p(x)>s\}}(x) ds dx\\ &=\int_{\Omega} u^p(x) dx \end{align*} The last step holds since $$ \int_0^\infty \chi_{\{u^p(x)>s\}}(x) ds = m\left(s\in [0, \infty) : s<u^p(x)\right) = m([0, u^p(x))) = u^p(x) $$