This is probably a very easy question, but I'm failing to understand it. Given a function $f \in L^{p}(\mathbb{R}^n)$, $1<p\leq \infty$, we define the uncentered maximal function of $f$ as $$ \tilde{M}f(x) = \sup_{x\in B}\frac{1}{\mu(B)}\int_{B}\left| f(y)\right|d\mu(y), $$
where the supremum is taken over all the balls that contain $x$, and $\mu$ can be the Lebesgue measure. Then, Stein claims (it's from the book "Harmonic Analysis, Real Variable Theory") that $$ \int_{\mathbb{R}^n}\left(\tilde{M}f\right)^{p}d\mu = p\int_{0}^{\infty}\mu\left( \{x\in \mathbb{R}^n; \tilde{M}f(x) > \alpha \}\right)\alpha^{p-1}d\alpha. $$
I can't prove this identity. If you make the variable change $\alpha^{p}=y$, then it becomes $$ \int_{\mathbb{R}^n}\left(\tilde{M}f\right)^{p}d\mu = \int_{0}^{\infty}\mu\left( \{x\in \mathbb{R}^n; \tilde{M}f(x)^p > y\}\right)dy, $$
but is this identity obvious? What am I missing? I appreciate any help!
Let $f \in L^p$. Then we have \begin{align*}p \int_0^\infty \alpha^{p-1} \mu(\{x \in X : \lvert f(x) \rvert > \alpha\}) \mathrm{d}\alpha &= p \int_0^\infty \alpha^{p-1} \int_X \chi_{\{x \in X : \lvert f(x) \rvert > \alpha\}} \mathrm{d}\mu \mathrm{d}\alpha \\ &= \int_X \int_0^\infty p \alpha^{p-1} \chi_{\{x \in X : \lvert f(x) \rvert > \alpha\}} \mathrm{d}\alpha \mathrm{d}\mu \\ &= \int_X \int_0^{\lvert f(x)\rvert} p \alpha^{p-1} \mathrm{d}\alpha \mathrm{d}\mu = \int_X \lvert f(x) \rvert^p \mathrm{d}\mu, \end{align*} where we used Fubini and $\chi_M$ is the characteristic function of the set $M$.