In my textbook, there is a theorem
Let $(X,\mu)$ be a measurable theorem.
For $f$ in $L^p(X,\mu )$, $0 < p < \infty$, we have $$ ||f||_{L^p}^p = p \int\limits_{0}^{\infty} \alpha^{p-1} d_f(\alpha ) \mathrm{d}\alpha . $$
Where $||.||_{L^p}^p$ is the ordinary $L^p$ norm and $d_f$ is the ditribution function.
The first step, they did $$ p \int\limits_{0}^{\infty} \alpha^{p-1} d_f(\alpha ) \mathrm{d}\alpha = p \int\limits_{0}^{\infty} \alpha^{p-1} \int_{X} \chi_{[|f| > \alpha]} \mathrm{d} \mu (x) \mathrm{d} \alpha$$ which is normal, as we just apply the definition of a distribution function.
But in the next step, they said they applied thẻ Fubini's theorem and obtain the following $$\int_X \int\limits_0^{|f(x)|} p \alpha ^{p-1} \mathrm{d} \alpha \mathrm{d} \mu (x) $$ which doesn't make sense to me.
As far as I know, the Fubini's theorem states that if $f$ is integrable on the product measure space then the order of integrating can be changed.
But how did it change the interval $[0;+\infty )$ into $[0; |f(x)|]$ and not $[|f(x)|, +\infty )$?
Please give me a clarification. Thank you.