Let $f\in L^p(\mathbb{R}^d)$ for $p\in [1,\infty)$. Show that $$\|f\|_p=\left(\int^{\infty}_{0}pa^{p-1}m\{|f|>a\} \right)^{1/p}$$
My attempt:
We can use Fubini's theorem in the following way: \begin{align*} \int^{\infty}_{0}pa^{p-1}m(\{|f|>a\})da&= \int^{\infty}_{0}pa^{p-1}\left(\int_{\mathbb{R}^d}1_{|f|>a}(x)dx\right)da \\ &= \int_{\mathbb{R}^d}\int^{\infty}_{0}pa^{p-1}1_{\{|f|>a\}}(x)dadx \\ &= \int_{\mathbb{R}^d}\left(\int^{|f(x)|}_{0}pa^{p-1}da\right)dx \\ &= \int_{\mathbb{R}^d}|f|^p \end{align*} Then taking the $p$th root of both sides gives us the desired inequality.
My main concern is if changing the bounds as I did is correct. Any help/critique is much appreciated!