Prove that $\int_Xf^p d\mu=\lim_{t\to \infty}\int_{0}^{t}p \alpha^{p-1}\lambda_f(\alpha)d\alpha$ where $\lambda_f(\alpha)=\mu({x:f(x) > \alpha})$
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$