I have the following question: Let $(X, \mathcal M, \mu)$ be a measure space, and let $f \in L^p(X,\mu)$ be a nonnegative, real-valued function. Let $\lambda_f: [0,\infty) \to [0,\infty)$ be the distribution function of $f$, defined by $$\lambda_f(\alpha) = \mu \left\{ x : f(x)> \alpha \right\}.$$
(i) First assume $p=1$. Show that the integral of $f$ is equal to $$\int_{0}^{\infty} \lambda_f(\alpha) d\alpha.$$ Hint: consider the product measure space $(X \times \mathbb R, \mathcal M \times \mathcal L, \mu \times \lambda)$ where ($\mathbb R, \mathcal L, \lambda$) is the real line with the $\sigma$-algebra of Lebesgue measurable sets and the Lebesgue measure. Relate the integral of $f$ to the set $$\left\{ (x, \alpha) \in X \times \mathbb R : 0 \leq \alpha \leq f(x) \right\}$$ and use Fubini's theorem.
(ii) In a similar fashion, show that for $1 \leq p < \infty$ $$\| f \|^p_p = \int_0^\infty p \alpha^{p-1} \lambda_f(\alpha) d \alpha.$$
Attempt: I know for part (i) we're aiming for something resembling $$\int_{X \times \mathbb R} \boldsymbol{1}_{f(x)>\alpha} d\mu(x)d\lambda(\alpha),$$ then we express this as a double integral, use Fubini, and the result follows. However I can't seem to start off, and specifically am having trouble relating the integral of $f$ to the set mentioned above. Any help would be appreciated. Thanks.
Hint: $$\int_0^{\infty} p \alpha^{p-1} 1_{\{f(x)>\alpha\}}d\alpha = \int_0^{f(x)} p\alpha^{p-1} d\alpha = f(x)^p$$