p-norm inequality for two random variables I read the following result in a book, however I believe that there is a mistake in the proof. Do you know of any book that proves this result, or do you have an idea on how to prove it?
Let $X$ and $Y$ be two nonnegative random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $X \in L^p$ for some $p \in (1,\infty)$. If it holds that for every $\alpha >0$, $$\alpha \mathbb{P}(Y \geq \alpha) \leq \int_{\{Y \geq \alpha\}} X d \mathbb{P},$$
then $$|| Y ||_p \leq q ||X||_p,$$ where $p^{-1}+q^{-1} =1$.
 A: Let's first assume that $Y \in L^p$. Using the identity
$$\mathbb{E}(Z) = \int_{(0,\infty)} \mathbb{P}(Z \geq r) \, dr \tag{1}$$
which holds for any non-negative random variable $Z$, we find that
\begin{align*} \mathbb{E}(Y^p) &= \int_{(0,\infty)} \mathbb{P}(Y^p \geq r) ~ dr \\ &= p \int_{(0,\infty)} r^{p-1} \mathbb{P}(Y \geq r) \, dr. \end{align*}
It follows form the given inequality and Tonelli's theorem that
\begin{align*} \mathbb{E}(Y^p) &\leq p \int_{(0,\infty)} r^{p-2} \int_{\{Y \geq r\}} X \, d\mathbb{P} \, dr \\ &=p  \int_{\Omega} \int_{(0,\infty)} r^{p-2} 1_{[0,Y]}(r) \cdot X \, dr \, d\mathbb{P} \\ &= p \int_{\Omega} \int_0^Y r^{p-2} X \, dr \, d\mathbb{P} \\ &= \frac{p}{p-1} \int_{\Omega} Y^{p-1} X \, d\mathbb{P}. \end{align*}
Hence, by Hölder's inequality,
$$\mathbb{E}(Y^p) \leq \frac{p}{p-1} \|X\|_p (\mathbb{E}(Y^p))^{1-1/p}.$$
Rearranging this inequality, we conclude that $\|Y\|_p \leq q \|X\|_p$ with $q=p/(p-1)$.
In the general case, i.e. $Y \notin L^p$, we can apply the above reasoning to $\min\{Y,n\}$. Note that the inequality
$$r \mathbb{P}(Y \geq r) \leq \int_{\{Y \geq r\}} X \, d\mathbb{P}$$
remains valid if we replace $Y$ by $\min\{Y,n\}$.
Remark: The result is often used to prove Doob's maximal inequality for martingales. The idea of proof is taken from Measures, integrals and martingales by R. Schilling (he doesn't state the result independently, though).
