Show that $p\mapsto ||X||_p$ is nondecreasing for $p\in (0,\infty]$

Question

Why is this function nondecreasing for $p\in (0,\infty]$, where $X$ is a random variable I suspect that I'm supposed to use the Monotone Convergence Theorem. That is, let $Y_n$ be a sequence of random variables such that $Y_n=\sup\{\text{random varible $Y$}|Y\leq X\text{ and }|Y|\leq n\}$. Then by monotone convergence theorem, $Y_n\to X$. So then we need to show that $p\mapsto ||Y||_p$ is nondecreasing. But that doesn't seem to help...

Answer 1

Monotone Convergence Theorem can be used to prove this if you already know the result for simple functions. An easier way is to use Holder's inequality: let $p<q$. Then $E|X|^{p} \leq (E|X|^{pr})^{1/r} )({E1^s})^{1/s}$ where $r=\frac q p$ and $s=\frac q {q-p}$. Just raise both sides to power $\frac 1 p$ and you get $(E|X|^{p})^{1/p} \leq (E|X|^{q})^{1/q}$.