If $\mathbb E(|X+Y|^p) <\infty $ where $X$ and $Y$ are independent, how to prove $\mathbb E(|X|^p) < \infty$ and $\mathbb E(|Y|^p) < \infty$?

Question

Suppose $X$ and $Y$ are independent and $\mathbb E(|X+Y|^p) <\infty $ for some $p>0$, then how to prove $\mathbb E(|X|^p) < \infty$ and $\mathbb E(|Y|^p) < \infty$??

What I thought is that when $X, Y$ are independent, $$\mathbb E(|X+Y|^p) = \mathbb E(|X|^p) + \mathbb E(|Y|^p)+ \text{extra terms},$$ so if left side is finite then right side is also finite.

Answer 1

Your argument is not valid. By Fubini's Theorem $E|X+y|^{p} <\infty$ for some $y$ and $|X|^{p} \leq 2^{p} (|X+y|^{2}+|y|^{p})$. Hence $E|X|^{p} <\infty$.