Show that $X \in L^p$

Question

Let $X$ and $Y$ be independent variables and $p \geq 1$. Show that $X + Y \in L^p$ $\implies$ $X, Y \in L^p$. I tried using the inequalities $$|X + Y|^p \leq 2^{p}(|X|^p + |Y|^p),$$ and $$|X|^p \leq 2^{p}(|X + Y|^p + |Y|^p),$$ but I didn't get anything useful.

Answer 1

This is proved using Fubini's Theorem. We have $\int(E|x+Y|^{p})dF_X(x) <\infty$ and this implies that $E|x+Y|^{p} <\infty$ for at least one $x$. ( In fact for almost all $x$ w.r.t. $F_X$!). Now you can see that $E|Y|^{p} <\infty$? from the inequality $|Y|^{p} \leq 2^{p} (|x+Y|^{p}+|x|^{p})$.