If the sum of two independent random variables is $ L^{p} $, does it imply that each is $ L^{p} $?

Question

Let $ X $ and $ Y $ be two independent random variables, i.e., $$ \forall a,b \in \Bbb{R}: \quad \textbf{Pr}(X < a,Y < b) = \textbf{Pr}(X < a) ~ \textbf{Pr}(Y < b). $$ Let $ p > 0 $ (not necessarily $ > 1 $). If $ \Bbb{E}[|X + Y|^{p}] < \infty $, how can we prove that $ \Bbb{E}[|X|^{p}] < \infty $?

Answer 1

There is a constant $C_p\in(0,\infty)$ such that if $a,b\ge 0$ then $(a+b)^p\le C_p(a^p+b^p)$. If $E|X+Y|^p<\infty$ then (Fubini) $E|X+y|^p<\infty$ for all $y\in G$ where $P(Y\in G)=1$. Fix one such $y$. Then $$ E|X|^p\le E|X+y-y|^p\le E(|X+y|+|y|)^p\le C_p(E|X+y|^p+|y|^p)<\infty. $$