Expected value inequality with zero-mean variable

Question

Let $X,Y$ be independent real valued random variables. In addition $\mathbb{E}[Y] = 0$ and $p\geq 1$. Then $$ \mathbb{E}[ |X|^p] \leq \mathbb{E}[ |X+Y|^p] $$ How can I prove this? I tried to use Jensen $$ \mathbb{E}[ |X+Y|^p] \geq |\mathbb{E}[X+Y]|^p = |\mathbb{E}[X]|^p $$ but I don't know how to incorporate independency.

Answer 1

\begin{align*} E|X+Y|^{p}&=\int_{{\bf{R}}\times{\bf{R}}}|u+v|^{p}d\mu_{(X,Y)}(u,v)\\ &=\int_{{\bf{R}}\times{\bf{R}}}|u+v|^{p}d\mu_{Y}(v)d\mu_{X}(u)\\ &\geq\int_{{\bf{R}}}\left|\int_{{\bf{R}}}vd\mu_{Y}(v)+u\right|^{p}d\mu_{X}(u)\\ &=\int_{{\bf{R}}}\left|E(Y)+u\right|^{p}d\mu_{X}(u)\\ &=\int_{{\bf{R}}}|u|^{p}d\mu_{X}(u)\\ &=E|X|^{p}. \end{align*}