This inequality is actually exercise 11, chapter 3.3 in Kai Lai CHUNG's A Course in Probability Theory:

If $X$ and $Y$ are independent, $E(|X|^p)<\infty$ for some $p\ge1$, and $E(Y)=0$, then $E(|X+Y|^p)\ge E(|X|^p)$.

As I understand, Fubini's theorem could be used here: $$E(|X+Y|^p)=\iint |x+y|^p \mu^2(dx,dy)=\int\left[\int |x+y|^p \mu_y(dy)\right]\mu_x(dx)$$

Since $E(|X|^p)=\int |x|^p \mu_x(dx)$, it would be sufficient to prove: $$|x|^p \le \int |x+y|^p \mu_y(dy).$$

However, I don't know how to use the $E(Y)=0$ condition here. Did I use Fubini's theorem properly?

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Notice that $$\int|x+y|^p\mu_Y(\mathrm dy)\geqslant \left|\int(x+y)\mu_Y(\mathrm dy)\right|^p,$$ by Jensen's inequality.