Prove that $\mathbb{E}|X-Y|\geq \mathbb{E}|Y|$ if $X$ and $Y$ are independent and $\mathbb{E}{X} = 0$

Question

Given that $X$ and $Y$ are independent random variables and $\mathbb{E}{X} = 0$, how to prove that this inequality $$\mathbb{E}|X-Y|\geq \mathbb{E}|Y|$$ holds?

Surely, it's true that $\mathbb{E}|X-Y|\geq \mathbb{E}|X| - \mathbb{E}|Y|$, but somehow the independence and $\mathbb{E}{X} = 0$ condition should sharpen this lower bound. However, I have no idea, how to use these conditions.

Any help would be appreciated.

Answer 1

In the original question, $E|X|$ should be replaced by $E|Y|$. Independence of $X$ and $Y$ means that $E|X-Y|=\int E[|X-y|] P_Y(dy)\geq \int |E(X-y)|P_Y(dy)=\int|y|P_Y(dy)=E|Y|$