$E[X] = 0 \implies |E[XI_{\{|X|>n\}}]| = |E[XI_{\{|X|\leq n\}}]|$

Question

Let $X: \Omega \to \mathbb{R}$ be a random variable. In a proof I'm reading, they use that:

$E[X] = 0 \implies |E[XI_{\{|X|>n\}}]| = |E[XI_{\{|X|\leq n\}}]|$

Here, $I_A$ is the indicator function on the Borel set $A$

Does anyone have an idea why this would be true?

I might be missing something trivial here.

Answer 1

$E[XI_{\{|X|>n\}}]=-E[XI_{\{|X|\leq n\}}]$ (because if $a+b=0$ then $a =-b$).