Proving $E(|X| \mid \mathscr{F})\geqslant |E(X \mid \mathscr{F})|$

Question

Let $(\Omega,\mathscr{A},P)$ be a probability space and $X\in\mathscr{L}_1(\Omega,\mathscr{A},P)$ a random variable on that space. Let $\mathscr{F}\subset\mathscr{A}$ be a sigma-algebra. Prove $E(|X| \mid \mathscr{F})\geqslant |E(X \mid \mathscr{F})|$, using linearity and monotonicity of the conditional expectation.

Consider $|X|=X^+ -X^-$

Then $E(|X| \mid \mathscr{F})=E(X^+ -X^- \mid \mathscr{F})=E(X^+ \mid \mathscr{F})-E(X^- \mid \mathscr{F})=?$

I do not know how to proceed. I should use monotonicity, but I do not see how.

Question:

How do I solve the problem?

Thanks in advance!

Answer 1

Hint: Use $X \leq |X|$ and $-X \leq |X|$ to prove that $$\mathbb{E}(X \mid \mathcal{F}) \leq \mathbb{E}(|X| \mid \mathcal{F})$$ and $$-\mathbb{E}(X \mid \mathcal{F}) \leq \mathbb{E}(|X| \mid \mathcal{F}).$$