This is an exercise of conditional expectations:
Let $Y$ be an integrable random variable on the space $(\Omega,{\mathcal A},{\bf P})$ and $\mathcal{G}$ be a sub $\sigma$-algebra of $\mathcal{A}$. Show that $|Y|\leq c$ implies $|E[Y\mid{\mathcal G}]|\leq c$.
With Jensen's inequality, one immediately has $|E[Y\mid{\mathcal G}]|\leq E(|Y|\mid{\mathcal G})$.
I am trying to show that $E[|Y|\mid{\mathcal G}]\leq |Y|$, which is not necessarily true though. If $Y$ is $\mathcal{G}$-measurable, then $E[|Y|\mid {\mathcal G}]= |Y|$. But I have no idea for the general case.