This is an exercise of Shiryaev's Probability on page 233:
Let $\xi$ and $\eta$ be independent identically distributed random variables with $E\xi$ defined. Show that $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2} \text{(a.s.)}$
How can I say that for any $A\in\sigma(\xi+\eta)$ we have $E\xi I_A=E\eta I_A$. Thank you!