Proof of some statements involving conditional expectation

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $X$ and $Y$ be i.i.d. nonnegative random variables. Show if the following is true:

  1. $E(X|X+Y)=(X+Y)/2$
  2. $E(X|XY)=\sqrt{XY}$

My thoughts:

  1. Since $\sigma(X)$ is equal to $\sigma(Y)$ is equal to $\sigma(X+Y)$ and because of $\mathcal{G}\subset\mathcal{F}$ it follows that $E(X|\mathcal{G})=X$.

$$\int E(X|X+Y)d\mu) = \int E(X|X)d\mu) = \int E(X|Y)d\mu) = \left(\int E(X|Y)d\mu)+\int E(X|X)d\mu)\right)/2 = (X+Y)/2 $$

Is this correct so far? For the second point I am lacking an idea how to proof that. Any inspiration is welcome. Thanks!

2022-07-25 20:40:20
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