Q:

Compute $E[X|\sigma (\mathbf 1_{X+Y=0})]$ for $X,Y$ iid Bernoulli

Let $X, Y$ be iid Bernoulli distributed with parameter $p\in[0,1]$. Define $Z=\mathbf 1_{X+Y=0}$ and $\mathcal G=\sigma (Z)$
Compute $E[X|\mathcal G]$ ( and $E[Y|\mathcal G]$ - are they independent?)

So we want $E[X|\sigma (\mathbf 1_{X+Y=0})]$ and
of course we have $\mathbb P[Z=1]=(1-p)^2 $ and $\mathbb P[Z=0]=1-(1-p)^2$.
But I can't really see the next rigorous step and struggle because one random variable is  discrete and one is continuous.

A:

If $X,Y\overset{iid}\sim\mathcal B(p)$ and  $Z=\mathbf 1_{X+Y=0}$, then:
$$\mathsf E(X\mid \sigma(Z)) {~=~ \mathsf P(X{=}1\mid Z{=}1)\cdot\mathbf 1_{Z=1} + \mathsf P(X{=}1\mid Z{=}0)\cdot\mathbf 1_{Z=0}\\~=~ \mathsf P(X{=}1\mid X{+}Y{=}0)\cdot\mathbf 1_{Z=1} + \mathsf P(X{=}1\mid X{+}Y{\neq}0)\cdot\mathbf 1_{Z=0} \\ ~\ddots}$$

