Let X, Y be idependent, identically distributed random variables with
$${P}(X=1) = {P}(Y=1) = p$$ $${P}(X=-1) = {P}(Y=-1) = 1-p$$ and set $$Z=\mathbf{1}_{\{X+Y=0\}}$$ $$\mathcal{G} = \sigma(Z)$$
Compute $\mathbb{E}[X|\mathcal{G}]$ and $\mathbb{E}[Y|\mathcal{G}]$. Are these random variables still independent?
I am confused because, I dont know on which space X and Y are on. Also because all examples I have seen just use a random variable as condition, now it is the generated sigma algebra of a RV and I dont know what $\mathbb{E}[-|\mathcal{G}]$ really means and how I can compute it.
We have $\mathcal{G} = \sigma(Z) = \{\emptyset, Z, Z^c,\Omega \}$. Here $\Omega = \Omega_X\times\Omega_Y$, i.e the direct product of the sample spaces corresponding to the two r.v.'s is your actual sample space $\Omega$.
To be more explicit $$\Omega = \{\{X=Y=1\},\{X=Y=-1\},\{X=-1,Y=1\},\{X=1,Y=-1\}\}$$
You can try computing the probability of each of these sets, and then tackle $E[\ \cdot|\sigma(Z)]$, which is basically computing $E[\ \cdot|A]$ for each $ A\in\sigma(Z)$, where $$Z=\{X=-1,Y=1\}\cup\{X=1,Y=-1\}$$