Let $\Omega = [0,1] \times [0,1]$ and let $\mathcal {B}([0,1] \times [0,1])$ be our sigma algebra.
Define two random variables: $\xi(x, y) = x \ $ and $ \ \eta (x,y) =y$.
Find $\mathbb{E}(\xi | \eta)$ and $\mathbb{E}(\xi - \eta \ | \ \sigma (\xi + \eta))$.
My main problem is finding the densities of $\xi$ and $\eta$.
The variables are independent, is that correct?
Here is what I have so far:
$$\xi ^{-1}([a,b]) = \{ [a,b] \times [c,d]: \ \ c, d \in [0,1] \} $$ and it seems to me that the density of $\xi$ and $\eta$ is $1$ on $[0,1] \times [0,1]$.
Could you tell me if I am right?