Consider $\Omega = [0,1] \times [0,1]$ with sigma algebra of borel sets on $[0,1]^2$. Let $P$ be the Lebesgue measure on $\Omega$. Let $$\xi(x, y) = x, \ \ \ \eta(x,y) = y.$$
How can I find $\mathbb{E}(\xi - \eta| \xi +\eta)$?
I tried to find out what $\sigma( \xi + \eta)$ and I've found that $$(\xi + \eta)^{-1}(a) = \{ (x, -a+x) : x \in \mathbb{R}\}$$ so it is a line.
But how do I do it in terms of general borel sets?
Could you help me with that?