Conditional expectation, cartesian product

Question

Let $\Omega=X\times Y$ be a sample space of pairs $(x,y)$.

Let $\zeta$ be a random variable on $\Omega$.

How would you interpret notation $\mathbb{E}(\zeta \mid x)$? Does this have any sense? I've never come across this. Any ideas?

Answer 1

This is what I think they mean. The random variable $\zeta$ is a function $X\times Y\rightarrow \mathbb{R}$. Given that we choose a certain $x\in X$ we have the function $\zeta|_{x\times Y}:x\times Y\rightarrow \mathbb{R}$. I guess they mean the expectation of this random variable (You might need to rescale to actually make it a random variable).