I am confused about conditional expectations. Let $(\Omega, \mathcal{F},P)$ be a probability space. Next, let $X$ and $Y$ be random variables on this space. Next, let $E[X|\sigma(Y)]$ be conditional expectation, which is $\sigma(Y)$-measurable random variable, and, therefore, also $\mathcal{F}$-measurable.
The question: can we define joint distribution of $E[X|\sigma(Y)]$, $Y$ and $X$?
Ok, for example, $X$ and $Y$ are jointly Normal. What would be a distribution of $(E[X|\sigma(Y)], X, Y)$?
Suppose $X$ and $Y$ are jointly normal; then
$$Z = E(X | \sigma (Y)) = \mu_X + \frac{\sigma_X}{\sigma_Y}\rho (Y - \mu_Y)$$
which is readily seen to be normally distributed as well, and the vector $(Z,X,Y)$ is jointly Gaussian. It's a simple exercise from here to compute its mean and covariance.