Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables.
I have a question comparing the conditional expectations $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$.
1.) Does $\mathbb{E}[Y|X=x]$ mean that $X$ is fixed to the value $x$? I.e., that it only varies with $Y$?
2.) From which space to which spaces do the two conditional expectations map, respectively?
3.) If $X$ and $Y$ are absolutely continuous real valued random variables and have a joint density $f_{X, Y}$ with respect to the (product) Lebesgue-measure $\lambda \otimes \lambda =\lambda^2$, that is, $\mathbb{P}_{X, Y}(B, C)=\int_{B \times C}f_{X, Y}(x, y)\lambda^2(dx, dy)=\int_B \int_C f_{X, Y}(x, y)\lambda(dx) \lambda(dy) $ with Borel sets $B$ and $C$. Then:
$$\mathbb{E}[Y|X=x] = \int y \, f_{Y|X=x} \lambda(dy) = \int y \, \frac{f_{X, Y}(x,y)}{f_X(x)}\lambda(dy)$$.
My question is now, how would that equation look like for $\mathbb{E}[Y|X]$? Does it make sense to write something like $f_{Y|X=X}(y)$ and $f_{X,Y}(X,y)$ and do these quantity even exist? I.e., should I write $$\mathbb{E}[Y|X] = \int y \, f_{Y|X=X} \lambda(dy) = \int y \, \frac{f_{X, Y}(X,y)}{f_X(X)}\lambda(dy)$$?
4.) Following the commments under the answer: Under which circumstances do the above formulations of the conditional expectation hold? And how do I compute the conditional expectation if these conditions do not hold, i.e., in the general case?
