I have a bit of trouble understanding how I can work with moving that $X$ in and out of the expectation. Suppose $X$ and $Y$ are random variables in the same probability space.
Suppose $\mathbb{E}[Y\mid X] = 0$ (I don't think this actually matters for my question).
Question: $\mathbb{E}[Y\mid X] = \frac{\mathbb{E}[YX\mid X]}{X}$, yes? (If $X$ and $Y$ are in the same probability space, this is always true? We need no additional assumptions?)
Is it also true that:
$$\mathbb{E}[Y_i\mid X_1,\cdots,X_N] = \frac{\mathbb{E}[Y_i X_i\mid X_1,\cdots,X_N] }{X_i}$$
where $Y_i$ is a RV, and so is $X_i$. Some clarification of this will help my intuition.