$X,Y,Z$ are all random variables on the same probability space.
I think the title is false in general, but direction of a proof would be helpful. I also have some similar questions:
$$ \text{If } \mathbb{E}[Y \mid X] = 0 \ \& \ \mathbb{E}[Y \mid Z] = 0 \Rightarrow \mathbb{E}[Y \mid X, Z ] = 0 $$
Let $\beta$ be a constant,
$$\mathbb{E}[Y \mid \beta X] = \mathbb{E}[Y \mid X]$$
(not $=\beta\mathbb{E}[Y \mid X]$)
Lastly, is $\mathbb{E}[Y\mid X = \mu]$ the same thing as $\mathbb{E}[Y\mid X]$? If not, what is different between saying the expected value of $Y$ given a realization of $X$ oppose to the expected value of $Y$, given the random variable $X$?
While I'm at it, any suggestions of (somewhat rigorious) books that cover expectation well is appreciated. I am trying to get a very good understanding of it on a fundamental level. I have analysis texts, but something geared more toward just probability space may be better for me.