My question is concerned with some notation I came across and can't seem to figure out. Let me give you the background:
Let $(\Omega, \mathcal{F}, P)$ be our probability space, $X, Y, Z : \Omega \rightarrow \mathbb{R}$ random variables, where $X$ is integrable. Then, we define the conditional expectation of $X$ on $Y, Z$, written $\mathbb{E}(X \, | \, Y, Z)$, as the random variable (defined on $\Omega$ with codomain $\mathbb{R}$) such that
$\mathbb{E}(X \, | \, Y, Z)$ is $\sigma(Y, Z)$ - measurable;
For every $A \in \sigma(Y, Z)$,
\begin{equation*} \int_A \mathbb{E}(X \, | \, Y, Z) \; dP = \int_A X \; dP. \end{equation*}
Notice that the Doob - Dynkin factorization lemma implies that, we can find a function $f : \mathbb{R} \rightarrow \mathbb{R}^2$ such that $\mathbb{E}(X \, | \, Y, Z) = f \circ (Y, Z)$, where $(Y, Z)$ is the random vector with coordinates $Y$ and $Z$. It is standard to write $\mathbb{E}(X \, | \, Y = y, Z = z)$ for the function $f$.
With these definitions in mind my question is: what does the notation $\mathbb{E}(X \, | \, Y, Z = z)$ mean?
As a follow up: Does anyone have a good reference for these concepts? I have good measure theory books, but none dive into the concept I'm asking.
I think you mean that $\mathbb{E}\left(X \, \middle|\, Y,Z\right)$ is $\sigma(Y,Z)$ - measurable, not $\sigma(X,Y)$ - measurable.
The notation $\mathbb{E}\left(X \, \middle|\, Y,Z=z\right)$ or $f(Y,z)$ is just a random variable which depends on $z$ and is a function of $Y$ (i.e. $\sigma(Y)$ - measurable).
For further reference, have a look at any of
Measure theory books may discuss this stuff but usually not in as much details, seeing as conditional expectation is more of an application of a measure theoretic concept (the Lebesgue-Radon-Nikodym derivative).