I'm struggling to understand the correct interpretation of conditional expectation of the form $ E[X \mid Y, Z ]. $ I know that $E[X \mid Y]$ is itself a random variable $f(y) = E[X \mid Y=y]$. Does this mean that the above is a random variable $g(Y,Z)$ where $g(y,z) = E[X \mid Y = y, Z = z]\ ?$
On the other hand, $E[X \mid Y,Z ]$ is nothing but $E[X \mid \sigma(Y,Z)]$. Clearly $\sigma(Y) \subseteq \sigma(Y,Z)$, so $E[X\mid Y,Z]$ is a constant when conditioning on some event $\{Y=y\} \in \sigma(Y)$, which seems to contradict the above interpretation as a r.v. depending on $Y$ and $Z$.
What am I missing here?