Here is one of the properties of conditional expectations stated in "The Theory of Stochastic Processes I" by Gikhman and Skorohod (on page 33):
The proof relies on the fact that there is a sequence of functions of a certain form converging to $g(\zeta,\eta)$ a.s (and in $L^1$). However, I'm not sure why this claim is true.
Also, if for all $z$, $|\mathsf{E}[g(\zeta,z)\mid \mathcal{B}]|\le X$ a.s. for some r.v. $X$ which does not depend on $z$, does the property imply that $$ |\mathsf{E}[g(\zeta,\eta)\mid \mathcal{B}]|<X \quad\text{a.s.}? $$
(it seems that for the second inequality to hold, the first must hold w.p.1 for all $z$ simultaneously)
Property $(g)$ mentioned in the proof implies that $\mathsf{E}[\zeta\eta\mid \mathcal{B}]=\eta\mathsf{E}[\zeta\mid \mathcal{B}]$ a.s.