Consider two binary random variables $G,Z$ and a continuous random variable $\eta$. Assume that $$ \begin{aligned} & (A) \quad E(\eta|Z=1)=E(\eta|Z=0)=0\\ & (B) \quad Pr(G=1| Z=1, \eta)=1 \quad \text{a.s.} \end{aligned} $$
Assuming also that the following expectations exist, which of them must be $0$? \begin{aligned} &(1)\quad E(\eta G| Z=1)\\ &(2) \quad E(\eta G Z)=0\\ &(3) \quad E(\eta Z| G=1)\\ & (4) \quad E(\eta| Z=1, G=1)\\ \end{aligned}
MY ATTEMPTS (The question has been revised thanks to the comments below)
For simplicity of notation, I assume that $\eta$ is discrete.
(1): Let $\mathcal{M}_1\equiv \{a\in \mathbb{R}: \Pr(\eta=a|Z=1)>0\}$.
We have that $$ \begin{aligned} E(\eta G |Z=1)& =\sum_{a\in \mathcal{M}_1} a \Pr(\eta=a, G=1|Z=1)\\ & =\sum_{a\in \mathcal{M}_1} a \underbrace{\Pr(G=1|\eta=a, Z=1)}_{=1}\times \underbrace{\Pr(\eta=a|Z=1)}_{\text{$>0$ because $a\in \mathcal{M}_1$}}\\ &=E(\eta|Z=1)=0 \end{aligned} $$
(2): We have that
$$ \begin{aligned} E(\eta G Z)& = E(E(\eta G Z|Z))\\ &=E(\eta G Z|Z=1)\Pr(Z=1)+E(\eta G Z|Z=0)\Pr(Z=0)\\ &=1\times E(\eta G |Z=1)\Pr(Z=1)+0\times E(\eta G |Z=0)\Pr(Z=0)\\ &=E(\eta G |Z=1)\Pr(Z=1)\\ &\overbrace{=}^{(1)}0 \end{aligned} $$
I'm not sure (3) and (4) hold.