We show that under these assumptions, $Z$ itself is a version of the conditional expectation $\mathbb{E}(Z \mid \mathcal{G})$ by confirming that the required properties hold. We have assumed that $Z$ is measurable in $\mathcal{G}$ which takes care of the the first condition. Next, fixing an event $G \in \mathcal{G}$ and applying the results

(i)	R2150: Expectation of conditional expectation for a random euclidean real number
(ii)	R2549: Conditional expectation of random complex product when factor is known

we have \begin{equation} \begin{split} \mathbb{E}(Z I_G) = \mathbb{E}(\mathbb{E}(Z I_G \mid \mathcal{G})) = \mathbb{E}(\mathbb{E}(Z \mid \mathcal{G}) I_G) \end{split} \end{equation} This completes the proof. $\square$