Result R2150: Expectation of conditional expectation for a random euclidean real number shows that $\mathbb{E} (\mathbb{E}(X \mid \mathcal{G})) = \mathbb{E} X$. Thus, applying result R3582: Signed expectation finite iff absolutely integrable random basic number twice, we have the chain of equivalencies \begin{equation} \begin{split} \mathbb{E}|\mathbb{E}(X \mid \mathcal{G})| < \infty \quad & \iff \quad \mathbb{E} (\mathbb{E}(X \mid \mathcal{G})) = \mathbb{E} X \in (-\infty, \infty) \quad & \iff \quad \mathbb{E} |X| < \infty \end{split} \end{equation} We have assumed the rightmost expression and hence the proof is complete. $\square$