P3275
Since $\mathcal{G}$ is a sigma-algebra on $\Omega$, then $\Omega \in \mathcal{G}$. Since $\mathbb{E}(X \mid \mathcal{G})$ is the conditional expectation of $X$ given $\mathcal{G}$, then
\begin{equation}
\mathbb{E}(\mathbb{E}(X \mid \mathcal{G}) I_G)
= \mathbb{E}(X I_G)
\end{equation}
for all $G \in \mathcal{G}$. Thus, in particular, we have
\begin{equation}
\mathbb{E}(\mathbb{E}(X \mid \mathcal{G}))
= \mathbb{E}(\mathbb{E}(X \mid \mathcal{G}) I_{\Omega})
= \mathbb{E}(X I_{\Omega})
= \mathbb{E}(X)
\end{equation}
$\square$
