I have been looking at the version of the tower property in this image.
I'm struggling to derive the version of the tower property that I'm more familiar with from this version.
How does the above version imply that $E[E[X|F]]=E[X]$? What is the sub-$\sigma$-algebra $G$ in this case?
In $E[X]$, all you know of the event $X$ is that it could be any event, so it means $X$ is given only under the knowledge that the entire space $\Omega$ has probability $1$ and nothing else. Hence the $\mathcal{G}$ in your example is given by $\{ \emptyset, \Omega\}$.