Law of total expectation
= Law of total expectation
{title2=$\mathbb E[\mathbb E[X\mid\mathcal G]]=\mathbb E[X]$}
{wiki}
Taking expectation after conditioning recovers the original expectation. More generally, if $\mathcal H\subseteq\mathcal G$, then
$$
\mathbb E[\mathbb E[X\mid\mathcal G]\mid\mathcal H]
=\mathbb E[X\mid\mathcal H].
$$
= Tower property of conditional expectation
{synonym}
= Law of iterated expectation
{synonym}