Double Expectation of a Conditional Expectation If there are two sub-sigma algebras $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$, neither a subset of the other from a probability space $(Y,\mathcal{F},P)$ and a random variable $X$ which is not measurable with respect to either $\mathcal{G}$ or $\mathcal{H}$, can I apply double expectation on the conditional expectation of $X|\mathcal{G}$ like this:
$$
E[E[X|\mathcal{G}]|\mathcal{H}] = E[X|\mathcal{G}]
$$
Thanks.

The property doesn't hold when $\Omega=\{a,b,c\}$, $\cal F:= 2^\Omega$, $\cal G:=\{\emptyset,\{a,b\},\{c\},\Omega\}$, $\cal H:=\{\emptyset,\{a\},\{b,c\},\Omega\}$ and $X:=\chi_{\{b\}}$.
