Why is $E[X\mid \mathcal{G}]=E[X\mid \sigma(\mathcal{G},\mathcal{H})]$ when $\mathcal{H}$ is indepedent of $\sigma(\sigma(X),\mathcal{G})$
I tried to use the definition of conditional expectation but I couldnt use the assumption of independence in a proper way which might help me finish the proof. Any hints to get me started?