Given that $\mathcal{G}$ is a sub-sigma field. $Z=\mathbb E(X|\mathcal{G})$, how can we show that $X$ is independent of $\mathcal{G}$ given $Z$?
I am struggling about the interpretation of this result.
By definition, we only need to show that given $A\in \sigma(X)$, $B\in \mathcal{G}$ that $\mathbb P(AB|\sigma(Z))=\mathbb P(A|\sigma(Z))\mathbb P(B|\sigma(Z))$ , but I don't know how to deal with it then..