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I want to prove:

$$E(X|\sigma (G_1,G_2))=E(X|G_1)$$

when $G_2,\sigma(\sigma(X),G_1)$ are independent. It's quite elusive expression to unpack, what would you say is the meaning? Previously I have proven that:

$L=\{A\in F| E(I_AX)=E(I_AE(X|G_1)\}$ is a $\lambda$-system and contains the set: $\{ A\cap B| A\in G_1,B\in G_2\}$. Note that $F$ is the simga algebra of the space which $G_1,G_2$ are subsets of.

How to continue from here?

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