I have a random variable $X \in L^1(\Omega, F,P)$ and I am required to study "the family of random variables of the type $\mathbb{E}\left[X | G \right]$,where $G$ spans the set of all sub-$\sigma$ algebras of $F$". Could you tell me what it means ? I know how the conditional expectation is defined and I have read that "span" means "generate", in the context of vector spaces and linear combinations, but that sentence is not so clear to me.
Thank you very much for any help you can provide !