I posted this question on MSE here but was encouraged to ask here instead. I sincerely hope that it is not inappropriate for this board.

Let $ (\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Consider the space $ V=L^2(\Omega,\mathcal{F},\mathbb{P})$ . For $ \chi\in V$ , let $ \mathcal{G}=\sigma(\chi)\subset\mathcal{F}$ . Then we have a chain of subspaces $ $ U=\mathrm{span}(\chi)\subset W=L^2(\Omega,\mathcal{G},\mathbb{P})\subset V.$ $

I am interested in the relationship between $ U$ and $ W$ . More specifically, I am looking for criteria for when an operator vanishing on $ U$ implies that it already vanishes on $ W$ . I certainly know that this is not generally the case but I was wondering if there might exist any piece of theory about anything related to this. However, I am not sure what key words to look for.

I apologize for the rather unspecific question.