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.