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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 $Read more