TITLE: Mapping $\sigma$-fields to $L^2$
QUESTION [4 upvotes]: Let $(\Omega,\mathcal A, P)$ be a probability space, and let $\Sigma$ be the set of sub $\sigma$-fields of $\mathcal A$, that is,
$$
\Sigma = \{\mathcal F \subset \mathcal A:\; \mathcal F \; \text{is a $\sigma$-field} \}.
$$
For $\mathcal F \in \Sigma$, let $L^2(\mathcal F)$ be the space of equivalence classes of random variables that are $\mathcal F$-measurable and have a finite second moment. We know that $L^2(\mathcal F)$ is closed linear subspace of $L^2(\mathcal A)$, the latter viewed as a Hilbert space with the usual $L^2$ norm.
Let $\text{Lin}(L^2(\mathcal A))$ be the set of all closed linear subspaces of $L^2(\mathcal A)$. 

Now consider the map $\mathcal F \mapsto L^2(\mathcal F)$. Is this map a bijection from $\Sigma$ to $\text{Lin}(L^2(\mathcal A))$?

REPLY [1 votes]: Assuming you can get access to it, look at theorem 1 in "On Relations Between Strict-Sense and Wide-Sense Conditional Expectations": https://epubs.siam.org/doi/10.1137/1102020
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