Suppose that we have distributions $F_1 $ and $F_2$. Under what conditions on $F_1,F_2$ is it possible to construct random variables $X\sim F_1,Y\sim F_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ is the conditional expectation of $X$ with respect to some $\sigma$-field $\mathscr{G}$?
If $X∼F1,Y∼F2$, what conditions on $F1,F2$ are needed s.t. we can construct $Y=E(X|\mathscr{G})$ for some $\mathscr{G}$?
Anthony Lee Zhang
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