In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.

Let $(\Omega,\mathcal{F},\mu)$ be a probability space and $\mathcal{F}_1,...,\mathcal{F}_n$ be sub-$\sigma$-algebras of $\mathcal{F}$. We will say that the collection $\mathcal{F}_1,...,\mathcal{F}_n$ possesses the inverse marginal property (IMP) if for arbitrary random variables $\xi_1,...,\xi_n$ such that

(1) $\xi_k$ is $\mathcal{F}_k$-measurable, $k=1,2,...,n$;

(2) $E|\xi_k|^2<\infty$, $k=1,2,...,n$;

(3) $E\xi_1=E\xi_2=...=E\xi_n$,

there exists a random variable $\xi$ such that $E|\xi|^2<\infty$ and $E(\xi|\mathcal{F}_k)=\xi_k$ for all $k=1,2,...,n$.

The simplest example of a collection of sub-$\sigma$-algebras which possesses the IMP is a system of pairwise independent sub-$\sigma$-algebras. In this case a needed random variable $\xi$ can be defined by $\xi:=\xi_1+...+\xi_n-(n-1)a$, where $a:=E\xi_1=E\xi_2=...=E\xi_n$.

**Question**: is the IMP a new notion or it is well-known?
Have you seen this property or something similar in the literature?

I will be very grateful for any comments on the IMP.

is notequivalent to the linear independence of the marginal subspaces $L^2_0(\mathcal{F}_1),...,L^2_0(\mathcal{F}_n)$. One can show that a collection of $\sigma$-algebras $\mathcal{F}_1,...,\mathcal{F}_n$ possesses the IMP if and only if the marginal subspaces $L^2_0(\mathcal{F}_1),...,L^2_0(\mathcal{F}_n)$ are linearly independent and their sum is closed in $L^2(\mathcal{F})$. For details see my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" (available on ArXiv). $\endgroup$2more comments