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Timeline for Inverse marginal property of a collection of $\sigma$-algebras

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Sep 25, 2021 at 17:05 comment added Pietro Majer So if this property has been used somewhere, I think the definition is more likely to be given in the geometric form, that does not require introducing new terms.
Sep 25, 2021 at 16:58 comment added Pietro Majer I think this condition implies that for $i<j$ one has $\mathcal F_i\cap \mathcal F_j$ only consists of sets of measure either $0$ or $1$ (because if $A\in \mathcal F_i\cap \mathcal F_j $ one can consider $\xi_i:=\chi_A-\mathbb P(A)$ and $\xi_k=0$ for all $k\ne i$, and deduce $0=\mathbb E \xi_i^2=\mathbb P(A)(1-\mathbb P(A)) $. From this one gets $$\oplus_kL^2(\mathcal F_k)=L^2(\sigma(\cup_k \mathcal F_k)). $$
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May 3, 2020 at 4:34 comment added S. Carnahan♦ Please choose between "article" and "paper" once and for all, and stop making these needless changes.
May 2, 2020 at 23:56 history edited Ivan Feshchenko CC BY-SA 4.0
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Apr 30, 2020 at 21:15 comment added Ivan Feshchenko No, IMP is not equivalent 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).
Apr 30, 2020 at 20:12 comment added RaphaelB4 Just that the vector space generated by the variable $\mathcal{F}_i$ measurable are in direct sum : $\forall \xi_1,\cdots,\xi_n\in L^2(\Omega,\mathcal{F_1},\mu)\times...\times L^2(\Omega,\mathcal{F_n},\mu)$. $\xi_1+\cdots +\xi_n=0\Rightarrow \xi_1=0,\cdots,\xi_n=0$.
Apr 30, 2020 at 18:48 comment added Ivan Feshchenko Unfortunately, I do not understand your question. Please specify the question.
Apr 30, 2020 at 15:38 comment added RaphaelB4 If one restrict to $\xi$ such that $\mathbb{E}(\xi)=0$. Is your IMP equivalent to $$ L^2(\Omega,\mathcal{F}_1,\mu)\oplus L^2(\Omega,\mathcal{F}_2,\mu)\oplus\cdots \oplus L^2(\Omega,\mathcal{F}_n,\mu)$$?
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S Apr 27, 2020 at 19:27 history bounty started Ivan Feshchenko
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