Sufficiency and completeness for classes of probability distributions generated by a single probability distribution (Q5956182)

The aim of the paper is to show that under finiteness assumptions it is possible to represent conditional expectations explicitly. Let \((\Omega, A,P_0)\) be a probability space, let \(B\) be a finite sub-\(\sigma\)-algebra of \(A\), and consider \({\mathbf P}(P_0)\) the set of all probability distributions of the conditional expectations restricted to \(B\) but well-defined on \(A\). Then \(B\) is proved to be sufficient and (totally) complete for \({\mathbf P}(P_0)\) and an element \(P\in{\mathbf P}(P_0)\) is an extremal point if and only if its conditional restriction to \(B\) is \(\{0,1\}\)-valued.