Proposition: Finite backward observation construction

Fix terminal observation spaces ${\mathcal{O}}_{T,g}$. Define backwards

These are observation spaces. They are the smallest family containing the terminal spaces and satisfying (eq:time-branch-inclusion).

Agreement on the initial direct sum is equivalent to agreement of all complete length-$T$ history-conditioned terminal expectations. It consequently implies agreement of all shorter history laws.

Proof. Adjoints of completely positive maps preserve adjoints. Hence the displayed span is adjoint closed and contains $I$. Any family satisfying the inclusions must contain the indicated span at each backward step, proving minimality by induction.

The pullbacks of all length-$T$ history-conditioned terminal observables span the recursively constructed spaces. The inserted identity adds no independent expectation: at each step it equals ${\sum }_h{{\Psi }^{(t)\ast }}_{h|g}(I)$. Thus agreement of all those expectations is exactly agreement on the initial spaces. Summing extensions of a shorter history gives its probability.