On the σ-Algebraic Realization Problem

Given two \(\sigma \) -algebras \(\mathcal {M}_1\) and \(\mathcal {M}_2\) , necessary and sufficient conditions are given for a \(\sigma \) -algebra \(\mathcal {M}_3\) to minimally make \(\mathcal {M}_1\) and \(\mathcal {M}_2\) conditionally independent. Representations in terms of projections among \(\sigma \) -algebras are derived from those conditions. A constructive algorithm is sketched. The concepts of weak and strong identification among \(\sigma \) -algebras are shown to be a crucial tool; in particular, minimal splitting is in-between.