On the σ-Algebraic Realization Problem

Given two σ-algebras M1 and M2, necessary and sufficient conditions are given for a σ-algebra M3 to minimally make M1 and M2 conditionally independent; representations in terms of projections among σ-algebras are derived from those conditions. A constructive algorithm is sketched. the concept of week and strong identification among σ-algebras are shown to be a crucial tool; in particular, minimal splitting is in-between.