Invariance Properties of the Conditional Independence Relation

Abstract

The conditional independence relation for a triple of $\sigma$-algebras is investigated. For certain operations on this relation, necessary and sufficient conditions are derived such that these operations leave the relation invariant. Examples of such operations are the enlargement or reduction of the $\sigma$-algebras, and an absolute continuous change of measure. A projection operator for $\sigma$-algebras is defined and some of its properties are stated. The $\sigma$-algebraic realization problem is briefly discussed.