Marginals and the product strong lifting problem

We study the permanence of strong liftings for products of topological probability spaces. We get results to the positive for the 'usual product' or for the product σ-algebra obtained from it by adjoining the two-sided nil-null sets under assumptions implying that the product topology is contained in this product σ-algebras. For the construction of strong liftings in the product the application of marginals is basic as well as that of the fundamental properties of lifting topologies.