Product liftings and densities with lifting invariant and density invariant sections

Given two measure spaces equipped with liftings or densities (complete if liftings are considered) the existence of product liftings and densities with lifting invariant or density invariant sections is investigated. It is proved that if one of the marginal liftings is admissibly generated (a subclass of consistent liftings), then one can always find a product lifting which has the property that all sections determined by one of the marginal spaces are lifting invariant (Theorem 2.13). For a large class of measures Theorem 2.13 is the best possible (Theorem 4.3). When densities are considered, then one can always have a product density with measurable sections, but in the case of non-atomic complete marginal measures there exists no product density with all sections being density invariant. The results are then applied to stochastic processes.