Various products of category densities and liftings

We extend earlier work [M.R. Burke, N.D. Macheras, K. Musiał, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164–1191] of the authors on the existence of category liftings in the product of two topological spaces X and Y   such that X×YX×Y is a Baire space. For given densities ρ, σ on X and Y  , respectively, we introduce two ‘Fubini type’ products ρ⊙σρ⊙σ and ρ⊡σρ⊡σ on X×YX×Y. We present a necessary and sufficient condition for ρ⊙σρ⊙σ to be a density. Provided (X,Y)(X,Y) and (Y,X)(Y,X) have the Kuratowski–Ulam property, we prove for given category liftings ρ, σ on the factors the existence of a category lifting π   on the product, dominating the density ρ⊡σρ⊡σ and such thatπ(A×B)=ρ(A)×σ(B)for Baire subsets A of X and B of Y,andρ([π(E)]y)=[π(E)]yfor all y∈Y and Baire subsets E of X×Y.We show that further properties of consistency with the product structure cannot be expected.We prove also that contrary to measure theoretical liftings, in case of Baire spaces there might exist countably additive liftings. This answers, assuming the existence of a compact cardinal, a question from [M.R. Burke, N.D. Macheras, K. Musiał, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164–1191]. The example we present is a version of an example of D.H. Fremlin of a space whose category algebra has a countably additive lifting.