We prove that if $(\Omega,\Sigma,\mathcal{I}),\; (\Theta,T,\mathcal{J})$ and $(\Omega \times\Theta,\Xi,\mathcal{K})$ are measurable spaces with $\sigma$-ideals satisfying some natural Fubini type conditions then there is no density on $(\Omega\times\Theta,\Xi,\mathcal{K})$ with density invariant sections.
References
D. Maharam, Category, Boolean algebras and measures, Lecture Notes in Math., 609 (1977), 124–135, Springer-Verlag. MR457665 0404.54024 D. Maharam, Category, Boolean algebras and measures, Lecture Notes in Math., 609 (1977), 124–135, Springer-Verlag. MR457665 0404.54024
K. Musiał, W. Strauss and N. D. Macheras, Product liftings and densities with lifting invariant and density invariant sections, Fund. Math., 166 (2000), 281–303. MR1809420 0966.28001 K. Musiał, W. Strauss and N. D. Macheras, Product liftings and densities with lifting invariant and density invariant sections, Fund. Math., 166 (2000), 281–303. MR1809420 0966.28001
