Non-existence of certain types of liftings and densities in product spaces with σ-ideals.

Abstract

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.