Existence of linear liftings with invariant sections in product measure spaces

Abstract

D e d i c a t e d t o P r o f. D r. D. Kö l z o w o n t h e o c c a s i o n o f h i s 75 th b i r t h d a yAbstract. We prove that on a complete product of two arbitrary probability spaces, one of which is endowed with a particular linear lifting (we call it admissible), there exists a linear lifting possessing the property that all its sections determined by the marginal space without lifting are invariant under corresponding, given a priori, marginal linear lifting. If both marginal spaces are endowed with linear liftings (at least one of them to be admissible) then the lifting in the product space may have additional product properties. We prove also that for non-atomic marginal spaces there exist no marginal linear liftings and no linear lifting in the product possessing all its sections invariant under both marginal linear liftings.

Various products for Lebesgue densities

“…Note that (Q 2 ) is related to the Question 3.3 raised in [13] as well as to Question 2.6 from [9] which asks whether there exist for probability spaces (X, , μ) and (Y, T, ν) densities υ ∈ ϑ(μ), τ ∈ ϑ(ν), and η ∈ ϑμ ⊗ν) such that υ ([η(E)…”

“…In [9, Question 2.6.] as well as in [13], for arbitrary products the question was raised concerning the existence of triples of densities satisfying the two-sided section property. In Remark 2.7 we resolve this problem for the triples (ϕ λ k , ϕ λ d−k , ϕ λ d ) to the negative.…”