Existence of linear liftings with invariant sections in product measure spaces
The authors continue their work on the relation of (measure theoretic) liftings on product measure spaces and their relations to liftings on the factor spaces. This paper deals with linear liftings which is a more general concept. In view of previous work the natural question is whether given linear liftings \(\rho, \sigma\) on the factor spaces \(\Omega_1, \Omega_2\) there is a linear lifting \(\pi\) on the product space satisfying \[ \pi (f\otimes g)= \rho (f) \otimes \sigma (g)\tag{P} \] and \[ \pi (h)_{\omega} = \sigma [\pi (h)_{\omega}] \tag{S} \] for all \( f\in L^\infty (\Omega_1), g \in L^\infty (\Omega_2), h\in L^\infty (\Omega_1 \times \Omega_2), \omega \in \Omega_1.\) They prove that the standard transfinite construction of linear liftings yields liftings \(\rho\) such that for every \(\sigma\) there is a \(\pi\) with (P) such that (S) holds for each \(h\) almost everywhere on \(\Omega_1.\)
scientific article; zbMATH DE number 2138525