Marginals and the product strong lifting problem
In general a lifting problem asks for a map from a quotient structure back to the original structure whose composition with the quotient map is the identity. The present paper deals with liftings for topological probability measure spaces. If \((X,\Sigma, \mathcal{T},\mu)\) is such a space then the quotient is usually modulo the ideal \(\mathcal{N}\) of \(\mu\)-null sets. A lifting \(\lambda:\Sigma/\mathcal{N}\to\Sigma\) usually replaced by \(\sigma=\lambda\circ q\), where \(q:\Sigma\to\Sigma/\mathcal{N}\) is the quotient map. The map~\(\sigma:\Sigma\to\Sigma\) has two properties: 1)~\(A=_\mu\sigma(A)\), and 2)~``if \(A=_\mu B\) then \(\sigma(A)=\sigma(B)\)''. Quite often a map like~\(\sigma\) is itself already called a lifting and in case \(\lambda\)~is a homomorphism then \(\sigma\) is a homomorphism `modulo null-sets'. The main result of the paper is an improvement of Theorem~4.3 of [\textit{K. Musiał} et al., J. Theor. Probab. 20, No. 3, 545--560 (2007; Zbl 1143.28003)] concerning liftings for product spaces, given liftings for the factors. Unfortunately the detail in the assumptions and conclusions make it impracticable give a summary that would do the theorem justice; we refer the interested reader to the paper itself for more information.
scientific article; zbMATH DE number 7187668
scientific article