The product-lifting for arbitrary products of complete probability spaces (Q2785347)

This interesting paper is a continuation and generalization of the paper published earlier by the same authors in J. Aust. Mat. Soc. 60, No. 3, 311-333 (1996; Zbl 0879.28008). The main result is the following theorem:NEWLINENEWLINENEWLINELet \(I= \{\alpha<\kappa\}\) be a set of ordinals and let \((\Omega_i,\Sigma_i,\mu_i)_{i\in I}\) and \((\Omega,\Sigma,\mu)\) be complete probability spaces such that \(\Omega= \prod_{i\in I}\Omega_i\), \(\Sigma\) contains the completed product \(\prod_{i\in I}\Sigma_i\) and \(\mu\) restricted to this completion coincides with the completion of \(\prod_{i\in I}\mu_i\). Then, for each lifting \(\rho_{i_0}\) on \((\Omega_{i_0},\Sigma_{i_0},\mu_{i_0})\) there exist liftings \(\rho_i\) on \((\Omega_i,\Sigma_i, \mu_i)\), \(i\neq i_0\), and a lifting \(\pi\) on \((\Omega,\Sigma,\mu)\) such that \(\pi\) is the product of all \(\rho_i\) on \(\prod_{i\in I}\Sigma_i\). Moreover, for each \(\alpha<\kappa\), if a set \(A\in\prod_{i\in I}\Sigma_i\) depends only on the coordinates less than \(\alpha\), then the same holds true for \(\pi(A)\).NEWLINENEWLINENEWLINEApplying the above result, the authors prove then that the normalized Haar measure on a compact group of an uncountable weight admits a lifting that is the product lifting of a family of liftings supported by compact Lie groups.NEWLINENEWLINENEWLINEThe rest of the paper is devoted to the examination of relations between strong product liftings and families of strong liftings on the coordinate spaces.