Liftings for Haar measure on \(\{0,1\}^ k\) (Q1178339)

A result of Shelah that consistently there is no Borel lifting for the Lebesgue measure is generalized. It is shown that consistently for no infinite cardinal \(\kappa\) there is a projective lifting for the Lebesgue measure on \(2^ \kappa\). Here a subset \(E\) of \(2^ \kappa\) is projective if \(E=E_ A\times 2^{\kappa\backslash A}\) for some countable \(A\subseteq\kappa\) and projective \(E_ A\subseteq 2^ \kappa\).