Linear liftings for non-complete probability spaces (Q1802766)

In a previous paper the second author showed that it is consistent with ZFC that the Lebesgue measure restricted to the Borel \(\sigma\)-algebra on \([0,1]\) has no lifting [see \textit{S. Shelah}, Isr. J. Math. 45, 90-96 (1983; Zbl 0549.03041)]. Modifying the technique of the above paper the authors establish the main result of the present paper, i.e., that it is also consistent with ZFC that the space \(L^ \infty([0,1],\Sigma,\mu)\), \(\Sigma\) the Borel \(\sigma\)-algebra on [0,1] and \(\mu\) Lebesgue measure restricted to \(\Sigma\), has no linear lifting. This result can be extended to all (not necessarily complete) probability spaces allowing a measurable inverse-measure-preserving function into [0,1] together with a Borel disintegration for the probability measure. The main result also settles to the negative the long-standing problem (not mentioned in this paper) whether the existence of a lower density for a probability space implies the existence of a linear lifting for \(L^ \infty\), the space of all bounded measurable functions with respect to that probability space.