Lifting problem for incomplete measure spaces

Determine whether, for a measure space (X, M, μ) that is not complete, there exists a Boolean algebra homomorphism s: M/M0 → M that is a right-inverse of the natural projection π: M ↠ M/M0 (where M0 is the μ-null ideal); equivalently, ascertain whether such incomplete measure spaces admit a lifting.

Background

The first existence problem concerns whether the natural projection π: M ↠ M/M0 from a σ-algebra M to its measure algebra (the quotient by the μ-null ideal M0) admits a right-inverse that is a Boolean algebra homomorphism. Such a right-inverse is a lifting.

Classical results resolve several cases positively: von Neumann proved existence for Lebesgue measure on ℝ, and Maharam showed existence when μ is complete and σ-finite. The present paper establishes an equivalence (Theorem 1.10) between the existence of such liftings and certain limiting operators for the differentiation of integrals in complete measure spaces. However, for measure spaces that are not complete, the existence of a lifting remains unresolved, with subtle set-theoretic issues implicated (see references [45], [53]).