Lifting theory
Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by Alfréd Haar), followed later by Dorothy Maharam’s (1958) paper, and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper. Lifting Theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph, now a standard reference in the field. Lifting Theory continued to develop after 1969, yielding significant new results and applications.
A lifting on a measure space (X, Σ, μ) is a linear and multiplicative inverse
of the quotient map
In other words, a lifting picks from every equivalence class [f] of bounded measurable functions modulo negligible functions a representative— which is henceforth written T([f]) or T[f] or simply Tf — in such a way that
Liftings are used to produce disintegrations of measures, for instance conditional probability distributions given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function.