In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar.
[1] The theory was further developed by Dorothy Maharam (1958)[2] and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961).
[3] Lifting theory was motivated to a large extent by its striking applications.
Its development up to 1969 was described in a monograph of the Ionescu Tulceas.
[4] Lifting theory continued to develop since then, yielding new results and applications.
A lifting on a measure space
is a linear and multiplicative operator
is the seminormed Lp space of measurable functions and
In other words, a lifting picks from every equivalence class
of bounded measurable functions modulo negligible functions a representative— which is henceforth written
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.
admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in
is the completion of a σ-finite[6] measure or of an inner regular Borel measure on a locally compact space, then
admits a lifting.The proof consists in extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.
is equipped with a completely regular Hausdorff topology
is σ-finite or comes from a Radon measure.
can be defined as the complement of the largest negligible open subset, and the collection
of bounded continuous functions belongs to
is the desired strong lifting.
of positive σ-additive measures on
such that Disintegrations exist in various circumstances, the proofs varying but almost all using strong liftings.
Its short proof gives the general flavor.
a separable Hausdorff space, both equipped with their Borel σ-algebras.
be a σ-finite Borel measure on
Then there exists a σ-finite Borel measure
that are mutually disjoint, whose union has negligible complement, and on which
This observation reduces the problem to the case that both
and fix a strong lifting
To show that this defines a disintegration is a matter of bookkeeping and a suitable Fubini theorem.
To see how the strongness of the lifting enters, note that