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. The theory was further developed by Dorothy Maharam and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea. 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. Lifting theory continued to develop since then, yielding new results and applications.
Definitions
A lifting on a measure space is a linear and multiplicative inverse of the quotient map where is the seminormed Lp space of measurable functions and is its usual normed quotient. In other words, a lifting picks from every equivalence class of bounded measurable functions modulo negligible functions a representative— which is henceforth written T or T 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 measureon the level sets of a function.
Existence of liftings
Theorem. Suppose is complete. Then admits a lifting if and only ifthere exists a collection of mutually disjoint integrable sets in Σ whose union is X. In particular, if is the completion of a σ-finite 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.
Strong liftings
Suppose is complete and X is equipped with a completely regularHausdorff topology τ ⊂ Σ such that the union of any collection of negligible open sets is again negligible - this is the case if is σ-finite or comes from a Radon measure. Then the support of μ, Supp, can be defined as the complement of the largest negligible open subset, and the collection Cb of bounded continuous functionsbelongs to. A strong lifting for is a lifting such that Tφ = φ on Supp for all φ in Cb. This is the same as requiring that TU ≥ for all open sets U in τ.
Theorem. If is σ-finite and complete and τ has a countable basis then admits a strong lifting.
Proof. Let T0 be a lifting for and a countable basis for τ. For any point p in the negligible set let Tp be any character on L∞ that extends the character φ ↦ φ of Cb. Then for p in X and in L∞ define: T is the desired strong lifting.
Application: disintegration of a measure
Suppose, are σ-finite measure spaces and π : X → Y is a measurable map. A disintegration of μ along π with respect to ν is a slew of positive σ-additive measures on such that
λy is carried by the fiber of π over y:
for every μ-integrable function f,
Disintegrations exist in various circumstances, the proofs varying but almost all using strong liftings. Here is a rather general result. Its short proof gives the general flavor.
Theorem. Suppose X is a Polish space and Y a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a σ-finite Borel measure on X and π : X → Y a Σ, Φ-measurable map. Then there exists a σ-finite Borel measure ν on Y and a disintegration. If μ is finite, ν can be taken to be the pushforward π∗μ, and then the λy are probabilities.
Proof. Because of the polish nature of X there is a sequence of compact subsets of X that are mutually disjoint, whose union has negligible complement, and on which π is continuous. This observation reduces the problem to the case that both X and Y are compact and π is continuous, and ν = π∗μ. Complete Φ under ν and fix a strong lifting T for. Given a bounded μ-measurable function f, let denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of π∗ with respect to π∗μ. Then set, for every y in Y, 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 and take the infimum over all positive φ in Cb with φ = 1; it becomes apparent that the support of λy lies in the fiber over y.
