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 (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). 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 operator which is a right 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 [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 T[1] = 1 and for all p \in X and all 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.

Existence of liftings

Theorem. Suppose is complete. Then admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in \Sigma 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.

<!-- Here are the details. Henceforth write Tf := T[f] = T([f]). is σ-finite if there exists a countable collection of sets of finite measure in \Sigma whose union has negligible complement. This permits a reduction to the case that the measure \mu is finite, in fact, it may be taken to be a probability. The proof uses [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma) together with the following order on pairs of sub-σ-algebras \mathfrak A of \Sigma and liftings for them: if and is the restriction of to. It is to be shown that a chain \mathfrak C of such pairs has an upper bound, and that a maximal pair, which then exists by Zorn's lemma, has \Sigma for its first entry. If \mathfrak C has no countable [cofinal](https://bliptext.com/articles/cofinal-mathematics) subset, then the union is a σ-algebra and there is an obvious lifting for it that restricts to the liftings of the chain; is the sought upper bound of the chain. The argument is more complicated when the chain \mathfrak C has a countable cofinal subset. In this case let \mathfrak U be the σ-algebra generated by the union which is generally only an [algebra of sets](https://bliptext.com/articles/field-of-sets). For the construction of it is convenient to identify a set with its indicator function and to write For let A_n denote the [conditional expectation](https://bliptext.com/articles/conditional-expectation) of A under. By [Doob's martingale convergence theorem](https://bliptext.com/articles/doob-s-martingale-convergence-theorems) the set \theta(A) of points where A_n converges to 1 differs negligibly from A. Here are a few facts that are straightforward to check (some use the completeness and finiteness of ): is a topology whose only negligible open set is the empty set and such that every is almost everywhere continuous, to wit, on and on. Then every, being the uniform limit of a sequence of step functions over \mathfrak U, is almost everywhere continuous in this topology. For p in X is a proper ideal of, contained (by another application of Zorn's lemma) in some maximal proper ideal which has codimension 1. The quotient map can be viewed as a character Tp. Defining provides the upper bound for the chain \mathfrak C. In either case the chain \mathfrak C therefore has an upper bound. By Zorn's lemma there is a maximal pair , and a small additional calculation shows that. END OF DETAILED PROOF-->

Strong liftings

Suppose is complete and X is equipped with a completely regular Hausdorff topology 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 \mu, can be defined as the complement of the largest negligible open subset, and the collection of bounded continuous functions belongs to A strong lifting for is a lifting such that on for all \varphi in This is the same as requiring that for all open sets U in \tau. "Theorem. If is σ-finite and complete and \tau has a countable basis then admits a strong lifting." Proof. Let T_0 be a lifting for and a countable basis for \tau. For any point p in the negligible set let T_p be any character on that extends the character of Then for p in X and [f] in define: T is the desired strong lifting.

Application: disintegration of a measure

Suppose and are σ-finite measure spaces (\mu, \mu positive) and is a measurable map. A disintegration of \mu along \pi with respect to \nu is a slew of positive σ-additive measures on such that 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 \mu be a σ-finite Borel measure on X and a measurable map. Then there exists a σ-finite Borel measure \nu on Y and a disintegration (). If \mu is finite, \nu can be taken to be the pushforward \pi_ \mu, and then the \lambda_y 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 \pi is continuous. This observation reduces the problem to the case that both X and Y are compact and \pi is continuous, and Complete \Phi under \nu and fix a strong lifting T for Given a bounded \mu-measurable function f, let denote its conditional expectation under \pi, that is, the Radon-Nikodym derivative of with respect to \pi_* \mu. 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 \varphi in C_b(Y) with it becomes apparent that the support of \lambda_y lies in the fiber over y.