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

which is a right inverse of the quotient map

where is the seminormed L^{p} 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 or or simply — in such a way that and for all 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.

Theorem.Suppose is complete.^{[5]}Then admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in whose union is In particular, if is the completion of aσ-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.

Suppose is complete and 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 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 in This is the same as requiring that

Theorem.If isσ-finite and complete and has a countable basis then admits a strong lifting.

**Proof.** Let be a lifting for and a countable basis for For any point in the negligible set

let be any character

is the desired strong lifting.

Suppose and are *σ*-finite measure spaces ( positive) and is a measurable map. A **disintegration of along with respect to ** is a slew of positive *σ*-additive measures on such that

- is carried by the fiber of over , i.e. and for almost all
- for every -integrable function in the sense that, for -almost all in is -integrable, the functionis -integrable, and the displayed equality holds.

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 is a Polish space^{[9]}and a separable Hausdorff space, both equipped with their Borelσ-algebras. Let be aσ-finite Borel measure on and a measurable map. Then there exists a σ-finite Borel measure on and a disintegration (*). If is finite, can be taken to be the pushforward^{[10]}and then the are probabilities.

**Proof.** Because of the polish nature of there is a sequence of compact subsets of that are mutually disjoint, whose union has negligible complement, and on which is continuous. This observation reduces the problem to the case that both and are compact and is continuous, and Complete under and fix a strong lifting for Given a bounded -measurable function let denote its conditional expectation under that is, the Radon-Nikodym derivative of^{[11]} with respect to Then set, for every in 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 in with it becomes apparent that the support of lies in the fiber over