Prince Rupert's cube

Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by A. 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

$T:L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\to {\mathcal{ℒ}}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$

of the quotient map

$\{\begin{array}{l}{\mathcal{ℒ}}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\to L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\\ f\mapsto [f]\end{array}$

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

$T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),\mathrm{\forall }p\in X,r,s\in \mathbf{R};$

$T([f]\times [g])(p)=T[f](p)\times T[g](p),\mathrm{\forall }p\in X;$

$T[1]=1.$

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 (X, Σ, μ) is complete. Then (X, Σ, μ) admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in Σ whose union is X. In particular, if (X, Σ, μ) is the completion of a σ-finite measure or of an inner regular Borel measure on a locally compact space, then (X, Σ, μ) 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 (X, Σ, μ) is complete and X is equipped with a completely regular Hausdorff topology τ ⊂ Σ such that the union of any collection of negligible open sets is again negligible – this is the case if (X, Σ, μ) is σ-finite or comes from a Radon measure. Then the support of μ, Supp(μ), can be defined as the complement of the largest negligible open subset, and the collection C_b(X, τ) of bounded continuous functions belongs to ${\mathcal{ℒ}}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$.

A strong lifting for (X, Σ, μ) is a lifting

$T:L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\to {\mathcal{ℒ}}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$

such that Tφ = φ on Supp(μ) for all φ in C_b(X, τ). This is the same as requiring that TU ≥ (U ∩ Supp(μ)) for all open sets U in τ.

Theorem. If (Σ, μ) is σ-finite and complete and τ has a countable basis then (X, Σ, μ) admits a strong lifting.

Proof. Let T₀ be a lifting for (X, Σ, μ) and {U₁, U₂, ...} a countable basis for τ. For any point p in the negligible set

$N:=\bigcup _n\{p\in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\mu ):(T_0{U}_n)(p)<U_n(p)\}$

let T_p be any character on L^∞(X, Σ, μ) that extends the character φ ↦ φ(p) of C_b(X, τ). Then for p in X and [f] in L^∞(X, Σ, μ) define:

$(T[f])(p):=\{\begin{array}{ll}(T_0[f])(p)& p\notin N\\ T_p[f]& p\in N.\end{array}$

T is the desired strong lifting.

Application: disintegration of a measure

Suppose (X, Σ, μ), (Y, Φ, ν) are σ-finite measure spaces (μ, ν positive) and π : X → Y is a measurable map. A disintegration of μ along π with respect to ν is a slew $Y\ni y\mapsto {\lambda }_y$ of positive σ-additive measures on (X, Σ) such that

1. λ_y is carried by the fiber ${\pi }^{-1}(\{y\})$ of π over y:

$\{y\}\in \mathrm{\Phi }\mathrm{a}\mathrm{n}\mathrm{d}{\lambda }_y((X\setminus {\pi }^{-1}(\{y\}))=0\mathrm{\forall }y\in Y$

1. for every μ-integrable function f,

$\underset{X}{\int }f(p)\mu (dp)=\underset{Y}{\int }(\underset{{\pi }^{-1}(\{y\})}{\int }f(p){\lambda }_y(dp))\nu (dy)(*)$

in the sense that, for ν-almost all y in Y, f is λ_y-integrable, the function

$y\mapsto \underset{{\pi }^{-1}(\{y\})}{\int }f(p){\lambda }_y(dp)$

is ν-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 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 λ_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 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 (Y, Φ, ν). Given a bounded μ-measurable function f, let $\lfloor f\rfloor$ denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of π_∗(fμ) with respect to π_∗μ. Then set, for every y in Y, ${\lambda }_y(f):=T(\lfloor f\rfloor )(y).$ 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

${\lambda }_y(f\cdot \phi \circ \pi )=\phi (y){\lambda }_y(f)\mathrm{\forall }y\in Y,\phi \in C_b(Y),f\in L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$

and take the infimum over all positive φ in C_b(Y) with φ(y) = 1; it becomes apparent that the support of λ_y lies in the fiber over y.