Lifting theory

Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by Alfréd Haar),^[1] followed later by Dorothy Maharam’s (1958) paper,^[2] and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper.^[3] 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,^[4] 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

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

of the quotient map

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

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

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

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

$T[1]=1.$

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.^[5] 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^[6] 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 $\mathcal L^\infty(X,\Sigma,\mu)$.

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

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

such that Tφ = φ on Supp(μ) for all φ in C_b(X, τ). This is the same as requiring that^[7] 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

$N:=\bigcup\nolimits _n \left\{p\in \mathrm{Supp}(\mu): (T_0U_n)(p)<U_n(p) \right\}$

let T_p be any character^[8] 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:

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

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:

$\{y\}\in\Phi\;\;\mathrm{ and }\;\; \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right)=0 \qquad \forall y\in Y$

1. for every μ-integrable function f,

$\int_X f(p)\;\mu(dp)= \int_Y \left(\int_{\pi^{-1}(\{y\})}f(p)\,\lambda_y(dp)\right) \nu(dy) \qquad (*)$

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

$y\mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp)$

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^[9] 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^[10] 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^[11] 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

$\lambda_y(f\cdot\varphi\circ\pi)=\varphi(y) \lambda_y(f) \qquad \forall y\in Y, \varphi\in C_b(Y), f\in L^\infty(X,\Sigma,\mu)$

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.

References

1. von Neumann, J.: Algebraische Repräsentanten der Funktionen bis auf eine Menge von Maße Null. J. Crelle 165, 109-115 (1931)
2. Maharam, D.: On a theorem of von Neumann. Proc. Amer. Math. Soc. 9, 987-995 (1958)
3. A. Ionescu Tulcea and C. Ionescu Tulcea: On the lifting property, I., J. Math. Anal. App. 3, 537-546 (1961)
4. Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, Topics in the Theory of Lifting, Ergebnisse der Mathematik, Vol. 48, Springer-Verlag, Berlin, Heidelberg, New York (1969)
5. A subset N ⊂ X is locally negligible if it intersects every integrable set in Σ in a subset of a negligible set of Σ. (X, Σ, μ) is complete if every locally negligible set is negligible and belongs to Σ.
6. i.e., there exists a countable collection of integrable sets –sets of finite measure in Σ– that covers the underlying set X.
7. U, Supp(μ) are identified with their indicator functions.
8. A character on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.
9. A separable space is Polish if its topology comes from a complete metric. In the present situation it would be sufficient to require that X is Suslin, i.e., is the continuous Hausdorff image of a polish space.
10. The pushforward π_∗μ of μ under π, also called the image of μ under π and denoted π(μ), is the measure ν on Φ defined by $\nu(A):=\mu\left(\pi^{-1}(A)\right)$ for A in Φ.
11. fμ is the measure that has density f with respect to μ