Lifting theory

Short description: Notion in measure theory

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 [math]\displaystyle{ (X, \Sigma, \mu) }[/math] is a linear and multiplicative operator [math]\displaystyle{ T : L^\infty(X, \Sigma, \mu) \to \mathcal{L}^\infty(X, \Sigma, \mu) }[/math] which is a right inverse of the quotient map [math]\displaystyle{ \begin{cases} \mathcal L^\infty(X,\Sigma,\mu) \to L^\infty(X,\Sigma,\mu) \\ f \mapsto [f] \end{cases} }[/math]

where [math]\displaystyle{ \mathcal{L}^\infty(X,\Sigma,\mu) }[/math] is the seminormed L^p space of measurable functions and [math]\displaystyle{ L^\infty(X, \Sigma, \mu) }[/math] is its usual normed quotient. In other words, a lifting picks from every equivalence class [math]\displaystyle{ [f] }[/math] of bounded measurable functions modulo negligible functions a representative— which is henceforth written [math]\displaystyle{ T([f]) }[/math] or [math]\displaystyle{ T[f] }[/math] or simply [math]\displaystyle{ Tf }[/math] — in such a way that [math]\displaystyle{ T[1] = 1 }[/math] and for all [math]\displaystyle{ p \in X }[/math] and all [math]\displaystyle{ r, s \in \Reals, }[/math] [math]\displaystyle{ T(r[f]+s[g])(p) = rT[f](p) + sT[g](p), }[/math] [math]\displaystyle{ T([f]\times[g])(p) = T[f](p) \times T[g](p). }[/math]

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

A strong lifting for [math]\displaystyle{ (X, \Sigma, \mu) }[/math] is a lifting [math]\displaystyle{ T : L^\infty(X, \Sigma, \mu) \to \mathcal{L}^\infty(X, \Sigma, \mu) }[/math] such that [math]\displaystyle{ T\varphi = \varphi }[/math] on [math]\displaystyle{ \operatorname{Supp}(\mu) }[/math] for all [math]\displaystyle{ \varphi }[/math] in [math]\displaystyle{ C_b(X, \tau). }[/math] This is the same as requiring that [math]\displaystyle{ T U \geq (U \cap \operatorname{Supp}(\mu)) }[/math] for all open sets [math]\displaystyle{ U }[/math] in [math]\displaystyle{ \tau. }[/math]

Theorem. If [math]\displaystyle{ (\Sigma, \mu) }[/math] is σ-finite and complete and [math]\displaystyle{ \tau }[/math] has a countable basis then [math]\displaystyle{ (X, \Sigma, \mu) }[/math] admits a strong lifting.

Proof. Let [math]\displaystyle{ T_0 }[/math] be a lifting for [math]\displaystyle{ (X, \Sigma, \mu) }[/math] and [math]\displaystyle{ U_1, U_2, \ldots }[/math] a countable basis for [math]\displaystyle{ \tau. }[/math] For any point [math]\displaystyle{ p }[/math] in the negligible set [math]\displaystyle{ N := \bigcup\nolimits_n \left\{p \in \operatorname{Supp}(\mu) : (T_0U_n)(p) \lt U_n(p)\right\} }[/math] let [math]\displaystyle{ T_p }[/math] be any character on [math]\displaystyle{ L^\infty(X, \Sigma, \mu) }[/math] that extends the character [math]\displaystyle{ \phi \mapsto \phi(p) }[/math] of [math]\displaystyle{ C_b(X, \tau). }[/math] Then for [math]\displaystyle{ p }[/math] in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ [f] }[/math] in [math]\displaystyle{ L^\infty(X, \Sigma, \mu) }[/math] define: [math]\displaystyle{ (T[f])(p):= \begin{cases} (T_0[f])(p)& p\notin N\\ T_p[f]& p\in N. \end{cases} }[/math] [math]\displaystyle{ T }[/math] is the desired strong lifting.

Application: disintegration of a measure

Suppose [math]\displaystyle{ (X, \Sigma, \mu) }[/math] and [math]\displaystyle{ (Y, \Phi, \nu) }[/math] are σ-finite measure spaces ([math]\displaystyle{ \mu, \mu }[/math] positive) and [math]\displaystyle{ \pi : X \to Y }[/math] is a measurable map. A disintegration of [math]\displaystyle{ \mu }[/math] along [math]\displaystyle{ \pi }[/math] with respect to [math]\displaystyle{ \nu }[/math] is a slew [math]\displaystyle{ Y \ni y \mapsto \lambda_y }[/math] of positive σ-additive measures on [math]\displaystyle{ (\Sigma, \mu) }[/math] such that

1. [math]\displaystyle{ \lambda_y }[/math] is carried by the fiber [math]\displaystyle{ \pi^{-1}(\{y\}) }[/math] of [math]\displaystyle{ \pi }[/math] over [math]\displaystyle{ y }[/math], i.e. [math]\displaystyle{ \{y\} \in \Phi }[/math] and [math]\displaystyle{ \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right) = 0 }[/math] for almost all [math]\displaystyle{ y \in Y }[/math]
2. for every [math]\displaystyle{ \mu }[/math]-integrable function [math]\displaystyle{ f, }[/math][math]\displaystyle{ \int_X f(p)\;\mu(dp)= \int_Y \left(\int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp)\right) \nu(dy) \qquad (*) }[/math] in the sense that, for [math]\displaystyle{ \nu }[/math]-almost all [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y, }[/math] [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \lambda_y }[/math]-integrable, the function [math]\displaystyle{ y \mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp) }[/math] is [math]\displaystyle{ \nu }[/math]-integrable, and the displayed equality [math]\displaystyle{ (*) }[/math] 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 [math]\displaystyle{ X }[/math] is a Polish space and [math]\displaystyle{ Y }[/math] a separable Hausdorff space, both equipped with their Borel σ-algebras. Let [math]\displaystyle{ \mu }[/math] be a σ-finite Borel measure on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \pi : X \to Y }[/math] a [math]\displaystyle{ \Sigma, \Phi- }[/math]measurable map. Then there exists a σ-finite Borel measure [math]\displaystyle{ \nu }[/math] on [math]\displaystyle{ Y }[/math] and a disintegration (*). If [math]\displaystyle{ \mu }[/math] is finite, [math]\displaystyle{ \nu }[/math] can be taken to be the pushforward [math]\displaystyle{ \pi_* \mu, }[/math] and then the [math]\displaystyle{ \lambda_y }[/math] are probabilities.

Proof. Because of the polish nature of [math]\displaystyle{ X }[/math] there is a sequence of compact subsets of [math]\displaystyle{ X }[/math] that are mutually disjoint, whose union has negligible complement, and on which [math]\displaystyle{ \pi }[/math] is continuous. This observation reduces the problem to the case that both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are compact and [math]\displaystyle{ \pi }[/math] is continuous, and [math]\displaystyle{ \nu = \pi_* \mu. }[/math] Complete [math]\displaystyle{ \Phi }[/math] under [math]\displaystyle{ \nu }[/math] and fix a strong lifting [math]\displaystyle{ T }[/math] for [math]\displaystyle{ (Y, \Phi, \nu). }[/math] Given a bounded [math]\displaystyle{ \mu }[/math]-measurable function [math]\displaystyle{ f, }[/math] let [math]\displaystyle{ \lfloor f\rfloor }[/math] denote its conditional expectation under [math]\displaystyle{ \pi, }[/math] that is, the Radon-Nikodym derivative of [math]\displaystyle{ \pi_*(f \mu) }[/math] with respect to [math]\displaystyle{ \pi_* \mu. }[/math] Then set, for every [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y, }[/math] [math]\displaystyle{ \lambda_y(f) := T(\lfloor f\rfloor)(y). }[/math] 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 [math]\displaystyle{ \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) }[/math] and take the infimum over all positive [math]\displaystyle{ \varphi }[/math] in [math]\displaystyle{ C_b(Y) }[/math] with [math]\displaystyle{ \varphi(y) = 1; }[/math] it becomes apparent that the support of [math]\displaystyle{ \lambda_y }[/math] lies in the fiber over [math]\displaystyle{ y. }[/math]