Lifting theory: Difference between revisions

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 $(X,\Sigma ,\mu )$ is a linear and multiplicative inverse

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

of the quotient map

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

where ${\mathcal {L}}^{\infty }(X,\Sigma ,\mu )$ is the seminormed L^p space of measurable functions and $L^{\infty }(X,\Sigma ,\mu )$ 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

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

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

$T[1]=1.$

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. 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 ${\mathcal {L}}^{\infty }(X,\Sigma ,\mu )$.

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 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

$N:=\bigcup \nolimits _{n}\left\{p\in \mathrm {Supp} (\mu ):(T_{0}U_{n})(p)<U_{n}(p)\right\}$

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:

$(T[f])(p):={\begin{cases}(T_{0}[f])(p)&p\notin N\\T_{p}[f]&p\in N.\end{cases}}$

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 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

$\lambda _{y}(f\cdot \varphi \circ \pi )=\varphi (y)\lambda _{y}(f)\qquad \forall y\in Y,\varphi \in C_{b}(Y),f\in L^{\infty }(X,\Sigma ,\mu )$

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.