Lifting theory

In mathematics, 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}}

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.^[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_{0}U_{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

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

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

↑ von Neumann, J.: Algebraische Repräsentanten der Funktionen bis auf eine Menge von Maße Null. J. Crelle 165, 109–115 (1931)
↑ Maharam, D.: On a theorem of von Neumann. Proc. Amer. Math. Soc. 9, 987–995 (1958)
↑ A. Ionescu Tulcea and C. Ionescu Tulcea: On the lifting property, I., J. Math. Anal. App. 3, 537–546 (1961)
↑ 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)
↑ 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.
↑ i.e., there exists a countable collection of integrable sets – sets of finite measure in Σ – that covers the underlying set X.
↑ U, Supp(μ) are identified with their indicator functions.
↑ A character on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.
↑ 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.
↑ 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.
↑ fμ is the measure that has density f with respect to μ

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[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] .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.

[character-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] μ is the measure that has density f with respect to μ