Lifting 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.[1] The theory was further developed by Dorothy Maharam (1958)[2] and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961).[3] 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.[4] Lifting theory continued to develop since then, yielding new results and applications.

Definitions

A lifting on a measure space ${\displaystyle (X,\Sigma ,\mu )}$ is a linear and multiplicative inverse

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

of the quotient map

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

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

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

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

${\displaystyle T[1]=1.}$

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

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

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

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

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

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:

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

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 ${\displaystyle Y\ni y\mapsto \lambda _{y}}$ of positive σ-additive measures on (X, Σ) such that

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

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

1. for every μ-integrable function f,

${\displaystyle \int _{X}f(p)\;\mu (dp)=\int _{Y}\left(\int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)\right)\nu (dy)\qquad (*)}$

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

${\displaystyle y\mapsto \int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)}$

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 ${\displaystyle \lfloor f\rfloor }$ 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, ${\displaystyle \lambda _{y}(f):=T(\lfloor f\rfloor )(y).}$ 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

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

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, John (1931). "Algebraische Repräsentanten der Funktionen "bis auf eine Menge vom Maße Null"". Journal für die reine und angewandte Mathematik (Crelle's Journal) (in German). 1931 (165): 109–115. doi:10.1515/crll.1931.165.109. MR 1581278.
2. Maharam, Dorothy (1958). "On a theorem of von Neumann". Proceedings of the American Mathematical Society. 9 (6): 987–987. doi:10.2307/2033342. MR 0105479.
3. Ionescu Tulcea, Alexandra; Ionescu Tulcea, Cassius (1961). "On the lifting property. I." Journal of Mathematical Analysis and Applications. 3 (3): 537–546. doi:10.1016/0022-247X(61)90075-0. MR 0150256.
4. Ionescu Tulcea, Alexandra; Ionescu Tulcea, Cassius (1969). Topics in the theory of lifting. Ergebnisse der Mathematik und ihrer Grenzgebiete. 48. New York: Springer-Verlag. MR 0276438. OCLC 851370324.
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 ${\displaystyle \nu (A):=\mu \left(\pi ^{-1}(A)\right)}$ for A in Φ.
11. fμ is the measure that has density f with respect to μ