Lifting theory

Lifting theory was first introduced in a series of pioneering papers by John von Neumann.^[1] A lifting on a measure space $(E,{\mathfrak {F}},\mu )$ is a linear and multiplicative inverse

T:L^{\infty }(E,{\mathfrak {F}},\mu )\to {\mathcal {L}}^{\infty }(E,{\mathfrak {F}},\mu )

of the quotient map

[\ ]:{\mathcal {L}}^{\infty }(E,{\mathfrak {F}},\mu )\to L^{\infty }(E,{\mathfrak {F}},\mu ),\qquad f\mapsto [f].

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 E,r,s\in \mathbf {R} ;

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

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 $(E,{\mathfrak {F}},\mu )$ is complete.^[2] Then $(E,{\mathfrak {F}},\mu )$ admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in ${\mathfrak {F}}$ whose union is E. In particular, if $(E,{\mathfrak {F}},\mu )$ is the completion of a σ-finite^[3] measure or of an inner regular Borel measure on a locally compact space, then $(E,{\mathfrak {F}},\mu )$ 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 that $(E,{\mathfrak {F}},\mu )$ is complete and that E is equipped with a completely regular Hausdorff topology $\tau \subset {\mathfrak {F}}$ such that the union of any collection of negligible open sets is again negligible – this is the case if $(E,{\mathfrak {F}},\mu )$ 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(E, τ) of bounded continuous functions belongs to ${\mathcal {L}}^{\infty }(E,{\mathfrak {F}},\mu )$ .

A strong lifting for $(E,{\mathfrak {F}},\mu )$ is a lifting $T:L^{\infty }(E,{\mathfrak {F}},\mu )\to {\mathcal {L}}^{\infty }(E,{\mathfrak {F}},\mu )$ such that Tφ = φ on Supp(μ) for all φ in C_b(E, τ). This is the same as requiring that^[4] $TU\geq {\big (}U\cap \mathrm {Supp} (\mu ){\big )}$ for all open sets U in τ.

Theorem. If $({\mathfrak {F}},\mu )$ is σ-finite and complete and τ has a countable basis then $(E,{\mathfrak {F}},\mu )$ admits a strong lifting.

Proof. Let T₀ be a lifting for $(E,{\mathfrak {F}},\mu )$ and $\{U_{1},U_{2},\cdots \}$ 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^[5] on $L^{\infty }(E,{\mathfrak {F}},\mu )$ that extends the character φ ↦ φ(p) of C_b(E, τ). Then for p in E and [f] in L^∞ 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 $(E,{\mathfrak {F}},\mu ),(X,{\mathfrak {X}},\nu )$ are σ-finite measure spaces (μ, ν positive) and π : E → X is a measurable map. A disintegration of μ along π with respect to ν is a slew $X\ni x\mapsto \lambda _{x}$ of positive σ-additive measures on $(E,{\mathfrak {F}})$ such that

λ_x is carried by the fiber $\pi ^{-1}(\{x\})$ of π over x:

\{x\}\in {\mathfrak {X}}\;\;\mathrm {and} \;\;\lambda _{x}\left((E\setminus \pi ^{-1}(\{x\})\right)=0\qquad \forall x\in X

for every μ-integrable function f,

\int _{E}f(p)\;\mu (dp)=\int _{X}\left(\int _{\pi ^{-1}(\{x\})}f(p)\,\lambda _{x}(dp)\right)\;\nu (dx)\qquad (*)

in the sense that, for ν-almost all x in X, f is λ_x-integrable, the function

x\mapsto \int _{\pi ^{-1}(\{x\})}f(p)\,\lambda _{x}(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 E is a polish^[6] space and X a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a σ-finite Borel measure on E and π : E → X an ${\mathfrak {F}},{\mathfrak {X}}$ –measurable map. Then there exists a σ-finite Borel measure ν on X and a disintegration (*). If μ is finite, ν can be taken to be the pushforward^[7] π_∗μ, and then the λ_x are probabilities.

Proof. Because of the polish nature of E there is a sequence of compact subsets of E 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 E and X are compact and π is continuous, and ν = π_∗μ. Complete ${\mathfrak {X}}$ under ν and fix a strong lifting T for $(X,{\mathfrak {X}},\nu )$ . Given a bounded μ-measurable function f, let $\lfloor f\rfloor$ denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of^[8] π_∗(fμ) with respect to π_∗μ. Then set, for every x in X, $\lambda _{x}(f):=T(\lfloor f\rfloor )(x).$ 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 _{x}(f\cdot \varphi \circ \pi )=\varphi (x)\lambda _{x}(f)\qquad \forall x\in X,\varphi \in C_{b}(X),f\in L^{\infty }(E,{\mathfrak {F}},\mu )

and take the infimum over all positive φ in C_b(X) with φ(x) = 1; it becomes apparent that the support of λ_x lies in the fiber over x.

References

Bellow, Alexandra (1969), Topics in the theory of lifting, Springer, Berlin, Heidelberg, New York

^ Bellow, Alexandra (1969), Topics in the theory of lifting, Springer, Berlin, Heidelberg, New York
^ A subset N ⊂ E is locally negligible if it intersects every integrable set in ${\mathfrak {F}}$ in a subset of a negligible set of ${\mathfrak {F}}$ . $(E,{\mathfrak {F}},\mu )$ is complete if every locally negligible set is negligible and belongs to ${\mathfrak {F}}$ .
^ i.e., there exists a countable collection of integrable sets – sets of finite measure in ${\mathfrak {F}}$ – that covers the underlying set E.
^ 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 E 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 ${\mathfrak {X}}$ defined by $\nu (A):=\mu \left(\pi ^{-1}(A)\right),A\in {\mathfrak {X}}$ .
^ fμ is the measure that has density f with respect to μ

Existence of liftings

Strong liftings

Application: disintegration of a measure

References

See also