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. 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 operator $$T : L^\infty(X, \Sigma, \mu) \to \mathcal{L}^\infty(X, \Sigma, \mu)$$ which is a right inverse 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 Lp 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&mdash; which is henceforth written $$T([f])$$ or $$T[f]$$ or simply $$Tf$$ &mdash; in such a way that $$T[1] = 1$$ and for all $$p \in X$$ and all $$r, s \in \Reals,$$ $$T(r[f]+s[g])(p) = rT[f](p) + sT[g](p),$$ $$T([f]\times[g])(p) = T[f](p) \times T[g](p).$$

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 $$(X, \Sigma, \mu)$$ is complete. Then $$(X, \Sigma, \mu)$$ admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in $$\Sigma$$ whose union is $$X.$$

In particular, if $$(X, \Sigma, \mu)$$ is the completion of a σ-finite measure or of an inner regular Borel measure on a locally compact space, then $$(X, \Sigma, \mu)$$ 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. <!-- Here are the details. Henceforth write Tf := T[f] = T([f]). $$(\Sigma, \mu)$$ is σ-finite if there exists a countable collection of sets of finite measure in $$\Sigma$$ whose union has negligible complement. This permits a reduction to the case that the measure $$\mu$$ is finite, in fact, it may be taken to be a probability. The proof uses Zorn's lemma together with the following order on pairs $$(\mathfrak A,T_{\mathfrak A})$$ of sub-σ-algebras $$\mathfrak A$$ of $$\Sigma$$ and liftings $$T_{\mathfrak A}$$ for them: $$ (\mathfrak A,T_{\mathfrak A})\le(\mathfrak B,T_{\mathfrak B}) $$ if $$\mathfrak A\subseteq\mathfrak B$$ and $$T_{\mathfrak A}$$ is the restriction of $$T_{\mathfrak B}$$ to $$L^\infty(X,\mathfrak A,\mu)$$. It is to be shown that a chain $$\mathfrak C$$ of such pairs has an upper bound, and that a maximal pair, which then exists by Zorn's lemma, has $$\Sigma$$ for its first entry.

If $$\mathfrak C$$ has no countable cofinal subset, then the union $$\mathfrak U := \bigcup\{\mathfrak A:\,(\mathfrak A,T_{\mathfrak A}) \in \mathfrak C\}$$ is a σ-algebra and there is an obvious lifting $$T_{\mathfrak U}$$ for it that restricts to the liftings of the chain; $$(\mathfrak U,T_{\mathfrak U})$$ is the sought upper bound of the chain.

The argument is more complicated when the chain $$ \mathfrak C$$ has a countable cofinal subset $$\left\{(\mathfrak A_n,T_{\mathfrak A_n}), n = 1, 2, \ldots\right\}$$. In this case let $$\mathfrak U$$ be the σ-algebra generated by the union $$\bigcup\{\mathfrak A_n: \, n = 1, 2, \ldots\},$$ which is generally only an algebra of sets. For the construction of $$T_{\mathfrak U}$$ it is convenient to identify a set $$A \subseteq X$$ with its indicator function and to write $$TA := TI_A=T[I_A].$$ For $$A \in \mathfrak U$$ let $$A_n$$ denote the conditional expectation of $$A$$ under $$\mathfrak A_n$$. By Doob's martingale convergence theorem the set $$\theta(A)$$ of points where $$A_n$$ converges to 1 differs negligibly from A.

Here are a few facts that are straightforward to check (some use the completeness and finiteness of $$(X, \mathfrak U, \mu)$$): $$\tau := \{\theta(A)\setminus N \ : \ A\in\mathfrak U, \mu(N) = 0\}\subset\mathfrak U$$ is a topology whose only negligible open set is the empty set and such that every $$ A=I_A\in\mathfrak U$$ is almost everywhere continuous, to wit, on $$ A\cap\theta(A)$$ and on $$ A^c\cap\theta(A^c)$$. Then every $$f \in\mathcal L^\infty(X,\mathfrak U,\mu)$$, being the uniform limit of a sequence of step functions over $$\mathfrak U$$, is almost everywhere continuous in this topology. For $$p$$ in $$X$$ $$I_p:=\{[f]: f \text{ is continuous at } p \text{ and }f(p) = 0\}.$$ is a proper ideal of $$ L^\infty(X,\mathfrak U,\mu)$$, contained (by another application of Zorn's lemma) in some maximal proper ideal $$J_p\subset L^\infty(X, \mathfrak U, \mu),$$ which has codimension 1. The quotient map $$L^\infty(X, \mathfrak U, \mu) \to L^\infty(X, \mathfrak U,\mu) / J_p$$ can be viewed as a character Tp. Defining $$\left(T_{\mathfrak U}[f]\right)(p):=T_p[f]\;\;,\;\;\;\;\;\;p\in E,$$ provides the upper bound $$(\mathfrak U,T_{\mathfrak U})$$ for the chain $$\mathfrak C$$.

In either case the chain $$ \mathfrak C$$ therefore has an upper bound. By Zorn's lemma there is a maximal pair $$(\mathfrak U,T_{\mathfrak U})$$, and a small additional calculation shows that $$ \mathfrak U=\mathfrak F$$. END OF DETAILED PROOF-->

Strong liftings
Suppose $$(X, \Sigma, \mu)$$ is complete and $$X$$ is equipped with a completely regular Hausdorff topology $$\tau \subseteq \Sigma$$ such that the union of any collection of negligible open sets is again negligible – this is the case if $$(X, \Sigma, \mu)$$ is σ-finite or comes from a Radon measure. Then the support of $$\mu,$$ $$\operatorname{Supp}(\mu),$$ can be defined as the complement of the largest negligible open subset, and the collection $$C_b(X, \tau)$$ of bounded continuous functions belongs to $$ \mathcal L^\infty(X, \Sigma, \mu).$$

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

"Theorem. If $(\Sigma, \mu)$ is σ-finite and complete and $\tau$ has a countable basis then $(X, \Sigma, \mu)$ admits a strong lifting."

Proof. Let $$T_0$$ be a lifting for $$(X, \Sigma, \mu)$$ and $$U_1, U_2, \ldots$$ a countable basis for $$\tau.$$ For any point $$p$$ in the negligible set $$N := \bigcup\nolimits_n \left\{p \in \operatorname{Supp}(\mu) : (T_0U_n)(p) < U_n(p)\right\}$$ let $$T_p$$ be any character on $$L^\infty(X, \Sigma, \mu)$$ that extends the character $$\phi \mapsto \phi(p)$$ of $$C_b(X, \tau).$$ Then for $$p$$ in $$X$$ and $$[f]$$ in $$L^\infty(X, \Sigma, \mu)$$ 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 $$(X, \Sigma, \mu)$$ and $$(Y, \Phi, \nu)$$ are σ-finite measure spaces ($$\mu, \mu$$ positive) and $$\pi : X \to Y$$ is a measurable map. A disintegration of $$\mu$$ along $$\pi$$ with respect to $$\nu$$ is a slew $$Y \ni y \mapsto \lambda_y$$ of positive σ-additive measures on $$(\Sigma, \mu)$$ such that


 * 1) $$\lambda_y$$ is carried by the fiber $$\pi^{-1}(\{y\})$$ of $$\pi$$ over $$y$$, i.e. $$ \{y\} \in \Phi $$ and $$ \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right) = 0 $$ for almost all $$ y \in Y $$
 * 2) for every $$\mu$$-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 $$\nu$$-almost all $$y$$ in $$Y,$$ $$f$$ is $$\lambda_y$$-integrable, the function $$y \mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp) $$ is $$\nu$$-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 $$\mu$$ be a σ-finite Borel measure on $$X$$ and $$\pi : X \to Y$$ a $$\Sigma, \Phi-$$measurable map. Then there exists a σ-finite Borel measure $$\nu$$ on $$Y$$ and a disintegration (*).

If $$\mu$$ is finite, $$\nu$$ can be taken to be the pushforward $$\pi_* \mu,$$ and then the $$\lambda_y$$ 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 $$\pi$$ is continuous. This observation reduces the problem to the case that both $$X$$ and $$Y$$ are compact and $$\pi$$ is continuous, and $$\nu = \pi_* \mu.$$ Complete $$\Phi$$ under $$\nu$$ and fix a strong lifting $$T$$ for $$(Y, \Phi, \nu).$$ Given a bounded $$\mu$$-measurable function $$f,$$ let $$\lfloor f\rfloor$$ denote its conditional expectation under $$\pi,$$ that is, the Radon-Nikodym derivative of $$\pi_*(f \mu)$$ with respect to $$\pi_* \mu.$$ 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 $$\varphi$$ in $$C_b(Y)$$ with $$\varphi(y) = 1;$$ it becomes apparent that the support of $$\lambda_y$$ lies in the fiber over $$y.$$