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 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 $T[1]=1$ and for all $p\in X$ and all $r,s\in \mathbb {R} ,$

$$T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),$$

$$T([f]\times [g])(p)=T[f](p)\times T[g](p).$$

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.

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 $TU\geq (U\cap \operatorname {Supp} (\mu ))$ 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

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

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 $y.$