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.

A lifting on a measure space ${\displaystyle (X,\Sigma ,\mu )}$is a linear and multiplicative operator $${\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )}$$which is a right inverse 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 ${\displaystyle [f]}$of bounded measurable functions modulo negligible functions a representative— which is henceforth written ${\displaystyle T([f])}$or ${\displaystyle T[f]}$or simply ${\displaystyle Tf}$— in such a way that ${\displaystyle T[1]=1}$and for all ${\displaystyle p\in X}$and all ${\displaystyle r,s\in \mathbb {R} ,}$$${\displaystyle T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),}$$$${\displaystyle T([f]\times [g])(p)=T[f](p)\times T[g](p).}$$

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.

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.

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

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

Proof. Let ${\displaystyle T_{0}}$be a lifting for ${\displaystyle (X,\Sigma ,\mu )}$and ${\displaystyle U_{1},U_{2},\ldots }$a countable basis for ${\displaystyle \tau .}$For any point ${\displaystyle p}$in the negligible set $${\displaystyle N:=\bigcup \nolimits _{n}\left\{p\in \operatorname {Supp} (\mu ):(T_{0}U_{n})(p)<U_{n}(p)\right\}}$$let ${\displaystyle T_{p}}$be any character on ${\displaystyle L^{\infty }(X,\Sigma ,\mu )}$that extends the character ${\displaystyle \phi \mapsto \phi (p)}$of ${\displaystyle C_{b}(X,\tau ).}$Then for ${\displaystyle p}$in ${\displaystyle X}$and ${\displaystyle [f]}$in ${\displaystyle L^{\infty }(X,\Sigma ,\mu )}$define: $${\displaystyle (T[f])(p):={\begin{cases}(T_{0}[f])(p)&p\notin N\\T_{p}[f]&p\in N.\end{cases}}}$$${\displaystyle T}$is the desired strong lifting.

Suppose ${\displaystyle (X,\Sigma ,\mu )}$and ${\displaystyle (Y,\Phi ,\nu )}$are σ-finite measure spaces (${\displaystyle \mu ,\nu }$positive) and ${\displaystyle \pi :X\to Y}$is a measurable map. A disintegration of ${\displaystyle \mu }$along ${\displaystyle \pi }$with respect to ${\displaystyle \nu }$is a slew ${\displaystyle Y\ni y\mapsto \lambda _{y}}$of positive σ-additive measures on ${\displaystyle (\Sigma ,\mu )}$such that

1. ${\displaystyle \lambda _{y}}$is carried by the fiber ${\displaystyle \pi ^{-1}(\{y\})}$of ${\displaystyle \pi }$over ${\displaystyle y}$, i.e. ${\displaystyle \{y\}\in \Phi }$and ${\displaystyle \lambda _{y}\left((X\setminus \pi ^{-1}(\{y\})\right)=0}$for almost all ${\displaystyle y\in Y}$
2. for every ${\displaystyle \mu }$-integrable function ${\displaystyle 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 ${\displaystyle \nu }$-almost all ${\displaystyle y}$in ${\displaystyle Y,}$${\displaystyle f}$is ${\displaystyle \lambda _{y}}$-integrable, the function $${\displaystyle y\mapsto \int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)}$$is ${\displaystyle \nu }$-integrable, and the displayed equality ${\displaystyle (*)}$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.

Proof. Because of the polish nature of ${\displaystyle X}$there is a sequence of compact subsets of ${\displaystyle X}$that are mutually disjoint, whose union has negligible complement, and on which ${\displaystyle \pi }$is continuous. This observation reduces the problem to the case that both ${\displaystyle X}$and ${\displaystyle Y}$are compact and ${\displaystyle \pi }$is continuous, and ${\displaystyle \nu =\pi _{*}\mu .}$Complete ${\displaystyle \Phi }$under ${\displaystyle \nu }$and fix a strong lifting ${\displaystyle T}$for ${\displaystyle (Y,\Phi ,\nu ).}$Given a bounded ${\displaystyle \mu }$-measurable function ${\displaystyle f,}$let ${\displaystyle \lfloor f\rfloor }$denote its conditional expectation under ${\displaystyle \pi ,}$that is, the Radon-Nikodym derivative of ${\displaystyle \pi _{*}(f\mu )}$with respect to ${\displaystyle \pi _{*}\mu .}$Then set, for every ${\displaystyle y}$in ${\displaystyle 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 ${\displaystyle \varphi }$in ${\displaystyle C_{b}(Y)}$with ${\displaystyle \varphi (y)=1;}$it becomes apparent that the support of ${\displaystyle \lambda _{y}}$lies in the fiber over ${\displaystyle y.}$