Lifting theory

Template:Short description 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 $(X,\mathrm{\Sigma },\mu )$ is a linear and multiplicative operator $$T:L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\to {ℒ}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$$ which is a right inverse of the quotient map $$\{\begin{array}{c}{ℒ}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\to L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )\\ f\mapsto [f]\end{array}$$

where ${ℒ}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$ is the seminormed L^p space of measurable functions and $L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$ 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 $r,s\in ℝ,$ $$T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),$$ $$T([f]\times [g])(p)=T[f](p)\times T[g](p).$$

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

$(X,\mathrm{\Sigma },\mu )$

is complete. Then

$(X,\mathrm{\Sigma },\mu )$

admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in

$\mathrm{\Sigma }$

whose union is

$X.$

In particular, if

$(X,\mathrm{\Sigma },\mu )$

is the completion of a σ-finite measure or of an inner regular Borel measure on a locally compact space, then

$(X,\mathrm{\Sigma },\mu )$

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 $(X,\mathrm{\Sigma },\mu )$ is complete and $X$ is equipped with a completely regular Hausdorff topology $\tau \subseteq \mathrm{\Sigma }$ such that the union of any collection of negligible open sets is again negligible – this is the case if $(X,\mathrm{\Sigma },\mu )$ is σ-finite or comes from a Radon measure. Then the support of $\mu ,$ $\mathrm{Supp}(\mu ),$ 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 ${ℒ}^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu ).$

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

Theorem. If

$(\mathrm{\Sigma },\mu )$

is σ-finite and complete and

$\tau$

has a countable basis then

$(X,\mathrm{\Sigma },\mu )$

admits a strong lifting.

Proof. Let $T_0$ be a lifting for $(X,\mathrm{\Sigma },\mu )$ and $U_1,U_2,\dots$ a countable basis for $\tau .$ For any point $p$ in the negligible set $$N:=\bigcup _n\{p\in \mathrm{Supp}(\mu ):(T_0{U}_n)(p)<U_n(p)\}$$ let $T_p$ be any character on $L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$ that extends the character $\varphi \mapsto \varphi (p)$ of $C_b(X,\tau ).$ Then for $p$ in $X$ and $[f]$ in $L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$ define: $$(T[f])(p):=\{\begin{array}{cc}(T_0[f])(p)& p\notin N\\ T_p[f]& p\in N.\end{array}$$ $T$ is the desired strong lifting.

Application: disintegration of a measure

Suppose $(X,\mathrm{\Sigma },\mu )$ and $(Y,\mathrm{\Phi },\nu )$ 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 $(\mathrm{\Sigma },\mu )$ such that

1. ${\lambda }_y$ is carried by the fiber ${\pi }^{-1}(\{y\})$ of $\pi$ over $y$, i.e. $\{y\}\in \mathrm{\Phi }$ and ${\lambda }_y((X\setminus {\pi }^{-1}(\{y\}))=0$ for almost all $y\in Y$
2. for every $\mu$-integrable function $f,$$$\underset{X}{\int }f(p)\mu (dp)=\underset{Y}{\int }(\underset{{\pi }^{-1}(\{y\})}{\int }f(p){\lambda }_y(dp))\nu (dy)(*)$$ in the sense that, for $\nu$-almost all $y$ in $Y,$ $f$ is ${\lambda }_y$-integrable, the function $$y\mapsto \underset{{\pi }^{-1}(\{y\})}{\int }f(p){\lambda }_y(dp)$$ 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

$\mathrm{\Sigma },\mathrm{\Phi }-$

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 $\mathrm{\Phi }$ under $\nu$ and fix a strong lifting $T$ for $(Y,\mathrm{\Phi },\nu ).$ 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 $${\lambda }_y(f\cdot \phi \circ \pi )=\phi (y){\lambda }_y(f)\mathrm{\forall }y\in Y,\phi \in C_b(Y),f\in L^{\mathrm{\infty }}(X,\mathrm{\Sigma },\mu )$$ and take the infimum over all positive $\phi$ in $C_b(Y)$ with $\phi (y)=1;$ it becomes apparent that the support of ${\lambda }_y$ lies in the fiber over $y.$