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

$[\ ]:\mathcal L^\infty(E,\mathfrak F,\mu)\to L^\infty(E,\mathfrak F,\mu)\;\;,\; f\mapsto [f].\;$

In other words, a lifting picks from every equivalence class $\underline{}[f]$ of bounded measurable functions modulo negligible functions a representative
— which is henceforth written $\underline{}T([f])$ or $\underline{}T[f]$ or simply $\underline{}Tf$ —
in such a way that

$T(r[f]%2Bs[g])(p)=rT[f](p) %2B sT[g](p)\;\;\forall p\in E\;,r,s\in \mathbf R\;,$

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

$\mathrm{one\ requires\ also\ that\ }T[1]=1\;.$

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.

1 Existence of liftings
2 Strong liftings
3 Application: disintegration of a measure
4 References
5 See also

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 sigma-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-sigma-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.

Strong liftings

Suppose now that $(E,\mathfrak F,\mu)$ is complete and that $\underline{}E$ comes 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 sigma-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 $\underline{}C_b(E,\tau)$ of bounded continuous functions belongs to $\mathcal L^\infty(E,\mathfrak F,\mu)$ .

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

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

Proof. Let $\underline{}T_0$ be a lifting for $(E,\mathfrak F,\mu)$ and $\{U_n\;:\;n=1,2\ldots\}$ a countable basis for $\tau$ . For any point $p$ in the negligible set $N:=\bigcup{}_n\left\{p\in \mathrm{supp}(\mu): (T_0U_n)(p)<U_n(p)\right\}$ let $\,T_p$ be any character^[5] on $L^\infty(E,\mathfrak F,\mu)$ that extends the character $\underline{}\phi\mapsto\phi(p)$ of $\underline{}C_b(E,\tau)$ . Then define, for $\underline{}p\in E$ and $[f]\in L^\infty$ ,

$\big(T[f]\big)(p):=\left\{ \begin{array}{lr} \big(T_0[f]\big)(p)&\mathrm{\ if\ }p\notin N\\ \;\;\;\;T_p[f]&\mathrm{\ \ \ if\ }p\in N\;.\\ \end{array} \right.$

$T$ is the desired strong lifting.

Application: disintegration of a measure

Suppose $(E,\mathfrak F,\mu)$ and $(X,\mathfrak X,\nu)$ are sigma-finite measure spaces ( $\mu,\nu$ positive) and $\pi:E\to X$ is a measurable map. A disintegration of $\;\mu$ along $\;\pi\;$ with respect to $\;\nu\;$ is a slew $X\ni x\mapsto \lambda_x$ of positive sigma-additive measures on $(E,\mathfrak F)$ such that (1) $\lambda_x$ is carried by the fiber $\;\pi^{-1}(\{x\})\;$ of $\,\pi$ over $\,x$ :

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

and (2) for every $\mu$ -integrable function $\;f$ ,

$\int_E f(p)\;\mu(dp)= \int_X \left(\int_{\pi^{-1}(\{x\})}f(p)\,\lambda_x(dp)\right) \;\nu(dx)$ (*)

in the sense that, for $\nu$ -almost all $x\in X\;, f$ is $\lambda_x$ -integrable, the function $x\mapsto \int_{\pi^{-1}(\{x\})} f(p)\,\lambda_x(dp)$ is $\nu$ -integrable, and the displayed equality (*) obtains.

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 sigma-algebras. Let $\mu$ be a sigma-finite Borel measure on $E$ and $\;\pi:E\to X\;$ an $\mathfrak F,\mathfrak X$ –measurable map. Then there exists a sigma-finite Borel measure $\nu$ on $X$ and a disintegration (*).
If $\mu$ is finite, $\nu$ can be taken to be the pushforward^[7] $\pi_\ast\mu$ , and then the $\lambda_x$ 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 $\,\pi$ is continuous. This observation reduces the problem to the case that both $E$ and $X$ are compact and $\,\pi$ is continuous, and $\nu=\pi_\ast\mu$ . Complete $\mathfrak X$ under $\nu$ and fix a strong lifting $T$ for $(X,\mathfrak X,\nu)$ . Given a bounded $\mu$ -measurable function $\;f$ , let $\lfloor f\rfloor$ denote its conditional expectation under $\pi$ , i.e., the Radon-Nikodym derivative of^[8] $\pi_\ast(f\mu)$ with respect to $\nu=\pi_\ast\mu$ . 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

$\lambda_x(f\cdot\phi\circ\pi)=\phi(x) \lambda_x(f) \;\;\;\; \forall x\in X\;,\;\phi\in C_b(X)\;,\;f\in L^\infty(E,\mathfrak F,\mu)\;$

and take the infimum over all positive $\phi\in C_b(X)$ with $\,\phi(x)=1$ ; it becomes apparent that the support of $\lambda_x$ 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 $N\subset E$ 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 $E$ .
^ $U,\mathrm{supp}(\mu)$ 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 $\pi_\ast\mu$ of $\mu$ under $\pi$ , also called the image of $\mu$ under $\pi$ and denoted $\pi(\mu)$ , is the measure $\nu$ on $\mathfrak X$ defined by $\nu(A):=\mu\big(\pi^{-1}(A)\big)\;,\;A\in\mathfrak X$ .
^ $\;f\mu$ is the measure that has density $\;f$ with respect to $\;\mu$

Lifting theory

Contents

Existence of liftings

Strong liftings

Application: disintegration of a measure

References

See also