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
$T : L^\infty(X, \Sigma, \mu) \to \mathcal^\infty(X, \Sigma, \mu)$
which is a right inverse of the quotient map
\begin
\mathcal L^\infty(X,\Sigma,\mu) \to L^\infty(X,\Sigma,\mu) \\
f \mapsto \end

where $\mathcal^\infty(X,\Sigma,\mu)$ 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 /math> of bounded measurable functions modulo negligible functions a representative— which is henceforth written T( or T /math> or simply $Tf$ — in such a way that T = 1 and for all $p \in X$ and all $r, s \in \Reals,$
T(r s (p) = rT p) + sT p),
T( times (p) = T p) \times T 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 $\mu,$ $\operatorname(\mu),$ 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
$T : L^\infty(X, \Sigma, \mu) \to \mathcal^\infty(X, \Sigma, \mu)$
such that $T\varphi = \varphi$ on $\operatorname(\mu)$ for all $\varphi$ in $C_b(X, \tau).$ This is the same as requiring that $T U \geq (U \cap \operatorname(\mu))$ 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\$
let $T_p$ be any characterA ''character'' on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1. 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 /math> in $L^\infty(X, \Sigma, \mu)$ define:
(T (p):= \begin (T_0 (p)& p\notin N\\
T_p p\in N.
\end
$T$ is the desired strong lifting.

Application: disintegration of a measure

Suppose $(X, \Sigma, \mu)$ and $(Y, \Phi, \nu)$ are ''σ''-finite measure spaces ($\mu, \nu$ 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

#$\lambda_y$ is carried by the fiber $\pi^(\)$ of $\pi$ over $y$, i.e. $\ \in \Phi$ and $\lambda_y\left((X\setminus \pi^(\)\right) = 0$ for almost all $y \in Y$
#for every $\mu$-integrable function $f,$$\int_X f(p)\;\mu(dp)= \int_Y \left(\int_ 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 $y \mapsto \int_ f(p)\,\lambda_y(dp)$ 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$f \mu$ is the measure that has density $f$ with respect to $\mu$ $\pi_*(f \mu)$ 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 $y.$

References

{{Measure theory

Measure theory