Lifting theory

A lifting on a measure space $(X,\; \backslash Sigma,\; \backslash mu)$ is a linear and multiplicative operator

$$T\; :\; L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)\; \backslash to\; \backslash mathcal\{L\}^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$$

which is a right inverse of the quotient map

$$\backslash begin\{cases\}$$

\mathcal L^\infty(X,\Sigma,\mu) \to L^\infty(X,\Sigma,\mu) \\

f \mapsto [f]

\end{cases}

where $\backslash mathcal\{L\}^\backslash infty(X,\backslash Sigma,\backslash mu)$ is the seminormed L^p space of measurable functions and $L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$ 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 $p\; \backslash in\; X$ and all $r,\; s\; \backslash in\; \backslash Reals,$

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

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

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.

Theorem. Suppose $(X,\; \backslash Sigma,\; \backslash mu)$ is complete.A subset $N\; \backslash subseteq\; X$ is locally negligible if it intersects every integrable set in $\backslash Sigma$ in a subset of a negligible set of $\backslash Sigma.$ $(X,\; \backslash Sigma,\; \backslash mu)$ is complete if every locally negligible set is negligible and belongs to $\backslash Sigma.$ Then $(X,\; \backslash Sigma,\; \backslash mu)$ admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in $\backslash Sigma$ whose union is $X.$

In particular, if $(X,\; \backslash Sigma,\; \backslash mu)$ is the completion of a σ-finitei.e., there exists a countable collection of integrable sets – sets of finite measure in $\backslash Sigma$ – that covers the underlying set $X.$ measure or of an inner regular Borel measure on a locally compact space, then $(X,\; \backslash Sigma,\; \backslash mu)$ 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.

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

A strong lifting for $(X,\; \backslash Sigma,\; \backslash mu)$ is a lifting

$$T\; :\; L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)\; \backslash to\; \backslash mathcal\{L\}^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$$

such that $T\backslash varphi\; =\; \backslash varphi$ on $\backslash operatorname\{Supp\}(\backslash mu)$ for all $\backslash varphi$ in $C\_b(X,\; \backslash tau).$ This is the same as requiring that$U,$ $\backslash operatorname\{Supp\}(\backslash mu)$ are identified with their indicator functions. $T\; U\; \backslash geq\; (U\; \backslash cap\; \backslash operatorname\{Supp\}(\backslash mu))$ for all open sets $U$ in $\backslash tau.$

Theorem. If $(\backslash Sigma,\; \backslash mu)$ is σ-finite and complete and $\backslash tau$ has a countable basis then $(X,\; \backslash Sigma,\; \backslash mu)$ admits a strong lifting.

Proof. Let $T\_0$ be a lifting for $(X,\; \backslash Sigma,\; \backslash mu)$ and $U\_1,\; U\_2,\; \backslash ldots$ a countable basis for $\backslash tau.$ For any point $p$ in the negligible set

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 $L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$ that extends the character $\backslash phi\; \backslash mapsto\; \backslash phi(p)$ of $C\_b(X,\; \backslash tau).$ Then for $p$ in $X$ and $[f]$ in $L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$ define:

T_p[f]& p\in N.

\end{cases}

$T$ is the desired strong lifting.

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

1. $\backslash lambda\_y$ is carried by the fiber $\backslash pi^\{-1\}(\backslash \{y\backslash \})$ of $\backslash pi$ over $y$, i.e. $\backslash \{y\backslash \}\; \backslash in\; \backslash Phi$ and $\backslash lambda\_y\backslash left((X\backslash setminus\; \backslash pi^\{-1\}(\backslash \{y\backslash \})\backslash right)\; =\; 0$ for almost all $y\; \backslash in\; Y$
2. for every $\backslash mu$-integrable function $f,$$$\backslash int\_X\; f(p)\backslash ;\backslash mu(dp)=\; \backslash int\_Y\; \backslash left(\backslash int\_\{\backslash pi^\{-1\}(\backslash \{y\backslash \})\}\; f(p)\backslash ,\backslash lambda\_y(dp)\backslash right)\; \backslash nu(dy)\; \backslash qquad\; (*)$$ in the sense that, for $\backslash nu$-almost all $y$ in $Y,$ $f$ is $\backslash lambda\_y$-integrable, the function $$y\; \backslash mapsto\; \backslash int\_\{\backslash pi^\{-1\}(\backslash \{y\backslash \})\}\; f(p)\backslash ,\backslash lambda\_y(dp)$$ is $\backslash nu$-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 spaceA separable space is Polish if its topology comes from a complete metric. In the present situation it would be sufficient to require that $X$ is Suslin, that is, is the continuous Hausdorff image of a Polish space. and $Y$ a separable Hausdorff space, both equipped with their Borel σ-algebras. Let $\backslash mu$ be a σ-finite Borel measure on $X$ and $\backslash pi\; :\; X\; \backslash to\; Y$ a $\backslash Sigma,\; \backslash Phi-$measurable map. Then there exists a σ-finite Borel measure $\backslash nu$ on $Y$ and a disintegration (*).

If $\backslash mu$ is finite, $\backslash nu$ can be taken to be the pushforwardThe pushforward $\backslash pi\_*\; \backslash mu$ of $\backslash mu$ under $\backslash pi,$ also called the image of $\backslash mu$ under $\backslash pi$ and denoted $\backslash pi(\backslash mu),$ is the measure $\backslash nu$ on $\backslash Phi$ defined by $\backslash nu(A)\; :=\; \backslash mu\backslash left(\backslash pi^\{-1\}(A)\backslash right)$ for $A$ in $\backslash Phi$. $\backslash pi\_*\; \backslash mu,$ and then the $\backslash lambda\_y$ 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 $\backslash pi$ is continuous. This observation reduces the problem to the case that both $X$ and $Y$ are compact and $\backslash pi$ is continuous, and $\backslash nu\; =\; \backslash pi\_*\; \backslash mu.$ Complete $\backslash Phi$ under $\backslash nu$ and fix a strong lifting $T$ for $(Y,\; \backslash Phi,\; \backslash nu).$ Given a bounded $\backslash mu$-measurable function $f,$ let $\backslash lfloor\; f\backslash rfloor$ denote its conditional expectation under $\backslash pi,$ that is, the Radon-Nikodym derivative of$f\; \backslash mu$ is the measure that has density $f$ with respect to $\backslash mu$ $\backslash pi\_*(f\; \backslash mu)$ with respect to $\backslash pi\_*\; \backslash mu.$ Then set, for every $y$ in $Y,$ $\backslash lambda\_y(f)\; :=\; T(\backslash lfloor\; f\backslash rfloor)(y).$ 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

$$\backslash lambda\_y(f\; \backslash cdot\; \backslash varphi\; \backslash circ\; \backslash pi)\; =\; \backslash varphi(y)\; \backslash lambda\_y(f)\; \backslash qquad\; \backslash forall\; y\backslash in\; Y,\; \backslash varphi\; \backslash in\; C\_b(Y),\; f\; \backslash in\; L^\backslash infty(X,\; \backslash Sigma,\; \backslash mu)$$

and take the infimum over all positive $\backslash varphi$ in $C\_b(Y)$ with $\backslash varphi(y)\; =\; 1;$ it becomes apparent that the support of $\backslash lambda\_y$ lies in the fiber over $y.$

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Category:Measure theory