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

:$T\colon L^\infty(X,\Sigma,\mu)\to \mathcal L^\infty(X,\Sigma,\mu)$

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 $\mathcal L^\infty(X,\Sigma,\mu)$ 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 '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(r s (p)=rT p) + sT p), \qquad \forall p\in X, r,s\in \mathbf R;
:T( times (p)=T p)\times T p), \qquad \forall p\in X;
:T 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.

Existence of liftings

Theorem. Suppose (''X'', Σ, ''μ'') is complete. Then (''X'', Σ, ''μ'') admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in Σ whose union is ''X''.

In particular, if (''X'', Σ, ''μ'') is the completion of a ''σ''-finite measure or of an inner regular Borel measure on a locally compact space, then (''X'', Σ, ''μ'') 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'', Σ, ''μ'') is complete and ''X'' is equipped with a completely regular Hausdorff topology τ ⊂ Σ such that the union of any collection of negligible open sets is again negligible – this is the case if (''X'', Σ, ''μ'') is ''σ''-finite or comes from a Radon measure. Then the ''support'' of ''μ'', Supp(''μ''), can be defined as the complement of the largest negligible open subset, and the collection ''C_b''(''X'', ''τ'') of bounded continuous functions belongs to $\mathcal L^\infty(X,\Sigma,\mu)$.

A strong lifting for (''X'', Σ, ''μ'') is a lifting
:$T:L^\infty(X,\Sigma,\mu)\to \mathcal L^\infty(X,\Sigma,\mu)$
such that ''Tφ'' = ''φ'' on Supp(''μ'') for all φ in ''C_b''(''X'', τ). This is the same as requiring that ''TU'' ≥ (''U'' ∩ Supp(''μ'')) for all open sets ''U'' in ''τ''.

Theorem. If (Σ, ''μ'') is ''σ''-finite and complete and ''τ'' has a countable basis then (''X'', Σ, ''μ'') admits a strong lifting.

Proof. Let ''T''₀ be a lifting for (''X'', Σ, ''μ'') and a countable basis for ''τ''. 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''^∞(''X'', Σ, ''μ'') that extends the character φ ↦ φ(''p'') of ''C_b''(''X'', τ). Then for ''p'' in ''X'' and 'f''in ''L''^∞(''X'', Σ, ''μ'') 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'', Σ, ''μ''), (''Y'', Φ, ν) are ''σ''-finite measure spaces (''μ'', ''ν'' positive) and ''π'' : ''X'' → ''Y'' is a measurable map. A disintegration of ''μ'' along ''π'' with respect to ''ν'' is a slew $Y\ni y\mapsto \lambda_y$ of positive ''σ''-additive measures on (''X'', Σ) such that

#λ_''y'' is carried by the fiber $\pi^(\)$ of π over ''y'':
:::$\\in\Phi\;\;\mathrm\;\; \lambda_y\left((X\setminus \pi^(\)\right)=0 \qquad \forall y\in Y$
#for every ''μ''-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 ''λ''_''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 ''μ'' be a ''σ''-finite Borel measure on ''X'' and π : ''X'' → ''Y'' a Σ, Φ–measurable map. Then there exists a σ-finite Borel measure $\nu$ on ''Y'' and a disintegration (*).

If ''μ'' is finite, $\nu$ can be taken to be the pushforward ''π''_∗''μ'', and then the ''λ''_''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 π is continuous. This observation reduces the problem to the case that both ''X'' and ''Y'' are compact and π is continuous, and $\nu$ = ''π''_∗''μ''. Complete Φ under $\nu$ and fix a strong lifting ''T'' for (''Y'', Φ, $\nu$). Given a bounded ''μ''-measurable function ''f'', let $\lfloor f\rfloor$ denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of''fμ'' is the measure that has density ''f'' with respect to ''μ'' ''π''_∗(''fμ'') with respect to ''π''_∗''μ''. 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 ''φ'' in ''C''_''b''(''Y'') with ''φ''(''y'') = 1; it becomes apparent that the support of ''λ''_''y'' lies in the fiber over ''y''.

References

Measure theory