Lifting theory

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Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by Alfréd Haar),[1] followed later by Dorothy Maharam’s (1958) paper,[2] and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper.[3] Lifting Theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph,[4] now a standard reference in the field. Lifting Theory continued to develop after 1969, yielding significant new results and applications.

A lifting on a measure space (X, Σ, μ) is a linear and multiplicative inverse

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

of the quotient map

\begin{cases}\mathcal L^\infty(X,\Sigma,\mu)\to L^\infty(X,\Sigma,\mu) \\
 f\mapsto [f]\end{cases}

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[f]+s[g])(p)=rT[f](p) + sT[g](p), \qquad \forall p\in X, r,s\in \mathbf R;
T([f]\times[g])(p)=T[f](p)\times T[g](p), \qquad \forall p\in X;
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.

Existence of liftings[edit]

Theorem. Suppose (X, Σ, μ) is complete.[5] 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[6] 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[edit]

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 Cb(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 Cb(X, τ). This is the same as requiring that[7] 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 T0 be a lifting for (X, Σ, μ) and {U1, U2, ...} a countable basis for τ. For any point p in the negligible set

N:=\bigcup\nolimits _n \left\{p\in \mathrm{Supp}(\mu): (T_0U_n)(p)<U_n(p) \right\}

let Tp be any character[8] on L(X, Σ, μ) that extends the character φ ↦ φ(p) of Cb(X, τ). Then for p in X and [f] in L(X, Σ, μ) define:

 (T[f])(p):= \begin{cases} (T_0[f])(p)& p\notin N\\
T_p[f]& p\in N.
\end{cases}

T is the desired strong lifting.

Application: disintegration of a measure[edit]

Suppose (X, Σ, μ), (Y, Φ, ν) are σ-finite measure spaces (μ, ν positive) and π : XY 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

  1. λy is carried by the fiber \pi^{-1}(\{y\}) of π over y:
 \{y\}\in\Phi\;\;\mathrm{ and }\;\; \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right)=0 \qquad \forall y\in Y
  1. for every μ-integrable function f,
 \int_X f(p)\;\mu(dp)= \int_Y \left(\int_{\pi^{-1}(\{y\})}f(p)\,\lambda_y(dp)\right) \nu(dy) \qquad (*)
in the sense that, for ν-almost all y in Y, f is λy-integrable, the function
 y\mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp)
is ν-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[9] 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 : XY a Σ, Φ–measurable map. Then there exists a σ-finite Borel measure ν on Y and a disintegration (*). If μ is finite, ν can be taken to be the pushforward[10] πμ, and then the λy 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 ν = πμ. Complete Φ under ν and fix a strong lifting T for (Y, Φ, ν). Given a bounded μ-measurable function f, let \lfloor f\rfloor denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of[11] π(fμ) with respect to πμ. 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 Cb(Y) with φ(y) = 1; it becomes apparent that the support of λy lies in the fiber over y.

References[edit]

  1. ^ von Neumann, J.: Algebraische Repräsentanten der Funktionen bis auf eine Menge von Maße Null. J. Crelle 165, 109-115 (1931)
  2. ^ Maharam, D.: On a theorem of von Neumann. Proc. Amer. Math. Soc. 9, 987-995 (1958)
  3. ^ A. Ionescu Tulcea and C. Ionescu Tulcea: On the lifting property, I., J. Math. Anal. App. 3, 537-546 (1961)
  4. ^ Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, Topics in the Theory of Lifting, Ergebnisse der Mathematik, Vol. 48, Springer-Verlag, Berlin, Heidelberg, New York (1969)
  5. ^ A subset NX is locally negligible if it intersects every integrable set in Σ in a subset of a negligible set of Σ. (X, Σ, μ) is complete if every locally negligible set is negligible and belongs to Σ.
  6. ^ i.e., there exists a countable collection of integrable sets –sets of finite measure in Σ– that covers the underlying set X.
  7. ^ U, Supp(μ) are identified with their indicator functions.
  8. ^ A character on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.
  9. ^ A 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, i.e., is the continuous Hausdorff image of a polish space.
  10. ^ The pushforward πμ of μ under π, also called the image of μ under π and denoted π(μ), is the measure ν on Φ defined by \nu(A):=\mu\left(\pi^{-1}(A)\right) for A in Φ.
  11. ^ fμ is the measure that has density f with respect to μ