Editing Lifting theory

{{Short description|Notion in measure 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]].<ref>
{{Cite journal|year=1931|first=John|last=von Neumann|authorlink=John von Neumann|title=Algebraische Repräsentanten der Funktionen "bis auf eine Menge vom Maße Null"| url=http://www.degruyter.com/view/j/crll.1931.1931.issue-165/crll.1931.165.109/crll.1931.165.109.xml|journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]]|language=de|volume=1931|issue=165|pages=109–115|doi=10.1515/crll.1931.165.109|mr=1581278}}</ref> The theory was further developed by [[Dorothy Maharam]] (1958)<ref>{{Cite journal|last=Maharam|first=Dorothy|authorlink=Dorothy Maharam|year=1958|title=On a theorem of von Neumann|url=https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9939-1958-0105479-6|journal=[[Proceedings of the American Mathematical Society]]|volume=9|issue=6|pages=987–994|doi=10.2307/2033342|jstor=2033342|mr=0105479|doi-access=free}}</ref> and by [[Alexandra Bellow|Alexandra Ionescu Tulcea]] and [[Cassius Ionescu-Tulcea|Cassius Ionescu Tulcea]] (1961).<ref>{{Cite journal|last1= Ionescu Tulcea|first1=Alexandra|author1-link=Alexandra Bellow|last2=Ionescu Tulcea|first2=Cassius|author2-link=Cassius Ionescu-Tulcea|year=1961|title=On the lifting property. I.|journal=[[Journal of Mathematical Analysis and Applications]]|volume=3|issue=3|pages=537–546|doi=10.1016/0022-247X(61)90075-0|mr=0150256|doi-access=free}}</ref> 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.<ref>{{cite book|last1= Ionescu Tulcea|first1=Alexandra|author1-link=Alexandra Bellow|last2=Ionescu Tulcea|first2=Cassius|author2-link=Cassius Ionescu-Tulcea|year=1969|title=Topics in the theory of lifting|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]|volume=48|publisher=[[Springer Science+Business Media|Springer-Verlag]]|location= New York|oclc=851370324|mr=0276438}}</ref> Lifting theory continued to develop since then, yielding new results and applications.

==Definitions==

A '''lifting''' on a [[measure space]] <math>(X, \Sigma, \mu)</math> is a linear and multiplicative operator
<math display=block>T : L^\infty(X, \Sigma, \mu) \to \mathcal{L}^\infty(X, \Sigma, \mu)</math>
which is a [[Inverse function#Right inverses|right inverse]] of the quotient map
<math display=block>\begin{cases}
\mathcal L^\infty(X,\Sigma,\mu) \to L^\infty(X,\Sigma,\mu) \\
f \mapsto [f]
\end{cases}</math>

where <math>\mathcal{L}^\infty(X,\Sigma,\mu)</math> is the seminormed [[Lp space|L<sup>p</sup> space]] of measurable functions and <math>L^\infty(X, \Sigma, \mu)</math> is its usual normed quotient. In other words, a lifting picks from every equivalence class <math>[f]</math> of bounded measurable functions modulo negligible functions a representative&mdash; which is henceforth written <math>T([f])</math> or <math>T[f]</math> or simply <math>Tf</math> &mdash; in such a way that <math>T[1] = 1</math> and for all <math>p \in X</math> and all <math>r, s \in \Reals,</math>
<math display=block>T(r[f]+s[g])(p) = rT[f](p) + sT[g](p),</math>
<math display=block>T([f]\times[g])(p) = T[f](p) \times T[g](p).</math>

Liftings are used to produce [[Disintegration theorem|disintegrations of measures]], for instance [[conditional probability distribution]]s given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function. 
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==Properties of liftings==
A lifting is necessarily positive:
<math display=block>[f]\ge0\implies T[f]\ge0 \mathrm{\ (since\ } [f] \mathrm{\ is\ a\ square)} </math>
and an isometry:<ref>The ''essential supremum'' of a class [''f''] of ''μ''-measurable functions is the smallest number α for which the set [''f'' > α] is ''μ''-negligible.</ref>
<math display=block>\big\Vert T[f]\big\Vert_\infty:= \sup_{p\in X}\,\big|T[f](p)\big|=\mathrm{ess.sup}\,\big|[f]\big|.</math>

For every point ''p'' in ''X'', the map <math>[f]\mapsto T_pf:= T[f](p)</math> is a character<ref name=character> A ''character'' on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.</ref> of <math>L^\infty(X, \Sigma, \mu).</math>
-->

==Existence of liftings==

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

In particular, if <math>(X, \Sigma, \mu)</math> is the completion of a ''σ''-finite<ref>i.e., there exists a countable collection of integrable sets &ndash; sets of finite measure in <math>\Sigma</math> &ndash; that covers the underlying set <math>X.</math></ref> measure or of an inner regular Borel measure on a [[locally compact space]], then <math>(X, \Sigma, \mu)</math> admits a lifting.</blockquote>

The proof consists in extending a lifting to ever larger sub-''σ''-algebras, applying [[Doob's martingale convergence theorems|Doob's martingale convergence theorem]] if one encounters a countable chain in the process.
<!-- 
Here are the details. Henceforth write ''Tf'' := ''T''[''f''] = ''T''([''f'']). <math>(\Sigma, \mu)</math> is ''σ''-finite if there exists a countable collection of sets of finite measure in <math>\Sigma</math> whose union has negligible complement. This permits a reduction to the case that the measure <math>\mu</math> is finite, in fact, it may be taken to be a probability. The proof uses [[Zorn's lemma]] together with the following order on pairs <math>(\mathfrak A,T_{\mathfrak A})</math> of sub-''σ''-algebras <math>\mathfrak A</math> of <math>\Sigma</math> and liftings <math>T_{\mathfrak A}</math> for them: <math> (\mathfrak A,T_{\mathfrak A})\le(\mathfrak B,T_{\mathfrak B}) </math> if <math>\mathfrak A\subseteq\mathfrak B</math> and <math>T_{\mathfrak A}</math> is the restriction of <math>T_{\mathfrak B}</math> to <math>L^\infty(X,\mathfrak A,\mu)</math>. It is to be shown that a chain <math>\mathfrak C</math> of such pairs has an upper bound, and that a maximal pair, which then exists by Zorn's lemma, has <math>\Sigma</math> for its first entry.

If <math>\mathfrak C</math> has no countable [[Cofinal (mathematics)|cofinal]] subset, then the union <math>\mathfrak U := \bigcup\{\mathfrak A:\,(\mathfrak A,T_{\mathfrak A}) \in \mathfrak C\}</math> is a ''σ''-algebra and there is an obvious lifting <math>T_{\mathfrak U}</math> for it that restricts to the liftings of the chain; <math>(\mathfrak U,T_{\mathfrak U})</math> is the sought upper bound of the chain.

The argument is more complicated when the chain <math> \mathfrak C</math> has a countable cofinal subset <math>\left\{(\mathfrak A_n,T_{\mathfrak A_n}), n = 1, 2, \ldots\right\}</math>. In this case let <math>\mathfrak U</math> be the  [[Sigma-algebra|''σ''-algebra generated]] by the union <math>\bigcup\{\mathfrak A_n: \, n = 1, 2, \ldots\},</math> which is generally only an [[Field of sets|algebra of sets]]. For the construction of <math>T_{\mathfrak U}</math> it is convenient to identify a set <math>A \subseteq X</math> with its indicator function and to write <math>TA := TI_A=T[I_A].</math> For <math>A \in \mathfrak U</math> let <math>A_n</math> denote the [[conditional expectation]] of <math>A</math> under <math>\mathfrak A_n</math>. By [[Doob's martingale convergence theorems|Doob's martingale convergence theorem]] the set <math>\theta(A)</math> of points where <math>A_n</math> converges to 1 differs negligibly from&nbsp;''A''.

Here are a few facts that are straightforward to check (some use the completeness and finiteness of <math>(X, \mathfrak U, \mu)</math>):
<math display=block>\tau := \{\theta(A)\setminus N \ : \ A\in\mathfrak U, \mu(N) = 0\}\subset\mathfrak U</math>
is a topology whose only negligible open set is the empty set and such that every <math> A=I_A\in\mathfrak U</math> is almost everywhere continuous, to wit, on <math> A\cap\theta(A)</math> and on <math> A^c\cap\theta(A^c)</math>. Then every <math>f \in\mathcal L^\infty(X,\mathfrak U,\mu)</math>, being the uniform limit of a sequence of step functions over <math>\mathfrak U</math>, is almost everywhere continuous in this topology. For <math>p</math> in <math>X</math>
<math display=block>I_p:=\{[f]: f \text{ is continuous at } p \text{ and }f(p) = 0\}.</math>
is a proper ideal of <math> L^\infty(X,\mathfrak U,\mu)</math>, contained (by another application of Zorn's lemma) in some maximal proper ideal <math>J_p\subset L^\infty(X, \mathfrak U, \mu),</math> which has codimension 1. The quotient map <math>L^\infty(X, \mathfrak U, \mu) \to L^\infty(X, \mathfrak U,\mu) / J_p</math> can be viewed as a character<ref name=character/>''T''<sub>''p''</sub>. Defining
<math display=block>\left(T_{\mathfrak U}[f]\right)(p):=T_p[f]\;\;,\;\;\;\;\;\;p\in E,</math>
provides the upper bound <math>(\mathfrak U,T_{\mathfrak U})</math> for the chain <math>\mathfrak C</math>.

In either case the chain <math> \mathfrak C</math> therefore has an upper bound. By Zorn's lemma there is a maximal pair <math>(\mathfrak U,T_{\mathfrak U})</math>,
and a small additional calculation shows that <math> \mathfrak U=\mathfrak F</math>. END OF DETAILED PROOF-->

==Strong liftings==

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

A '''strong lifting''' for <math>(X, \Sigma, \mu)</math> is a lifting 
<math display=block>T : L^\infty(X, \Sigma, \mu) \to \mathcal{L}^\infty(X, \Sigma, \mu)</math> 
such that <math>T\varphi = \varphi</math> on <math>\operatorname{Supp}(\mu)</math> for all <math>\varphi</math> in <math>C_b(X, \tau).</math> This is the same as requiring that<ref><math>U,</math> <math>\operatorname{Supp}(\mu)</math> are identified with their indicator functions.</ref> <math>T U \geq (U \cap \operatorname{Supp}(\mu))</math> for all open sets <math>U</math> in <math>\tau.</math>

<blockquote>'''Theorem.''' If <math>(\Sigma, \mu)</math> is ''σ''-finite and complete and <math>\tau</math> has a countable basis then <math>(X, \Sigma, \mu)</math> admits a strong lifting.</blockquote>

'''Proof.''' Let <math>T_0</math> be a lifting for <math>(X, \Sigma, \mu)</math> and <math>U_1, U_2, \ldots</math> a countable basis for <math>\tau.</math> For any point <math>p</math> in the negligible set
<math display=block>N := \bigcup\nolimits_n \left\{p \in \operatorname{Supp}(\mu) : (T_0U_n)(p) < U_n(p)\right\}</math>
let <math>T_p</math> be any character<ref name=character>A ''character'' on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.</ref> on <math>L^\infty(X, \Sigma, \mu)</math> that extends the character <math>\phi \mapsto \phi(p)</math> of <math>C_b(X, \tau).</math> Then for <math>p</math> in <math>X</math> and <math>[f]</math> in <math>L^\infty(X, \Sigma, \mu)</math> define:
<math display=block>(T[f])(p):= \begin{cases} (T_0[f])(p)& p\notin N\\
T_p[f]& p\in N.
\end{cases}</math>
<math>T</math> is the desired strong lifting.

==Application: disintegration of a measure==

Suppose <math>(X, \Sigma, \mu)</math> and <math>(Y, \Phi, \nu)</math> are ''σ''-finite measure spaces (<math>\mu, \mu</math> positive) and <math>\pi : X \to Y</math> is a measurable map. A '''disintegration of <math>\mu</math> along <math>\pi</math> with respect to <math>\nu</math>''' is a slew <math>Y \ni y \mapsto \lambda_y</math> of positive ''σ''-additive measures on <math>(\Sigma, \mu)</math> such that

#<math>\lambda_y</math> is carried by the fiber <math>\pi^{-1}(\{y\})</math> of <math>\pi</math> over <math>y</math>, i.e. <math> \{y\} \in \Phi </math> and <math> \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right) = 0 </math> for almost all <math> y \in Y </math>
#for every <math>\mu</math>-integrable function <math>f,</math><math display=block>\int_X f(p)\;\mu(dp)= \int_Y \left(\int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp)\right) \nu(dy) \qquad (*)</math> in the sense that, for <math>\nu</math>-almost all <math>y</math> in <math>Y,</math> <math>f</math> is <math>\lambda_y</math>-integrable, the function <math display=block>y \mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp) </math> is <math>\nu</math>-integrable, and the displayed equality <math>(*)</math> holds.

[[Disintegration theorem|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.

<blockquote>'''Theorem.''' Suppose <math>X</math> is a [[Polish space]]<ref>A separable space is ''Polish'' if its topology comes from a complete metric. In the present situation it would be sufficient to require that <math>X</math> is ''Suslin'', that is, is the continuous Hausdorff image of a Polish space.</ref> and <math>Y</math> a separable Hausdorff space, both equipped with their Borel ''σ''-algebras. Let <math>\mu</math> be a ''σ''-finite Borel measure on <math>X</math> and <math>\pi : X \to Y</math> a <math>\Sigma, \Phi-</math>measurable map. Then there exists a σ-finite Borel measure <math>\nu</math> on <math>Y</math> and a disintegration (*).

If <math>\mu</math> is finite, <math>\nu</math> can be taken to be the pushforward<ref>The ''pushforward'' <math>\pi_* \mu</math> of <math>\mu</math> under <math>\pi,</math> also called the image of <math>\mu</math> under <math>\pi</math> and denoted <math>\pi(\mu),</math> is the measure <math>\nu</math> on <math>\Phi</math> defined by <math>\nu(A) := \mu\left(\pi^{-1}(A)\right)</math> for <math>A</math> in <math>\Phi</math>.</ref> <math>\pi_* \mu,</math> and then the <math>\lambda_y</math> are probabilities.</blockquote>

'''Proof.''' Because of the polish nature of <math>X</math> there is a sequence of compact subsets of <math>X</math> that are mutually disjoint, whose union has negligible complement, and on which <math>\pi</math> is continuous. This observation reduces the problem to the case that both <math>X</math> and <math>Y</math> are compact and <math>\pi</math> is continuous, and <math>\nu = \pi_* \mu.</math> Complete <math>\Phi</math> under <math>\nu</math> and fix a strong lifting <math>T</math> for <math>(Y, \Phi, \nu).</math> Given a bounded <math>\mu</math>-measurable function <math>f,</math> let <small><math>\lfloor f\rfloor</math></small> denote its conditional expectation under <math>\pi,</math> that is, the [[Radon–Nikodym theorem|Radon-Nikodym derivative]] of<ref><math>f \mu</math> is the measure that has density <math>f</math> with respect to <math>\mu</math></ref> <math>\pi_*(f \mu)</math> with respect to <math>\pi_* \mu.</math> Then set, for every <math>y</math> in <math>Y,</math> <math>\lambda_y(f) := T(\lfloor f\rfloor)(y).</math> 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
<math display=block>\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)</math>
and take the infimum over all positive <math>\varphi</math> in <math>C_b(Y)</math> with <math>\varphi(y) = 1;</math> it becomes apparent that the support of <math>\lambda_y</math> lies in the fiber over <math>y.</math>

==References==

{{reflist}}

[[Category:Measure theory]]