Editing Lifting theory

Lifting theory was first introduced in a series of pioneering papers by [[John von Neumann]].<ref>Bellow, Alexandra (1969), Topics in the theory of lifting, Springer, Berlin, Heidelberg, New York</ref> A '''lifting''' on a [[measure space]] <math>(E,\mathfrak F,\mu)</math> is a linear and multiplicative inverse

:<math> T:L^\infty(E,\mathfrak F,\mu)\to \mathcal L^\infty(E,\mathfrak F,\mu)</math>

of the quotient map

: <math>[\ ]:\mathcal L^\infty(E,\mathfrak F,\mu)\to L^\infty(E,\mathfrak F,\mu), \qquad f\mapsto [f].</math>

In other words, a lifting picks from every equivalence class [''f''] of bounded measurable functions modulo negligible functions a representative&mdash; which is henceforth written ''T''([''f'']) or ''T''[''f''] or simply ''Tf'' &mdash; in such a way that

:<math>T(r[f]+s[g])(p)=rT[f](p) + sT[g](p), \qquad \forall p\in E, r,s\in \mathbf R;</math>
:<math>T([f]\times[g])(p)=T[f](p)\times T[g](p), \qquad \forall p\in E;</math>
:<math>T[1]=1.</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>[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> \big\Vert T[f]\big\Vert_\infty:= \sup_{p\in E}\,\big|T[f](p)\big|=\mathrm{ess.sup}\,\big|[f]\big|.</math>

For every point ''p'' in ''E'', 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(E,\mathfrak F,\mu)</math>.
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==Existence of liftings==
<blockquote>'''Theorem.''' Suppose <math>(E,\mathfrak F,\mu)</math> is complete.<ref>A subset ''N'' ⊂ ''E'' is locally negligible if it intersects every integrable set in <math>\mathfrak F</math> in a subset of a negligible set of <math>\mathfrak F</math>. <math>(E,\mathfrak F,\mu)</math> is ''complete'' if every locally negligible set is negligible and belongs to <math>\mathfrak F</math>.</ref> Then <math>(E,\mathfrak F,\mu)</math> admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in <math>\mathfrak F</math> whose union is ''E''.

In particular, if <math>(E,\mathfrak F,\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>\mathfrak F</math> &ndash; that covers the underlying set ''E''.</ref> measure or of an inner regular Borel measure on a locally compact space, then <math>(E,\mathfrak F,\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.

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Here are the details. Henceforth write ''Tf'' := ''T''[''f''] = ''T''([''f'']). <math>(\mathfrak F,\mu)</math> is σ-finite if there exists a countable collection of sets of finite measure in <math>\mathfrak F</math> whose union has negligible complement. This permits a reduction to the case that the measure μ 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>\mathfrak F</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(E,\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>\mathfrak F</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 ''A'' ⊆ ''E'' with its indicator function and to write <math>TA:=TI_A=T[I_A]</math>. For <math>A\in\mathfrak U</math> let ''A<sub>n</sub>'' denote the [[conditional expectation]] of ''A'' under <math>\mathfrak A_n</math>. By [[Doob's martingale convergence theorems|Doob's martingale convergence theorem]] the set θ(''A'')of points where ''A<sub>n</sub>'' converges to 1 differs negligibly from ''A''.

Here are a few facts that are straightforward to check (some use the completeness and finiteness of <math>(E,\mathfrak U,\mu)</math>):
:<math> \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(E,\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 ''p'' in ''E''

:<math> I_p:=\{[f]: f\mathrm{\ is\ continuous\ at\ }p\mathrm{\ and\ }f(p)=0\}.</math>

is a proper ideal of <math> L^\infty(E,\mathfrak U,\mu)</math>, contained (by another application of Zorn's lemma) in some maximal proper ideal <math> J_p\subset L^\infty(E,\mathfrak U,\mu)</math>, which has codimension 1. The quotient map <math>L^\infty(E,\mathfrak U,\mu)\to L^\infty(E,\mathfrak U,\mu)/J_p</math> can be viewed as a character<ref name=character/>''T<sub>p</sub>''. Defining

:<math> \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 that <math>(E,\mathfrak F,\mu)</math> is complete and that ''E'' is equipped with a completely regular Hausdorff topology <math>\tau\subset\mathfrak F</math> such that the union of any collection of negligible open sets is again negligible &ndash; this is the case if <math>(E,\mathfrak F,\mu)</math> 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<sub>b</sub>''(''E'', τ) of bounded continuous functions belongs to <math> \mathcal L^\infty(E,\mathfrak F,\mu)</math>.

A '''strong lifting''' for <math>(E,\mathfrak F,\mu)</math> is a lifting <math> T:L^\infty(E,\mathfrak F,\mu)\to \mathcal L^\infty(E,\mathfrak F,\mu)</math> such that ''T''φ = φ on Supp(μ) for all φ in ''C<sub>b</sub>''(''E'', τ). This is the same as requiring that<ref>''U'', Supp(μ) are identified with their indicator functions.</ref> <math> TU\ge \big(U\cap\mathrm{Supp}(\mu)\big)</math> for all open sets ''U'' in τ.

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

'''Proof.''' Let ''T''<sub>0</sub> be a lifting for <math> (E,\mathfrak F,\mu)</math> and <math> \{U_1, U_2, \cdots \}</math> a countable basis for τ. For any point ''p'' in the negligible set 

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

let ''T<sub>p</sub>'' 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(E,\mathfrak F,\mu)</math> that extends the character φ ↦ φ(''p'') of ''C<sub>b</sub>''(''E'', τ). Then for ''p'' in ''E'' and [''f''] in ''L''<sup>∞</sup> define:

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

''T'' is the desired strong lifting.

==Application: disintegration of a measure==
Suppose <math> (E,\mathfrak F,\mu), (X,\mathfrak X,\nu)</math> are σ-finite measure spaces (μ, ν positive) and π : ''E'' → ''X'' is a measurable map. A '''disintegration of μ along π with respect to ν''' is a slew <math> X\ni x\mapsto \lambda_x</math> of positive σ-additive measures on <math> (E,\mathfrak F)</math> such that 

#λ<sub>''x''</sub> is carried by the fiber <math>\pi^{-1}(\{x\})</math>of π over ''x'':
:::<math> \{x\}\in\mathfrak X\;\;\mathrm{ and }\;\; \lambda_x\left((E\setminus \pi^{-1}(\{x\})\right)=0 \qquad \forall x\in X </math>
#for every μ-integrable function ''f'',
:::<math> \int_E f(p)\;\mu(dp)= \int_X \left(\int_{\pi^{-1}(\{x\})}f(p)\,\lambda_x(dp)\right) \;\nu(dx) \qquad (*)</math>
::in the sense that, for ν-almost all ''x'' in ''X'', ''f'' is λ<sub>''x''</sub>-integrable, the function 
:::<math> x\mapsto \int_{\pi^{-1}(\{x\})} f(p)\,\lambda_x(dp) </math> 
::is ν-integrable, and the displayed equality (*) 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 ''E'' is a polish<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 ''E'' is ''Suslin'', i.e., is the continuous Hausdorff image of a polish space.</ref> space and ''X'' a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a σ-finite Borel measure on ''E'' and π : ''E'' → ''X'' an <math>\mathfrak F,\mathfrak X</math>&ndash;measurable map. Then there exists a σ-finite Borel measure ν on ''X'' and a disintegration (*).

If μ is finite, ν can be taken to be the pushforward<ref> The ''pushforward'' π<sub>∗</sub>μ of μ under π, also called the image of μ under π and denoted π(μ), is the measure ν on <math>\mathfrak X</math> defined by <math>\nu(A):=\mu\left(\pi^{-1}(A)\right), A\in\mathfrak X</math>.</ref> π<sub>∗</sub>μ, and then the λ<sub>''x''</sub> are probabilities.</blockquote>

'''Proof.''' Because of the polish nature of ''E'' there is a sequence of compact subsets of ''E'' 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 ''E'' and ''X'' are compact and π is continuous, and ν = π<sub>∗</sub>μ. Complete <math>\mathfrak X</math> under ν and fix a strong lifting ''T'' for <math>(X,\mathfrak X,\nu)</math>. Given a bounded μ-measurable function  ''f'', let <small><math>\lfloor f\rfloor</math></small> denote its conditional expectation under π, i.e., the [[Radon–Nikodym theorem|Radon-Nikodym derivative]] of<ref> ''f''μ is the measure that has density ''f'' with respect to μ</ref> π<sub>∗</sub>(''f''μ) with respect to π<sub>∗</sub>μ. Then set, for every ''x'' in ''X'', <math>\lambda_x(f):=T(\lfloor f\rfloor)(x).</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> \lambda_x(f\cdot\varphi\circ\pi)=\varphi(x) \lambda_x(f) \qquad \forall x\in X, \varphi\in C_b(X), f\in L^\infty(E,\mathfrak F,\mu)</math>

and take the infimum over all positive φ in ''C<sub>b</sub>''(''X'') with φ(''x'') = 1; it becomes apparent that the support of λ<sub>''x''</sub> lies in the fiber over ''x''.

== References ==
* {{citation|first=Alexandra|last=Bellow|title=Topics in the theory of lifting|publisher=Springer, Berlin, Heidelberg, New York|year=1969}}
<references />

==See also ==
*[[John von Neumann]]
*[[Alexandra Bellow]]

[[Category:Measure theory]]