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.[1] The theory was further developed by Dorothy Maharam (1958)[2] and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961).[3] 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.[4] Lifting theory continued to develop since then, yielding new results and applications.
Definitions
A lifting on a measure space
is a linear and multiplicative operator
![{\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L))^{\infty }(X,\Sigma ,\mu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70d04c1f72fdeba18fe5f2970dc7ab325e8758c8)
which is a right inverse of the quotient map
![{\displaystyle {\begin{cases}{\mathcal {L))^{\infty }(X,\Sigma ,\mu )\to L^{\infty }(X,\Sigma ,\mu )\\f\mapsto [f]\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ad6827ca4c062b6aa1cfbea130a90550574b92)
where
is the seminormed Lp space of measurable functions and
is its usual normed quotient. In other words, a lifting picks from every equivalence class
of bounded measurable functions modulo negligible functions a representative— which is henceforth written
or
or simply
— in such a way that
and for all
and all
![{\displaystyle T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59101820fb45d8ad985d12b512a99e7619b83e9d)
![{\displaystyle T([f]\times [g])(p)=T[f](p)\times T[g](p).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6619283b67dd67c37dc6e7ddccb31ba5e5d0b0b)
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
is complete.[5] Then
admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in
whose union is
In particular, if
is the completion of a σ-finite[6] measure or of an inner regular Borel measure on a locally compact space, then
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
is complete and
is equipped with a completely regular Hausdorff topology
such that the union of any collection of negligible open sets is again negligible – this is the case if
is σ-finite or comes from a Radon measure. Then the support of
can be defined as the complement of the largest negligible open subset, and the collection
of bounded continuous functions belongs to
A strong lifting for
is a lifting
![{\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L))^{\infty }(X,\Sigma ,\mu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70d04c1f72fdeba18fe5f2970dc7ab325e8758c8)
such that
on
for all
in
This is the same as requiring that[7]
for all open sets
in
Theorem. If
is σ-finite and complete and
has a countable basis then
admits a strong lifting.
Proof. Let
be a lifting for
and
a countable basis for
For any point
in the negligible set
![{\displaystyle N:=\bigcup \nolimits _{n}\left\{p\in \operatorname {Supp} (\mu ):(T_{0}U_{n})(p)<U_{n}(p)\right\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d06ce07c782f4068ec6314cbb9c632a67c6561)
let
be any character[8] on
that extends the character
of
Then for
in
and
in
define:
![{\displaystyle (T[f])(p):={\begin{cases}(T_{0}[f])(p)&p\notin N\\T_{p}[f]&p\in N.\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24eca46003522257a530ba74bd8fa1e55a1a9d3)
is the desired strong lifting.
Application: disintegration of a measure
Suppose
and
are σ-finite measure spaces (
positive) and
is a measurable map. A disintegration of
along
with respect to
is a slew
of positive σ-additive measures on
such that
is carried by the fiber
of
over
, i.e.
and
for almost all ![{\displaystyle y\in Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323)
- for every
-integrable function ![{\displaystyle f,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d)
![{\displaystyle \int _{X}f(p)\;\mu (dp)=\int _{Y}\left(\int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)\right)\nu (dy)\qquad (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d891c61f18b94f4f8540ab9923ee1d0c47b20515)
in the sense that, for
-almost all
in
is
-integrable, the function ![{\displaystyle y\mapsto \int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9efffc4b463a0bb5a88a25d03698790d4ef0b1)
is
-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
is a Polish space[9] and
a separable Hausdorff space, both equipped with their Borel σ-algebras. Let
be a σ-finite Borel measure on
and
a
measurable map. Then there exists a σ-finite Borel measure
on
and a disintegration (*).
If
is finite,
can be taken to be the pushforward[10]
and then the
are probabilities.
Proof. Because of the polish nature of
there is a sequence of compact subsets of
that are mutually disjoint, whose union has negligible complement, and on which
is continuous. This observation reduces the problem to the case that both
and
are compact and
is continuous, and
Complete
under
and fix a strong lifting
for
Given a bounded
-measurable function
let
denote its conditional expectation under
that is, the Radon-Nikodym derivative of[11]
with respect to
Then set, for every
in
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
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d64c29d221da2bd46e9be41a7d42014b5c0388)
and take the infimum over all positive
in
with
it becomes apparent that the support of
lies in the fiber over