Lifting theory - AbsoluteAstronomy.com

Lifting theory was first introduced in a series of pioneering papers by John von Neumann

John von Neumann

John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

. A lifting on a measure space

is a linear and multiplicative inverse

of the quotient map

In other words, a lifting picks from every equivalence class

of bounded measurable functions modulo negligible functions
a representative

— which is henceforth written

or simply

—

in such a way that

Liftings are used to produce
disintegrations of measures

Disintegration theorem

In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability 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.
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 sigma-finite
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-sigma-algebras,
applying Doob's martingale convergence theorem

Doob's martingale convergence theorems

In mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph Leo Doob....

if one encounters a countable chain in the process.

Strong liftings

Suppose now that

is complete
and that

comes 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 sigma-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

such that

for all

.
This is the same as requiring that

for all open sets

.

Theorem. If
is sigma-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

let

be any character
on

that extends the character

.
Then define, for

and

is the desired strong lifting.

Application: disintegration of a measure

Suppose

and

are sigma-finite measure spaces (

positive)
and

is a measurable map.
A disintegration of

along

with respect to

is a slew

of positive sigma-additive measures on

such that (1)

is carried by the fiber

over

and (2) for every

-integrable
function

(*)
in the sense that, for

-almost all

-integrable,
the function

-integrable,
and the displayed equality (*) obtains.

Disintegrations

Disintegration theorem

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 and
a separable Hausdorff space, both equipped with their Borel sigma-algebras.
Let

be a
sigma-finite Borel measure on
and

an –measurable map.
Then there exists a sigma-finite Borel measure

on

and a disintegration (*).

If

is finite,

can be taken to
be the pushforward

,
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

,
i.e., the Radon-Nikodym derivative

Radon–Nikodym theorem

In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...

with respect to

.
Then set, for every

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

and take the infimum over all positive

with

;
it becomes apparent that the support of

lies in the fiber over

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.