universally measurable

universally measurable

Given a

measurable space

(

Ω

,

ℱ

)

, a subset

S

⊆

Ω

is said to be

universally measurable

if it lies in the

completion

(

http://planetmath.org/CompleteMeasure

) of the

σ

-

algebra

ℱ

with respect to every finite

measure

μ

on

(

Ω

,

ℱ

)

.
That is, for every such

μ

, there exist

A

,

B

∈

ℱ

such that

A

⊆

S

⊆

B

and

μ

⁢

(

B

∖

A

)

=

0

.

If, for any finite measure

μ

we denote the completion of

ℱ

by

ℱ

μ

*

then, the

collection

of universally measurable sets is

ℱ

*

=

⋂

μ

ℱ

μ

*

where

μ

ranges over the finite measures on

(

Ω

,

ℱ

)

.
Being an

intersection

of

σ

-algebras, the collection

ℱ

*

of universally measurable sets is itself a

σ

-algebra, and is called the

universal completion

of

ℱ

.
The

σ

-algebra

ℱ

is called

universally complete

if it contains every set which is universally measurable with respect to

ℱ

. That is, if

ℱ

*

=

ℱ

. The universal completion of a

σ

-algebra is itself universally complete.

A

function

f

:

A

→

B

between measurable spaces

(

A

,

𝒜

)

and

(

B

,

ℬ

)

is said to be universally measurable if it is

𝒜

*

/

ℬ

*

-

measurable

, so that

f

-

1

⁢

(

S

)

∈

𝒜

*

for all

S

∈

ℬ

*

. In fact, this is

equivalent

to requiring that

f

be

𝒜

*

/

ℬ

-measurable so, in particular, every

measurable function

is universally measurable.

As any

σ

-finite measure is equivalent to a probability measure (

http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure

), the term “finite measure” may be replaced by “

σ

-finite measure” in the definitions above.
In particular, every

σ

-finite measure

μ

on

(

Ω

,

ℱ

)

can be uniquely extended to the universal completion of

ℱ

.

A subset of a

topological space

X

is said to be universally measurable if it is universally measurable with respect to the Borel

σ

-algebra on

X

. It can be shown, for example, that the image of any

continuous map

between

Polish spaces

is universally measurable.

An important example of the use of universally measurable functions comes from the theory of continuous-time stochastic processes. The first time that a

continuous

process hits a given value is a universally measurable time, as stated by the début

theorem

. However, it is not always measurable.

It is well known that the

axiom of choice

is required to

construct

non-Lebesgue measurable subsets of

ℝ

n

and, it can be shown that this statement extends to all universally measurable subsets of

ℝ

n

. That is, the

Zermelo-Fraenkel axioms

of

set theory

, without the axiom of choice, are

consistent

with the statement that all subsets of

ℝ

n

are universally measurable. This remains true even if we allow a weaker version of the AOC, in the form of the

axiom of dependent choice

.

As every universally measurable subset of the

real numbers

is

Lebesgue measurable

, the proof of Vitali’s Theorem gives an example of the use of the axiom of choice to construct a non-universally

measurable set

. It is also possible to construct universally but non-Borel measurable sets (

http://planetmath.org/ALebesgueMeasurableButNonBorelSet

) without using the axiom of choice.

Title

universally measurable

Canonical name

UniversallyMeasurable

Date of creation

2013-03-22 18:36:58

Last modified on

2013-03-22 18:36:58

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Definition

Classification

msc 28A20

Classification

msc 28A05

Related topic

CompleteMeasure

Related topic

AnalyticSet2

Defines

universal completion

Defines

universally complete

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