measure-preserving

measure-preserving

1

Definition

Definition

-

Let

(

X

1

,

𝔅

1

,

μ

1

)

and

(

X

2

,

𝔅

2

,

μ

2

)

be

measure spaces

, and

T

:

X

1

→

X

2

be a measurable

transformation

. The transformation

T

is said to be

measure-preserving

if for all

A

∈

𝔅

2

we have that

μ

1

⁢

(

T

-

1

⁢

(

A

)

)

=

μ

2

⁢

(

A

)

,

where

T

-

1

⁢

(

A

)

is, as usual, the set of points

x

∈

X

1

such that

T

⁢

(

x

)

∈

A

.

Additional Notation:

•

If

T

is

bijective

, measure-preserving, and its

inverse

T

-

1

is also measure-preserving, then

T

is said to be an

measure-preserving transformation

.

•

Measure-preserving transformations between the same measure space are sometimes called

of the measure space.

Remarks:

•

The fact that a map

T

:

X

1

⟶

X

2

is measure-preserving depends heavily on the

sigma-algebras

𝔅

i

and

measures

μ

i

involved. If other measures or sigma-algebras are also in consideration, one should make clear to which measure space is

T

:

X

1

⟶

X

2

measure-preserving.

•

Measure-preserving maps are the

morphisms

on the

category

whose objects are measure spaces. This should be clear from the next results and examples.

2

Properties

•

The

composition

of measure-preserving maps is again measure-preserving. Of course, we are supposing that the domains and codomains of the maps are such that the composition is possible.

•

Let

(

X

1

,

𝔅

1

,

μ

1

)

and

(

X

2

,

𝔅

2

,

μ

2

)

be measure spaces and

(

X

1

,

𝔅

1

¯

,

μ

1

¯

)

and

(

X

2

,

𝔅

2

¯

,

μ

2

¯

)

their

completions

. If

T

:

(

X

1

,

𝔅

1

,

μ

1

)

⟶

(

X

2

,

𝔅

2

,

μ

2

)

is measure-preserving, then so is

T

:

(

X

1

,

𝔅

1

¯

,

μ

1

¯

)

⟶

(

X

2

,

𝔅

2

¯

,

μ

2

¯

)

.

•

Let

(

X

1

,

𝔅

1

,

μ

1

)

and

(

X

2

,

𝔅

2

,

μ

2

)

be measure spaces and

T

1

:

X

1

⟶

X

1

,

T

2

:

X

2

⟶

X

2

be measure-preserving maps. Then, the

product map

T

1

×

T

2

:

X

1

×

X

2

⟶

X

1

×

X

2

, defined by

T

1

×

T

2

⁢

(

x

1

,

x

2

)

:=

(

T

1

⁢

(

x

1

)

,

T

2

⁢

(

x

2

)

)

is a measure-preserving transformation of

(

T

1

×

T

2

,

𝔅

1

×

𝔅

1

,

μ

1

×

μ

2

)

.

3

Examples

•

The

identity map

of a measure space

(

X

,

𝔅

,

μ

)

is always measure-preserving.

•

Let

G

be a

locally compact group

(

http://planetmath.org/TopologicalGroup

). For each

a

∈

G

, the transformation

T

⁢

(

g

)

:=

a

⁢

g

is measure-preserving relatively to any

left Haar measure

. Similarly, any right

translation

on

G

any right Haar measure.

•

Every

continuous

surjective

homomorphism

between compact Hausdorff is measure-preserving relatively to the normalized Haar measure (see this entry (

http://planetmath.org/ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure

)).

Title

measure-preserving

Canonical

name

Measurepreserving

Date of creation

2013-03-22 12:19:41

Last modified on

2013-03-22 12:19:41

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

17

Author

asteroid (17536)

Entry type

Definition

Classification

msc 28D05

Classification

msc 37A05

Synonym

measure preserving

Synonym

measure-preserving transformation

Synonym

measure-preserving map

Related topic

ErgodicTransformation

Defines

invertible measure-preserving transformation

Defines

endomorphism of a measure space

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