aarjaneiro/planetmath huggingface dataset:
completion of a measure space

completion of a measure space

If the measure space

(

X

,

𝒮

,

μ

)

is not

complete

, then it can be completed in the following way. Let

𝒵

=

⋃

E

∈

𝒮

,

μ

⁢

(

E

)

=

0

𝒫

⁢

(

E

)

,

i.e. the family of all subsets of sets whose

μ

-measure is zero.
Define

𝒮

¯

=

{

A

∪

B

:

A

∈

𝒮

,

B

∈

𝒵

}

.

We assert that

𝒮

¯

is a

σ

-

algebra

. In fact, it clearly contains the emptyset, and it is

closed under

countable

unions because both

𝒮

and

𝒵

are. We thus need to show that it is closed under

complements

. Let

A

∈

𝒮

,

B

∈

𝒵

and suppose

E

∈

𝒮

is such that

B

⊂

E

and

μ

⁢

(

E

)

=

0

.
Then we have

(

A

∪

B

)

c

=

A

c

∩

B

c

=

A

c

∩

(

E

-

(

E

-

B

)

)

c

=

A

c

∩

(

E

c

∪

(

E

-

B

)

)

=

(

A

c

∩

E

c

)

∪

(

A

c

∩

(

E

-

B

)

)

,

where

A

c

∩

E

c

∈

𝒮

and

A

c

∩

(

E

-

B

)

∈

𝒵

. Hence

(

A

∪

B

)

c

∈

𝒮

¯

.

Now we define

μ

¯

on

𝒮

¯

by

μ

¯

⁢

(

A

∪

B

)

=

μ

⁢

(

A

)

, whenever

A

∈

𝒮

and

B

∈

𝒵

. It is easily verified that this defines in fact a measure, and that

(

X

,

𝒮

¯

,

μ

¯

)

is the

completion

of

(

X

,

𝒮

,

μ

)

.

Title

completion of a measure space

Canonical name

CompletionOfAMeasureSpace

Date of creation

2013-03-22 14:06:59

Last modified on

2013-03-22 14:06:59

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Derivation

Classification

msc 28A12

Generated on Fri Feb 9 20:06:30 2018 by

LaTeXML
counting measure

counting measure

Let

(

X

,

𝔅

)

be a

measurable space

. The

measure

μ

on

X

defined by

μ

⁢

(

A

)

=

{

n

if

⁢

A

⁢

has exactly

⁢

n

⁢

elements

∞

otherwise.

for all

A

∈

𝔅

is called the

counting measure

on

X

.
Usually this is applied when

X

is countable, e.g.

ℕ

or

ℤ

.

Title

counting measure

Canonical name

CountingMeasure

Date of creation

2013-03-22 12:21:52

Last modified on

2013-03-22 12:21:52

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Definition

Classification

msc 28A12

Related topic

Measure

Generated on Fri Feb 9 20:06:38 2018 by

LaTeXML
existence of the Lebesgue measure

existence of the Lebesgue measure

Theorem

(Lebesgue).

Let

B

be the Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

) on the real number line. Then, there is a unique

measure

μ

on the

measurable space

(

R

,

B

)

satisfying

μ

⁢

(

(

a

,

b

)

)

=

b

-

a

for all

real numbers

a

<

b

.

Title

existence of the

Lebesgue measure

Canonical name

ExistenceOfTheLebesgueMeasure

Date of creation

2013-03-22 18:33:12

Last modified on

2013-03-22 18:33:12

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Classification

msc 26A42

Related topic

LebesgueMeasure

Related topic

Measure

Related topic

CaratheodorysExtensionTheorem

Generated on Fri Feb 9 20:06:42 2018 by

LaTeXML
extending a capacity to a Cartesian product

extending a capacity to a Cartesian product

A

capacity

on a set

X

can be extended to a

set function

on a

Cartesian product

X

×

K

simply by projecting any subset onto

X

, and then applying the original capacity.

Theorem

.

Suppose that

(

X

,

F

)

is a

paved space

such that

F

is

closed under

finite unions and finite

intersections

, and that

(

K

,

K

)

is a compact paved space.
Define

G

to be the

closure

under finite unions and finite intersections of the paving

F

×

K

on

X

×

K

.

If

I

is an

F

-capacity and

π

X

:

X

×

K

→

X

is the

projection map

, we can form the

composition

I

∘

π

X

:

𝒫

⁢

(

X

×

K

)

→

ℝ

,

I

∘

π

X

⁢

(

S

)

=

I

⁢

(

π

X

⁢

(

S

)

)

.

Then

π

X

⁢

(

S

)

∈

F

δ

for any

S

∈

G

δ

, and

I

∘

π

X

is a

G

-capacity.

This result justifies looking at capacities when considering projections from the Cartesian product

X

×

K

onto

X

. We see that the property of being a capacity is preserved under composing with such projections. However,

additivity

of set functions is not preserved, so the corresponding result would not be true if “capacity” was replaced by “

measure

” or “

outer measure

”.

Recall that if

S

⊆

X

×

K

is

(

𝒢

,

I

∘

π

X

)

-capacitable then, for any

ϵ

>

0

, there is an

A

∈

𝒢

δ

such that

A

⊆

S

and

I

∘

π

X

⁢

(

A

)

>

I

∘

π

X

⁢

(

S

)

-

ϵ

. However,

π

X

⁢

(

A

)

⊆

π

X

⁢

(

S

)

and, by the above

theorem

,

π

X

⁢

(

A

)

∈

ℱ

δ

. This has the following

consequence

.

Lemma

.

Let

S

⊆

X

×

K

be

(

G

,

I

∘

π

X

)

-capacitable. Then,

π

X

⁢

(

S

)

is

(

F

,

I

)

-capacitable.

Title

extending a capacity to a Cartesian product

Canonical name

ExtendingACapacityToACartesianProduct

Date of creation

2013-03-22 18:47:38

Last modified on

2013-03-22 18:47:38

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Classification

msc 28A05

Generated on Fri Feb 9 20:06:44 2018 by

LaTeXML
Hahn decomposition theorem

Hahn decomposition theorem

Let

μ

be a

signed measure

in the

measurable space

(

Ω

,

𝒮

)

. There are two

measurable sets

A

and

B

such that:

1.

A

∪

B

=

Ω

and

A

∩

B

=

∅

;

2.

μ

⁢

(

E

)

≥

0

for each

E

∈

𝒮

such that

E

⊂

A

;

3.

μ

⁢

(

E

)

≤

0

for each

E

∈

𝒮

such that

E

⊂

B

.

The pair

(

A

,

B

)

is called a

Hahn decomposition

for

μ

.
This decomposition is not unique, but any other such decomposition

(

A

′

,

B

′

)

satisfies

μ

⁢

(

A

′

⁢

△

⁢

A

)

=

μ

⁢

(

B

⁢

△

⁢

B

′

)

=

0

(where

△

denotes the

symmetric difference

), so the two decompositions differ in a set of

measure

0.

Title

Hahn decomposition theorem

Canonical name

HahnDecompositionTheorem

Date of creation

2013-03-22 13:26:59

Last modified on

2013-03-22 13:26:59

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A12

Defines

Hahn decomposition

Generated on Fri Feb 9 20:06:46 2018 by

LaTeXML
Jordan decomposition

Jordan decomposition

Let

(

Ω

,

𝒮

,

μ

)

be a

signed measure

space, and let

(

A

,

B

)

be a

Hahn decomposition

for

μ

. We define

μ

+

and

μ

-

by

μ

+

⁢

(

E

)

=

μ

⁢

(

A

∩

E

)

and

μ

-

⁢

(

E

)

=

-

μ

⁢

(

B

∩

E

)

.

This definition is easily shown to be

independent

of the chosen Hahn decomposition.

It is clear that

μ

+

is a

positive measure

, and it is called the

positive variation

of

μ

. On the other hand,

μ

-

is a positive finite

measure

, called the

negative variation

of

μ

.
The measure

|

μ

|

=

μ

+

+

μ

-

is called the

total variation

of

μ

.

Notice that

μ

=

μ

+

-

μ

-

. This decomposition of

μ

into its positive and negative parts is called the

Jordan decomposition

of

μ

.

Title

Jordan decomposition

Canonical name

JordanDecomposition

Date of creation

2013-03-22 13:27:02

Last modified on

2013-03-22 13:27:02

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 28A12

Defines

positive variation

Defines

negative variation

Defines

total variation

Generated on Fri Feb 9 20:06:48 2018 by

LaTeXML
Lebesgue decomposition theorem

Lebesgue decomposition theorem

Let

μ

and

ν

be two

σ

-finite

signed measures

in the

measurable space

(

Ω

,

𝒮

)

. There exist two

σ

-finite (

http://planetmath.org/SigmaFinite

) signed measures

ν

0

and

ν

1

such that:

1.

ν

=

ν

0

+

ν

1

;

2.

ν

0

≪

μ

(i.e.

ν

0

is

absolutely continuous

with respect to

μ

;)

3.

ν

1

⟂

μ

(i.e.

ν

1

and

μ

are

singular

.)

These two

measures

are uniquely determined.

Title

Lebesgue decomposition theorem

Canonical name

LebesgueDecompositionTheorem

Date of creation

2013-03-22 13:26:28

Last modified on

2013-03-22 13:26:28

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A12

Generated on Fri Feb 9 20:06:50 2018 by

LaTeXML
Lebesgue measure

Lebesgue measure

Let

S

⊆

ℝ

, and let

S

′

be the

complement

of

S

with respect to

ℝ

. The set

S

is said to be

Lebesgue measurable

if, for any

A

⊆

ℝ

,

m

*

⁢

(

A

)

=

m

*

⁢

(

A

∩

S

)

+

m

*

⁢

(

A

∩

S

′

)

where

m

*

⁢

(

S

)

is the

Lebesgue outer measure

of

S

. If

S

is Lebesgue measurable, then we define the

Lebesgue measure

of

S

to be

m

⁢

(

S

)

=

m

*

⁢

(

S

)

. The Lebesgue measurable sets include

open sets

,

closed sets

as well all the sets obtained from them by taking

countable

unions and

intersections

. However, with aid of the

axiom of choice

it is possible to construct non-measurable sets.

The Lebesgue measure on

ℝ

n

is the

completion

(

http://planetmath.org/CompletionOfAMeasureSpace

) of the

n

-fold

product measure

of the Lebesgue measure on

ℝ

.

The Lebesgue measure is a formalization of the intuitive notion of length of a set in

ℝ

, an area of a set in

ℝ

2

and volume in

ℝ

3

, etc. It obeys many properties one would expect from these intuitive notions, such as invariance under

translation

and

rotation

.

The Lebesgue measure was introduced by Henri Lebesgue in the first decade of the twentieth century. It became the prototypical example of what later became known simply as

measure

, a

concept

which unified such diverse objects as area, probability, and

function

.

Title

Lebesgue measure

Canonical name

LebesgueMeasure

Date of creation

2013-03-22 11:48:24

Last modified on

2013-03-22 11:48:24

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

16

Author

bbukh (348)

Entry type

Definition

Classification

msc 28A12

Classification

msc 26A42

Classification

msc 03B52

Classification

msc 03B50

Related topic

Measure

Related topic

LebesgueOuterMeasure

Related topic

Integral2

Related topic

MikowskiInequality

Related topic

VitalisTheorem

Related topic

BorelSigmaAlgebra

Related topic

HausdorffMeasure

Defines

Lebesgue measurable

Generated on Fri Feb 9 20:06:52 2018 by

LaTeXML
Lebesgue outer measure

Lebesgue outer measure

\PMlinkescapephrase

extension theorem

Let

S

be a subset of

ℝ

, let

L

⁢

(

I

)

be the traditional definition of the length of an

interval

I

⊆

ℝ

: If

I

=

(

a

,

b

)

, then

L

⁢

(

I

)

=

b

-

a

. Finally, let

M

be the set consisting of the values

∑

A

∈

C

L

⁢

(

A

)

for all possible

countable

collections

of open intervals

C

that covers

S

(that is,

S

⊆

∪

C

).
Then the

Lebesgue outer measure

of

S

is defined by:

m

*

⁢

(

S

)

=

inf

⁡

(

M

)

Note that

(

ℝ

,

𝒫

⁢

(

ℝ

)

,

m

*

)

is an

outer measure

space (

http://planetmath.org/OuterMeasure2

). In particular:

•

Lebesgue outer measure is defined for any subset of

ℝ

(and

𝒫

⁢

(

ℝ

)

is a

σ

-algebra).

•

m

*

⁢

(

A

)

≥

0

for any

A

⊆

ℝ

, and

m

*

⁢

(

∅

)

=

0

.

•

If

A

and

B

are

disjoint sets

, then

m

*

⁢

(

A

∪

B

)

≤

m

*

⁢

(

A

)

+

m

*

⁢

(

B

)

. More generally, if

⟨

A

i

⟩

is a countable

sequence

of disjoint sets, then

m

*

⁢

(

⋃

A

i

)

≤

∑

m

*

⁢

(

A

i

)

. This property is known as

countable subadditivity

and is weaker than

countable additivity

. In fact,

m

*

is

not

countably additive

.

Lebesgue outer measure has other nice properties:

•

The outer measure of an interval is its length:

m

*

⁢

(

(

a

,

b

)

)

=

b

-

a

.

•

m

*

is translation-invariant. That is, if we define

A

+

y

to be the set

{

x

+

y

:

x

∈

A

}

, we have

m

*

⁢

(

A

)

=

m

*

⁢

(

A

+

y

)

for any

y

∈

ℝ

.

The outer measure

satisfies

all the axioms of a

measure

except (countable) additivity. However, it is countably additive when one restricts to at least the

Borel sets

, as this is the usual construction of

Borel measure

. This result is roughly

contained

in the

Caratheodory Extension theorem

.

Title

Lebesgue outer measure

Canonical name

LebesgueOuterMeasure

Date of creation

2013-03-22 11:48:15

Last modified on

2013-03-22 11:48:15

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 28A12

Synonym

outer measure

Related topic

Infimum

Related topic

LebesgueMeasure

Related topic

ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength

Related topic

CaratheodorysLemma

Related topic

ConstructionOfOuterMeasures

Generated on Fri Feb 9 20:06:54 2018 by

LaTeXML
outer regular

outer regular

Let

X

be a

locally compact

Hausdorff topological space

with Borel

σ

–algebra

ℬ

, and suppose

μ

is a

measure

on

(

X

,

ℬ

)

. For any

Borel set

B

∈

ℬ

, the measure

μ

is said to be

outer regular

on

B

if

μ

⁢

(

B

)

=

inf

⁡

{

μ

⁢

(

U

)

∣

U

⊃

B

,

U

⁢

open

}

.

We say

μ

is

inner regular

on

B

if

μ

⁢

(

B

)

=

sup

⁡

{

μ

⁢

(

K

)

∣

K

⊂

B

,

K

⁢

compact

}

.

Title

outer regular

Canonical name

OuterRegular

Date of creation

2013-03-22 12:39:57

Last modified on

2013-03-22 12:39:57

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 28A12

Related topic

BorelSigmaAlgebra

Defines

inner regular

Generated on Fri Feb 9 20:06:56 2018 by

LaTeXML
quasi-invariant

quasi-invariant

Definition 1

.

Let

(

E

,

ℬ

)

be a

measurable space

, and

T

:

E

→

E

be a

measurable

map. A

measure

μ

on

(

E

,

ℬ

)

is said to be

quasi-invariant

under

T

if

μ

∘

T

-

1

is

absolutely continuous

with respect to

μ

. That is, for all

A

∈

ℬ

with

μ

⁢

(

A

)

=

0

, we also have

μ

⁢

(

T

-

1

⁢

(

A

)

)

=

0

. We also say that

T

leaves

μ

quasi-invariant.

As a example, let

E

=

ℝ

with

ℬ

the Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

), and

μ

be

Lebesgue measure

. If

T

⁢

(

x

)

=

x

+

5

, then

μ

is quasi-invariant under

T

. If

S

⁢

(

x

)

=

0

, then

μ

is not quasi-invariant under

S

. (We have

μ

⁢

(

{

0

}

)

=

0

, but

μ

⁢

(

T

-

1

⁢

(

{

0

}

)

)

=

μ

⁢

(

ℝ

)

=

∞

).

To give another example, take

E

to be the nonnegative integers and
declare every subset of

E

to be a

measurable set

. Fix

λ

>

0

.
Let

μ

⁢

(

{

n

}

)

=

λ

n

n

!

and extend

μ

to all subsets
by

additivity

. Let

T

be the shift

function

:

n

→

n

+

1

. Then

μ

is quasi-invariant under

T

and not

invariant

(

http://planetmath.org/HaarMeasure

).

Title

quasi-invariant

Canonical name

Quasiinvariant

Date of creation

2013-03-22 15:56:00

Last modified on

2013-03-22 15:56:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 28A12

Related topic

RepresentationsOfLocallyCompactGroupoids

Generated on Fri Feb 9 20:07:17 2018 by

LaTeXML
signed measure

signed measure

A

signed measure

on a

measurable space

(

Ω

,

𝒮

)

is a function

μ

:

𝒮

→

ℝ

∪

{

+

∞

}

which is

σ

-additive (

http://planetmath.org/Additive

) and such that

μ

⁢

(

∅

)

=

0

.

Remarks.

1.

The usual (

positive

)

measure

is a particular case of signed measure, in which

|

μ

|

=

μ

(see

Jordan decomposition

.)

2.

Notice that the value

-

∞

is not allowed. For some authors, a signed measure can only take finite values (so that

+

∞

is not allowed either). This is sometimes useful because it turns the space of all signed measures into a

normed vector space

, with the natural operations, and the norm given by

∥

μ

∥

=

|

μ

|

⁢

(

Ω

)

.

3.

An important example of signed measures arises from the usual measures in the following way: Let

(

Ω

,

𝒮

,

μ

)

be a

measure space

, and let

f

be a (real valued)

measurable function

such that

∫

{

x

∈

Ω

:

f

⁢

(

x

)

<

0

}

|

f

|

⁢

𝑑

μ

<

∞

.

Then a signed measure is defined by

A

↦

∫

A

f

⁢

𝑑

μ

.

Title

signed measure

Canonical name

SignedMeasure

Date of creation

2013-03-22 13:26:55

Last modified on

2013-03-22 13:26:55

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 28A12

Generated on Fri Feb 9 20:07:19 2018 by

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singular measure

singular measure

Two

measures

μ

and

ν

in a

measurable space

(

Ω

,

𝒜

)

are
called

singular

if there exist two disjoint sets

A

and

B

in

𝒜

such that

A

∪

B

=

Ω

and

μ

⁢

(

B

)

=

ν

⁢

(

A

)

=

0

.
This is denoted by

μ

⟂

ν

.

Title

singular measure

Canonical name

SingularMeasure

Date of creation

2013-03-22 13:26:26

Last modified on

2013-03-22 13:26:26

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 28A12

Defines

singular

Generated on Fri Feb 9 20:07:21 2018 by

LaTeXML
support of integrable function with respect to counting measure is countable

support of integrable function with respect to counting measure is countable

Let

(

X

,

𝔅

,

μ

)

be a

measure space

with

μ

the

counting measure

. If

f

is an integrable function,

∫

X

f

⁢

𝑑

μ

<

∞

, then it has

countable

(

http://planetmath.org/Countable

)

support

(

http://planetmath.org/Support6

).

Proof.

WLOG, we assume that

f

is real valued and is nonnegative. Let

S

0

denote the

preimage

of the interval

[

1

,

∞

)

and, for every

positive integer

n

, let

S

n

denote the preimage of the interval

[

1

n

+

1

,

1

n

)

. Since the

integral

of

f

is

bounded

, each

S

n

can be at most finite. Taking the union of all the

S

n

, we get the support

supp

⁡

f

=

⋃

n

=

0

∞

S

n

. Thus,

supp

⁡

f

is a union of countably many

finite sets

and hence is countable.
∎

Title

support of integrable function with respect to counting measure is countable

Canonical name

SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable

Date of creation

2013-03-22 14:59:34

Last modified on

2013-03-22 14:59:34

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Result

Classification

msc 28A12

Related topic

UncountableSumsOfPositiveNumbers

Related topic

SupportOfIntegrableFunctionIsSigmaFinite

Generated on Fri Feb 9 20:07:22 2018 by

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The proof of theorem is wrong

The proof of theorem is wrong

Let’s create a very simple

measurable space

:

X

=

{

a

,

b

}

,

𝒜

=

{

∅

,

{

a

}

,

{

b

}

,

X

}

.

Let’s take the

π

-system

P

=

{

{

a

}

}

containing only one subset of

X

.

Let’s create two

measures

μ

=

δ

a

+

δ

b

and

ν

=

δ

a

+

2

⁢

δ

b

. Then obviously

μ

and

ν

agree on

P

and are finite, but they obviously are not equal on

𝒜

.

The proof, however, claims that it is sufficient if

μ

and

ν

are finite. I believe that

μ

⁢

(

X

)

=

ν

⁢

(

X

)

is a necessary condition.

Title

The

proof of theorem is wrong

Canonical name

TheProofOfTheoremIsWrong

Date of creation

2013-03-22 19:16:05

Last modified on

2013-03-22 19:16:05

Owner

tomprimozic (26284)

Last modified by

tomprimozic (26284)

Numerical id

4

Author

tomprimozic (26284)

Entry type

Example

Classification

msc 28A12

Generated on Fri Feb 9 20:07:24 2018 by

LaTeXML
Hardy-Littlewood maximal theorem

Hardy-Littlewood maximal theorem

There is a constant

K

>

0

such that for each

Lebesgue integrable

function

f

∈

L

1

⁢

(

ℝ

n

)

, and each

t

>

0

,

m

⁢

(

{

x

:

M

⁢

f

⁢

(

x

)

>

t

}

)

≤

K

t

⁢

∥

f

∥

1

=

K

t

⁢

∫

ℝ

n

|

f

⁢

(

x

)

|

⁢

𝑑

x

,

where

M

⁢

f

is the

Hardy-Littlewood maximal function

of

f

.

Remark.

The theorem holds for the constant

K

=

3

n

.

Title

Hardy-Littlewood maximal theorem

Canonical name

HardyLittlewoodMaximalTheorem

Date of creation

2013-03-22 13:27:33

Last modified on

2013-03-22 13:27:33

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A15

Classification

msc 28A25

Related topic

HardyLittlewoodMaximalOperator

Defines

Hardy-Littlewood theorem

Generated on Fri Feb 9 20:07:28 2018 by

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Lebesgue differentiation theorem

Lebesgue differentiation theorem

Let

f

be a

locally integrable function

on

ℝ

n

with

Lebesgue measure

m

, i.e.

f

∈

L

loc

1

⁢

(

ℝ

n

)

.

Lebesgue’s

differentiation

theorem

basically says that for almost every

x

, the

averages

1

m

⁢

(

Q

)

⁢

∫

Q

|

f

⁢

(

y

)

-

f

⁢

(

x

)

|

⁢

𝑑

y

converge

to

0

when

Q

is a cube containing

x

and

m

⁢

(

Q

)

→

0

.

Formally, this means that there is a set

N

⊂

ℝ

n

with

μ

⁢

(

N

)

=

0

, such that for every

x

∉

N

and

ε

>

0

, there exists

δ

>

0

such that, for each cube

Q

with

x

∈

Q

and

m

⁢

(

Q

)

<

δ

, we have

1

m

⁢

(

Q

)

⁢

∫

Q

|

f

⁢

(

y

)

-

f

⁢

(

x

)

|

⁢

𝑑

y

<

ε

.

For

n

=

1

, this can be restated as an analogue of the

fundamental theorem of calculus for Lebesgue integrals

. Given a

x

0

∈

ℝ

,

d

d

⁢

x

⁢

∫

x

0

x

f

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

x

)

for almost every

x

.

Title

Lebesgue differentiation theorem

Canonical name

LebesgueDifferentiationTheorem

Date of creation

2013-03-22 13:27:36

Last modified on

2013-03-22 13:27:36

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A15

Generated on Fri Feb 9 20:07:30 2018 by

LaTeXML