aarjaneiro/planetmath huggingface dataset:
proof of Radon-Nikodym theorem

proof of Radon-Nikodym theorem

The following

proof of Radon-Nikodym theorem

is based on the original

argument

by John von Neumann.
We suppose that

μ

and

ν

are real, nonnegative, and finite.
The extension to the

σ

-finite case is a standard exercise,
as is

μ

-a.e. uniqueness of

Radon-Nikodym derivative

.
Having done this, the thesis also holds for signed and complex-valued

measures

.

Let

(

X

,

ℱ

)

be a

measurable space

and let

μ

,

ν

:

ℱ

→

[

0

,

R

]

two finite measures on

X

such that

ν

⁢

(

A

)

=

0

for every

A

∈

ℱ

such that

μ

⁢

(

A

)

=

0

.
Then

σ

=

μ

+

ν

is a finite measure on

X

such that

σ

⁢

(

A

)

=

0

if and only if

μ

⁢

(

A

)

=

0

.

Consider the

linear functional

T

:

L

2

⁢

(

X

,

ℱ

,

σ

)

→

ℝ

defined by

T

⁢

u

=

∫

X

u

⁢

𝑑

μ

⁢

∀

u

∈

L

2

⁢

(

X

,

ℱ

,

σ

)

.

(1)

T

is well-defined
because

μ

is finite and dominated by

σ

, so that

L

2

⁢

(

X

,

ℱ

,

σ

)

⊆

L

2

⁢

(

X

,

ℱ

,

μ

)

⊆

L

1

⁢

(

X

,

ℱ

,

μ

)

;

it is also linear and

bounded

because

|

T

⁢

u

|

≤

∥

u

∥

L

2

⁢

(

X

,

ℱ

,

σ

)

⋅

σ

⁢

(

X

)

.

By

Riesz representation theorem

, there exists

g

∈

L

2

⁢

(

X

,

ℱ

,

σ

)

such that

T

⁢

u

=

∫

X

u

⁢

𝑑

μ

=

∫

X

u

⋅

g

⁢

𝑑

σ

(2)

for every

u

∈

L

2

⁢

(

X

,

ℱ

,

σ

)

.
Then

μ

⁢

(

A

)

=

∫

A

g

⁢

𝑑

σ

for every

A

∈

ℱ

,
so that

0

<

g

≤

1

μ

- and

σ

-a.e.
(Consider the former with

A

=

{

x

∣

g

⁢

(

x

)

≤

0

}

or

A

=

{

x

⁢

∣

g

⁢

(

x

)

>

⁢

1

}

.)
Moreover, the second equality in (

LABEL:eq:q

)
holds when

u

=

χ

A

for

A

∈

ℱ

,
thus also when

u

is a simple

measurable function

by linearity of

integral

,
and finally when

u

is a (

μ

- and

σ

-a.e.)
nonnegative

ℱ

-measurable function
because of the

monotone convergence theorem

.

Now,

1

/

g

is

ℱ

-

measurable

and nonnegative

μ

- and

σ

-a.e.;
moreover,

1

g

⋅

g

=

1

σ

- and

μ

-a.e.
Thus, for every

A

∈

ℱ

,

∫

A

1

g

⁢

𝑑

μ

=

∫

A

𝑑

σ

=

σ

⁢

(

A

)

(3)

Since

σ

is finite,

1

/

g

∈

L

1

⁢

(

X

,

ℱ

,

μ

)

,
and so is

f

=

1

g

-

1

.
Then for every

A

∈

ℱ

ν

⁢

(

A

)

=

σ

⁢

(

A

)

-

μ

⁢

(

A

)

=

∫

A

(

1

g

-

1

)

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

.

Title

proof of Radon-Nikodym theorem

Canonical name

ProofOfRadonNikodymTheorem

Date of creation

2013-03-22 18:58:03

Last modified on

2013-03-22 18:58:03

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Proof

Classification

msc 28A15

Synonym

Hilbert spaces

proof of Radon-Nikodym’s theorem

Synonym

measure- theoretic proof of Radon-Nikodym theorem

Generated on Fri Feb 9 20:07:32 2018 by

LaTeXML
Radon-Nikodym theorem

Radon-Nikodym theorem

Let

μ

and

ν

be two

σ

-finite

measures

on the same

measurable space

(

Ω

,

𝒮

)

, such that

ν

≪

μ

(i.e.

ν

is

absolutely continuous

with respect to

μ

.)
Then there exists a

measurable function

f

, which is nonnegative
and finite, such that for each

A

∈

𝒮

,

ν

⁢

(

A

)

=

∫

A

f

⁢

𝑑

μ

.

This function is unique (any other function satisfying these
conditions is equal to

f

μ

-almost everywhere,) and it is called
the

Radon-Nikodym derivative

of

ν

with respect to

μ

,
denoted by

f

=

d

⁢

ν

d

⁢

μ

.

Remark.

The theorem also holds if

ν

is a

signed measure

. Even if

ν

is not

σ

-finite the theorem holds, with the exception that

f

is not necessarely finite.

Some properties of the Radon-Nikodym derivative

Let

ν

,

μ

, and

λ

be

σ

-finite measures in

(

Ω

,

𝒮

)

.

1.

If

ν

≪

λ

and

μ

≪

λ

, then

d

⁢

(

ν

+

μ

)

d

⁢

λ

=

d

⁢

ν

d

⁢

λ

+

d

⁢

μ

d

⁢

λ

⁢

μ

⁢

-almost everywhere

;

2.

If

ν

≪

μ

≪

λ

, then

d

⁢

ν

d

⁢

λ

=

d

⁢

ν

d

⁢

μ

⁢

d

⁢

μ

d

⁢

λ

⁢

μ

⁢

-almost everywhere

;

3.

If

μ

≪

λ

and

g

is a

μ

-integrable function, then

∫

Ω

g

⁢

𝑑

μ

=

∫

Ω

g

⁢

d

⁢

μ

d

⁢

λ

⁢

𝑑

λ

;

4.

If

μ

≪

ν

and

ν

≪

μ

, then

d

⁢

μ

d

⁢

ν

=

(

d

⁢

ν

d

⁢

μ

)

-

1

.

Title

Radon-Nikodym theorem

Canonical name

RadonNikodymTheorem

Date of creation

2013-03-22 13:26:15

Last modified on

2013-03-22 13:26:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A15

Related topic

AbsolutelyContinuous

Related topic

BoundedLinearFunctionalsOnLpmu

Related topic

MartingaleProofOfTheRadonNikodymTheorem

Related topic

BoundedLinearFunctionalsOnLinftymu

Defines

Radon-Nikodym derivative

Generated on Fri Feb 9 20:07:34 2018 by

LaTeXML
almost continuous function

almost continuous function

Let

m

denote

Lebesgue measure

,

A

be a Lebesgue measurable subset of

ℝ

, and

f

:

A

→

ℂ

(or

f

:

A

→

ℝ

). Then

f

is

almost

continuous

if, for every

ε

>

0

, there exists a

closed subset

F

of

ℝ

such that

F

⊆

A

,

m

⁢

(

A

-

F

)

<

ε

, and

f

|

F

is continuous.

Title

almost

continuous function

Canonical

name

AlmostContinuousFunction

Date of creation

2013-03-22 16:13:45

Last modified on

2013-03-22 16:13:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 28A20

Synonym

almost continuous

Related topic

LusinsTheorem2

Generated on Fri Feb 9 20:07:36 2018 by

LaTeXML
conditional expectations are uniformly integrable

conditional expectations are uniformly integrable

The

collection

of all

conditional expectations

of an

integrable

random variable

forms a

uniformly integrable

set. More generally, we have the following result.

Theorem

.

Let

S

be a uniformly integrable set of random variables defined on a

probability space

(

Ω

,

F

,

P

)

. Then, the set

{

𝔼

[

X

∣

𝒢

]

:

X

∈

S

and

𝒢

is a sub-

σ

-algebra of

ℱ

}

is also uniformly integrable.

To prove the result, we first use the fact that uniform integrability implies that

S

is

L

1

-bounded. That is, there is a

constant

L

>

0

such that

𝔼

⁢

[

|

X

|

]

≤

L

for every

X

∈

S

.
Also, choosing any

ϵ

>

0

, there is a

δ

>

0

so that

𝔼

⁢

[

|

X

|

⁢

1

A

]

<

ϵ

for all

X

∈

S

and

A

∈

ℱ

with

ℙ

⁢

(

A

)

≤

δ

.

Set

K

=

L

/

δ

. Then, if

Y

=

𝔼

[

X

∣

𝒢

]

for any

X

∈

S

and

𝒢

⊆

ℱ

, Jensen’s

inequality

gives

|

Y

|

≤

𝔼

[

|

X

|

∣

𝒢

]

.

So, applying Markov’s inequality,

ℙ

(

|

Y

|

>

K

)

≤

K

-

1

𝔼

[

|

Y

|

]

≤

K

-

1

𝔼

[

|

X

|

]

≤

L

/

K

=

δ

and, therefore

𝔼

⁢

[

|

Y

|

⁢

1

{

|

Y

|

>

K

}

]

≤

𝔼

⁢

[

|

X

|

⁢

1

{

|

Y

|

>

K

}

]

<

ϵ

.

Title

conditional expectations are uniformly integrable

Canonical name

ConditionalExpectationsAreUniformlyIntegrable

Date of creation

2013-03-22 18:40:08

Last modified on

2013-03-22 18:40:08

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A20

Classification

msc 60A10

Related topic

ConditionalExpectation

Related topic

UniformlyIntegrable

Generated on Fri Feb 9 20:07:39 2018 by

LaTeXML
criterion for interchanging summation and integration

criterion for interchanging summation and integration

The following criterion for interchanging integration and

summation

is often useful in practise: Suppose one has a

sequence

of

measurable
functions

f

k

:

M

→

ℝ

(The index

k

runs over
non-negative

integers

.) on some

measure space

M

and can find another sequence of measurable
functions

g

k

:

M

→

ℝ

such that

|

f

k

⁢

(

x

)

|

≤

g

k

⁢

(

x

)

for all

k

and almost all

x

and

∑

k

=

0

∞

g

k

⁢

(

x

)

converges for almost all

x

∈

M

and

∑

k

=

0

∞

∫

g

k

⁢

(

x

)

⁢

𝑑

x

<

∞

. Then

∫

M

∑

k

=

0

∞

f

k

⁢

(

x

)

⁢

d

⁢

x

=

∑

k

=

0

∞

∫

M

f

k

⁢

(

x

)

⁢

𝑑

x

This criterion is a

corollary

of the

monotone

and

dominated
convergence theorems

. Since the

g

k

’s are nonnegative, the
sequence of partial sums is increasing, hence, by the

monotone
convergence theorem

,

∫

M

∑

k

=

0

∞

g

k

⁢

(

x

)

⁢

d

⁢

x

<

∞

.
Since

∑

k

=

0

∞

g

k

⁢

(

x

)

converges for almost all

x

,

|

∑

k

=

0

n

f

k

⁢

(

x

)

|

≤

∑

k

=

0

n

|

f

k

⁢

(

x

)

|

≤

∑

k

=

0

n

g

k

⁢

(

x

)

≤

∑

k

=

0

∞

g

k

⁢

(

x

)

,

the dominated convergence theorem implies that we may

integrate

the sequence of partial sums term-by-term, which is tantamount to
saying that we may switch integration and summation.

As an example of this method, consider the following:

∫

-

∞

+

∞

∑

k

=

1

∞

cos

⁡

(

x

/

k

)

x

2

+

k

4

⁢

d

⁢

x

The idea behind the method is to pick our

g

’s as simple as possible
so that it is easy to integrate them and apply the criterion. A good
choice here is

g

k

⁢

(

x

)

=

1

/

(

x

2

+

k

4

)

. We then have

∫

-

∞

+

∞

g

k

⁢

(

x

)

⁢

𝑑

x

=

π

/

k

2

and, as

∑

k

=

1

∞

k

-

2

<

∞

, we can interchange summation
and integration:

∑

k

=

1

∞

∫

-

∞

+

∞

cos

⁡

(

x

/

k

)

x

2

+

k

4

⁢

𝑑

x

.

Doing the

integrals

, we obtain the answer

π

⁢

∑

k

=

1

∞

e

-

k

k

2

Title

criterion for interchanging summation and integration

Canonical

name

CriterionForInterchangingSummationAndIntegration

Date of creation

2013-03-22 16:20:05

Last modified on

2013-03-22 16:20:05

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Result

Classification

msc 28A20

Generated on Fri Feb 9 20:07:41 2018 by

LaTeXML
dominated convergence theorem

dominated convergence theorem

Let

X

be a

measure space

, and let

Φ

,

f

1

,

f

2

,

…

be

measurable functions

such that

∫

X

Φ

<

∞

and

|

f

n

|

≤

Φ

for each

n

.
If

f

n

→

f

almost everywhere, then

f

is integrable and

lim

n

→

∞

⁡

∫

X

f

n

=

∫

X

f

.

This theorem is a corollary of the

Fatou-Lebesgue theorem

.

Title

dominated convergence theorem

Canonical name

DominatedConvergenceTheorem

Date of creation

2013-03-22 13:12:47

Last modified on

2013-03-22 13:12:47

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

13

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A20

Synonym

Lebesgue’s dominated convergence theorem

Related topic

MonotoneConvergenceTheorem

Related topic

FatousLemma

Related topic

VitaliConvergenceTheorem

Generated on Fri Feb 9 20:07:42 2018 by

LaTeXML
Egorov’s theorem

Egorov’s theorem

Let

(

X

,

𝒮

,

μ

)

be a

measure space

, and let

E

be a subset of

X

of finite

measure

. If

f

n

is a sequence of

measurable functions

converging to

f

almost everywhere, then for each

δ

>

0

there exists a set

E

δ

such that

μ

⁢

(

E

δ

)

<

δ

and

f

n

→

f

uniformly (

http://planetmath.org/UniformConvergence

) on

E

-

E

δ

.

Title

Egorov’s theorem

Canonical name

EgorovsTheorem

Date of creation

2013-03-22 13:13:46

Last modified on

2013-03-22 13:13:46

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A20

Synonym

Egoroff’s theorem

Generated on Fri Feb 9 20:07:44 2018 by

LaTeXML
equivalent conditions for uniform integrability

equivalent conditions for uniform integrability

Let

(

Ω

,

ℱ

,

μ

)

be a

measure space

and

S

be a

bounded

subset of

L

1

⁢

(

Ω

,

ℱ

,

μ

)

. That is,

∫

|

f

|

⁢

𝑑

μ

is bounded over

f

∈

S

. Then, the following are

equivalent

.

1.

For every

ϵ

>

0

there is a

δ

>

0

so that

∫

A

|

f

|

⁢

𝑑

μ

<

ϵ

for all

f

∈

S

and

μ

⁢

(

A

)

<

δ

.

2.

For every

ϵ

>

0

there is a

K

>

0

satisfying

∫

|

f

|

>

K

|

f

|

⁢

𝑑

μ

<

ϵ

for all

f

∈

S

.

3.

There is a

measurable function

Φ

:

ℝ

→

[

0

,

∞

)

such that

Φ

⁢

(

x

)

/

|

x

|

→

∞

as

|

x

|

→

∞

and

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

is bounded over all

f

∈

S

. Moreover, the

function

Φ

can always be chosen to be

symmetric

and convex.

So, for bounded subsets of

L

1

, either of the above properties can be used to define

uniform integrability

.

Conversely

, when the measure space is finite, then conditions (

2

) and (

3

) are easily shown to imply that

S

is bounded in

L

1

.

To show the equivalence of these statements, let us suppose that

∫

|

f

|

⁢

𝑑

μ

<

L

for

f

∈

S

.

(

1

) implies (

2

)

For

ϵ

>

0

, property (

1

) gives a

δ

>

0

so that

∫

A

|

f

|

⁢

𝑑

μ

<

ϵ

whenever

f

∈

S

and

μ

⁢

(

A

)

<

δ

. Choosing

K

>

L

/

δ

, Markov’s

inequality

gives

μ

(

|

f

|

>

K

)

≤

K

-

1

∫

|

f

|

d

μ

≤

L

/

K

<

δ

and, therefore,

∫

|

f

|

>

K

|

f

|

⁢

𝑑

μ

<

ϵ

.

(

2

) implies (

3

)

For each

n

=

1

,

2

,

…

, property (

2

) gives a

K

n

satisfying

∫

(

|

f

|

-

K

n

)

+

⁢

𝑑

μ

≤

∫

|

f

|

>

K

n

|

f

|

⁢

𝑑

μ

≤

2

-

n

.

Without loss of generality, the

K

n

can be chosen to be

increasing

to

infinity

, so we can define

Φ

⁢

(

x

)

=

∑

n

(

|

x

|

-

K

n

)

+

. Then,

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

=

∑

n

∫

(

|

f

|

-

K

n

)

+

⁢

𝑑

μ

≤

∑

n

2

-

n

=

1

.

(

3

) implies (

1

)

First, suppose that

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

<

M

for

f

∈

S

.
For

ϵ

>

0

, the condition that

Φ

⁢

(

x

)

/

|

x

|

→

∞

as

|

x

|

→

∞

gives a

K

>

0

such that

Φ

⁢

(

x

)

/

|

x

|

≥

2

⁢

M

/

ϵ

whenever

|

x

|

>

K

.
Setting

δ

=

ϵ

/

2

⁢

K

,

∫

A

|

f

|

⁢

𝑑

μ

≤

∫

|

f

|

>

K

|

f

|

⁢

𝑑

μ

+

K

⁢

μ

⁢

(

A

)

<

(

ϵ

/

2

⁢

M

)

⁢

∫

|

f

|

>

K

Φ

⁢

(

f

)

⁢

𝑑

μ

+

K

⁢

δ

<

ϵ

/

2

+

ϵ

/

2

=

ϵ

.

whenever

μ

⁢

(

A

)

<

δ

and

f

∈

S

.

Title

equivalent conditions for uniform integrability

Canonical name

EquivalentConditionsForUniformIntegrability

Date of creation

2013-03-22 18:40:17

Last modified on

2013-03-22 18:40:17

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A20

Generated on Fri Feb 9 20:07:47 2018 by

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existence of the essential supremum

existence of the essential supremum

We state the existence of the

essential supremum

for a set

𝒮

of extended real valued functions on a

σ

-finite (

http://planetmath.org/SigmaFinite

)

measure space

(

Ω

,

ℱ

,

μ

)

.

Theorem

.

Suppose that the measure space

(

Ω

,

F

,

μ

)

is

σ

-finite. Then, the essential supremum of

S

exists. Furthermore, if

S

is nonempty then there exists a sequence

(

f

n

)

n

=

1

,

2

,

…

in

S

such that

esssup

⁡

𝒮

=

sup

n

⁡

f

n

.

(1)

Note that, by reversing the

inequalities

, this result also applies to the essential infimum, except that equation (

1

) is replaced by

essinf

⁡

𝒮

=

inf

n

⁡

f

n

.

Title

existence of the essential supremum

Canonical name

ExistenceOfTheEssentialSupremum

Date of creation

2013-03-22 18:39:22

Last modified on

2013-03-22 18:39:22

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A20

Related topic

EssentialSupremum

Generated on Fri Feb 9 20:07:49 2018 by

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Fatou-Lebesgue theorem

Fatou-Lebesgue theorem

Let

(

X

,

μ

)

be a

measure space

. If

Φ

:

X

→

ℝ

is a nonnegative function with

∫

Φ

⁢

𝑑

μ

<

∞

, and if

f

1

,

f

2

,

…

is a sequence of

measurable functions

such that

|

f

n

|

≤

Φ

for each

n

, then

g

=

lim inf

n

→

∞

⁡

f

n

⁢

and

⁢

h

=

lim sup

n

→

∞

⁡

f

n

are both integrable, and

-

∞

<

∫

g

⁢

𝑑

μ

≤

lim inf

n

→

∞

⁡

∫

f

n

⁢

𝑑

μ

≤

lim sup

k

→

∞

⁡

∫

f

n

⁢

𝑑

μ

≤

∫

h

⁢

𝑑

μ

<

∞

.

Title

Fatou-Lebesgue theorem

Canonical name

FatouLebesgueTheorem

Date of creation

2013-03-22 13:12:53

Last modified on

2013-03-22 13:12:53

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A20

Related topic

FatousLemma

Generated on Fri Feb 9 20:07:50 2018 by

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Fatou’s lemma

Fatou’s lemma

If

f

1

,

f

2

,

…

is a sequence of nonnegative

measurable functions

in a

measure space

X

, then

∫

X

lim inf

n

→

∞

⁡

f

n

≤

lim inf

n

→

∞

⁡

∫

X

f

n

Title

Fatou’s lemma

Canonical name

FatousLemma

Date of creation

2013-03-22 13:12:50

Last modified on

2013-03-22 13:12:50

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A20

Related topic

FatouLebesgueTheorem

Related topic

MonotoneConvergenceTheorem

Related topic

DominatedConvergenceTheorem

Generated on Fri Feb 9 20:07:52 2018 by

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Lusin’s theorem

Lusin’s theorem

Let

A

be a

Lebesgue measurable

subset of

ℝ

and

f

:

A

→

ℂ

(or

f

:

A

→

ℝ

). Then

f

is Lebesgue measurable (

http://planetmath.org/MeasurableFunctions

) if and only if

f

is almost

continuous

.

Title

Lusin’s

theorem

Canonical name

LusinsTheorem

Date of creation

2013-03-22 16:13:48

Last modified on

2013-03-22 16:13:48

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 28A20

Related topic

MeasurableFunctions

Related topic

AlmostContinuousFunction

Generated on Fri Feb 9 20:07:56 2018 by

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measurable function

measurable function

Let

(

X

,

ℬ

⁢

(

X

)

)

and

(

Y

,

ℬ

⁢

(

Y

)

)

be two

measurable spaces

. Then a

function

f

:

X

→

Y

is called a

measurable function

if:

f

-

1

⁢

(

ℬ

⁢

(

Y

)

)

⊆

ℬ

⁢

(

X

)

where

f

-

1

⁢

(

ℬ

⁢

(

Y

)

)

=

{

f

-

1

⁢

(

E

)

∣

E

∈

ℬ

⁢

(

Y

)

}

.

In other words, the

inverse image

of every

ℬ

⁢

(

Y

)

-

measurable set

is

ℬ

⁢

(

X

)

-

measurable

. The space of all measurable functions

f

:

X

→

Y

is denoted as

ℳ

⁢

(

(

X

,

ℬ

⁢

(

X

)

)

,

(

Y

,

ℬ

⁢

(

Y

)

)

)

.

Any measurable function into

(

ℝ

,

ℬ

⁢

(

ℝ

)

)

, where

ℬ

⁢

(

ℝ

)

is the

Borel sigma algebra

of the

real numbers

ℝ

, is called a

Borel measurable function

.

1

1

More generally, a measurable function is called

Borel measurable

if the range space

Y

is a

topological space

with

ℬ

⁢

(

Y

)

the

sigma algebra

generated by all

open sets

of

Y

.

The space of all Borel measurable functions from a measurable space

(

X

,

ℬ

⁢

(

X

)

)

is denoted by

ℒ

0

⁢

(

X

,

ℬ

⁢

(

X

)

)

.

Similarly, we write

ℒ

¯

0

⁢

(

X

,

ℬ

⁢

(

X

)

)

for

ℳ

⁢

(

(

X

,

ℬ

⁢

(

X

)

)

,

(

ℝ

¯

,

ℬ

⁢

(

ℝ

¯

)

)

)

, where

ℬ

⁢

(

ℝ

¯

)

is the Borel sigma algebra of

ℝ

¯

, the set of

extended real numbers

.

Remark

. If

f

:

X

→

Y

and

g

:

Y

→

Z

are measurable functions, then so is

g

∘

f

:

X

→

Z

, for if

E

is

ℬ

⁢

(

Z

)

-measurable, then

g

-

1

⁢

(

E

)

is

ℬ

⁢

(

Y

)

-measurable, and

f

-

1

⁢

(

g

-

1

⁢

(

E

)

)

is

ℬ

⁢

(

X

)

-measurable. But

f

-

1

⁢

(

g

-

1

⁢

(

E

)

)

=

(

g

∘

f

)

-

1

⁢

(

E

)

, which implies that

g

∘

f

is a measurable function.

Example:

•

Let

E

be a subset of a measurable space

X

. Then the

characteristic function

χ

E

is a measurable function if and only if

E

is measurable.

Title

measurable function

Canonical name

MeasurableFunction

Date of creation

2013-03-22 12:50:50

Last modified on

2013-03-22 12:50:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Definition

Classification

msc 28A20

Synonym

Borel measurable

Related topic

ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets

Related topic

LusinsTheorem2

Related topic

BorelGroupoid

Related topic

BorelMorphism

Defines

Borel measurable function

Generated on Fri Feb 9 20:07:59 2018 by

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modes of convergence of sequences of measurable functions

modes of convergence of sequences of measurable functions

Let

(

X

,

𝔅

,

μ

)

be a

measure space

,

f

n

:

X

→

[

-

∞

,

∞

]

be

measurable functions

for every

positive integer

n

, and

f

:

X

→

[

-

∞

,

∞

]

be a measurable function. The following are modes of convergence of

{

f

n

}

:

•

{

f

n

}

converges almost everywhere

to

f

if

μ

⁢

(

X

-

{

x

∈

X

:

lim

n

→

∞

⁡

f

n

⁢

(

x

)

=

f

⁢

(

x

)

}

)

=

0

•

{

f

n

}

converges almost uniformly

to

f

if, for every

ε

>

0

, there exists

E

ε

∈

𝔅

with

μ

⁢

(

X

-

E

ε

)

<

ε

and

{

f

n

}

converges uniformly

to

f

on

E

ε

•

{

f

n

}

converges in measure

to

f

if, for every

ε

>

0

, there exists a positive integer

N

such that, for every positive integer

n

≥

N

,

μ

⁢

(

{

x

∈

X

:

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

≥

ε

}

)

<

ε

.

•

If, in ,

f

and each

f

n

are also

Lebesgue integrable

,

{

f

n

}

converges

in

L

1

⁢

(

μ

)

to

f

if

lim

n

→

∞

⁡

∫

X

|

f

n

-

f

|

⁢

𝑑

μ

=

0

.

A lot of theorems in

real analysis

(

http://planetmath.org/BibliographyForRealAnalysis

) deal with these modes of convergence. For example, Fatou’s lemma, Lebesgue’s

monotone convergence theorem

, and Lebesgue’s

dominated convergence theorem

give conditions on

sequences

of measurable functions that converge almost everywhere under which they also converge in

L

1

⁢

(

μ

)

. Also, Egorov’s theorem that, if

μ

⁢

(

X

)

<

∞

, then convergence almost everywhere implies almost

uniform convergence

.

Title

modes of convergence of sequences of measurable functions

Canonical name

ModesOfConvergenceOfSequencesOfMeasurableFunctions

Date of creation

2013-03-22 16:14:05

Last modified on

2013-03-22 16:14:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 28A20

Related topic

TravelingHumpSequence

Related topic

VitaliConvergenceTheorem

Defines

converges almost everywhere

Defines

convergence almost everywhere

Defines

converges almost uniformly

Defines

almost uniform convergence

Defines

converges in measure

Defines

convergence in measure

Defines

converges in

L

1

⁢

(

μ

)

Defines

L

1

⁢

(

μ

)

convergence

Generated on Fri Feb 9 20:08:01 2018 by

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proof of dominated convergence theorem

proof of dominated convergence theorem

It is not difficult to prove that

f

is measurable. In fact we can write

f

⁢

(

x

)

=

sup

n

⁡

inf

k

≥

n

⁡

f

k

⁢

(

x

)

and we know that

measurable functions

are

closed under

the

sup

and

inf

operation

.

Consider the

sequence

g

n

⁢

(

x

)

=

2

⁢

Φ

⁢

(

x

)

-

|

f

⁢

(

x

)

-

f

n

⁢

(

x

)

|

.
Clearly

g

n

are nonnegative

functions

since

f

-

f

n

≤

2

⁢

Φ

.
So, applying Fatou’s Lemma, we obtain

lim

n

→

∞

⁡

∫

X

|

f

-

f

n

|

⁢

𝑑

μ

≤

lim sup

n

→

∞

⁡

∫

X

|

f

-

f

n

|

⁢

𝑑

μ

=

-

lim inf

n

→

∞

⁡

∫

X

-

|

f

-

f

n

|

⁢

d

⁢

μ

=

∫

X

2

⁢

Φ

⁢

𝑑

μ

-

lim inf

n

→

∞

⁡

∫

X

2

⁢

Φ

-

|

f

-

f

n

|

⁢

d

⁢

μ

≤

∫

X

2

⁢

Φ

⁢

𝑑

μ

-

∫

X

2

⁢

Φ

-

lim sup

n

→

∞

⁡

|

f

-

f

n

|

⁢

d

⁢

μ

=

∫

X

2

⁢

Φ

⁢

𝑑

μ

-

∫

X

2

⁢

Φ

⁢

𝑑

μ

=

0

.

Title

proof of dominated convergence theorem

Canonical name

ProofOfDominatedConvergenceTheorem

Date of creation

2013-03-22 13:30:02

Last modified on

2013-03-22 13:30:02

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Proof

Classification

msc 28A20

Related topic

SecondProofOfDominatedConvergenceTheorem

Related topic

SecondProofOfDominatedConvergenceTheorem2

Generated on Fri Feb 9 20:08:06 2018 by

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