proof of Vitali convergence theorem

proof of Vitali convergence theorem

Theorem

.

Let

f

1

,

f

2

,

…

be

L

p

-integrable functions on
a

measure space

(

X

,

μ

)

, for

1

≤

p

<

∞

.
The following conditions are necessary and

sufficient

for

f

n

to be a

Cauchy sequence

in the

L

p

⁢

(

X

,

μ

)

norm:

(i)

the

sequence

f

n

is Cauchy in

measure

;

(ii)

the

functions

{

|

f

n

|

p

}

are

uniformly integrable

; and

(iii)

for each

ϵ

>

0

, there is a set

A

of finite measure, with

∥

f

n

⁢

𝟏

⁢

(

X

∖

A

)

∥

<

ϵ

for all

n

.

Proof.

We abbreviate

|

f

n

-

f

m

|

by

f

m

⁢

n

.

Necessity of (i).

Fix

t

>

0

, and let

E

m

⁢

n

=

{

f

m

⁢

n

≥

t

}

.
Then

μ

⁢

(

E

m

⁢

n

)

1

/

p

=

1

t

⁢

∥

t

⁢

1

⁢

(

E

m

⁢

n

)

∥

≤

1

t

⁢

∥

f

m

⁢

n

∥

→

0

,

as

m

,

n

→

∞

.

Necessity of (ii).

Select

N

such that

∥

f

n

-

f

N

∥

<

ϵ

when

n

≥

N

.
The family

{

|

f

1

|

p

,

…

,

|

f

N

-

1

|

p

,

|

f

N

|

p

}

is uniformly integrable
because it consists of only

finitely

many integrable functions.

So for every

ϵ

>

0

,
there is

δ

>

0

such that

μ

⁢

(

E

)

<

δ

implies

∥

f

n

⁢

𝟏

⁢

(

E

)

∥

<

ϵ

for

n

≤

N

.
On the other hand, for

n

>

N

,

∥

f

n

⁢

𝟏

⁢

(

E

)

∥

≤

∥

(

f

n

-

f

N

)

⁢

𝟏

⁢

(

E

)

∥

+

∥

f

N

⁢

𝟏

⁢

(

E

)

∥

<

2

⁢

ϵ

for the same sets

E

,
and thus the entire

infinite

sequence

{

|

f

n

|

p

}

is
uniformly integrable too.

Necessity of (iii).

Select

N

such that

∥

f

n

-

f

N

∥

<

ϵ

for all

n

≥

N

.
Let

φ

be a

simple function

approximating

f

N

in

𝐋

p

norm up to

ϵ

.
Then

∥

f

n

-

φ

∥

<

2

⁢

ϵ

for all

n

≥

N

.
Let

A

N

=

{

φ

≠

0

}

be the

support

of

φ

, which must
have finite measure.
It follows that

∥

f

n

⁢

𝟏

⁢

(

X

∖

A

N

)

∥

=

∥

f

n

-

f

n

⁢

𝟏

⁢

(

A

N

)

∥

≤

∥

f

n

-

φ

∥

+

∥

φ

-

f

n

⁢

𝟏

⁢

(

A

N

)

∥

=

∥

f

n

-

φ

∥

+

∥

(

φ

-

f

n

)

⁢

𝟏

⁢

(

A

N

)

∥

<

2

⁢

ϵ

+

2

⁢

ϵ

.

For each

n

<

N

, we can similarly construct sets

A

n

of finite measure,
such that

∥

f

n

⁢

𝟏

⁢

(

X

∖

A

n

)

∥

<

4

⁢

ϵ

.
If we set

A

=

A

1

∪

⋯

∪

A

N

-

1

∪

A

N

, a finite union,
then

A

has finite measure, and clearly

∥

f

n

⁢

𝟏

⁢

(

X

∖

A

)

∥

<

4

⁢

ϵ

for any

n

.

Sufficiency.

We show

f

m

⁢

n

to be small for large

m

,

n

by a multi-step estimate:

∥

f

m

⁢

n

∥

≤

∥

f

m

⁢

n

⁢

𝟏

⁢

(

A

∖

E

m

⁢

n

)

∥

+

∥

f

m

⁢

n

⁢

𝟏

⁢

(

E

m

⁢

n

)

∥

+

∥

f

m

⁢

n

⁢

𝟏

⁢

(

X

∖

A

)

∥

.

Use condition (iii) to choose

A

of finite measure
such that

∥

f

n

⁢

𝟏

⁢

(

X

∖

A

)

∥

<

ϵ

for every

n

.
Then

∥

f

m

⁢

n

⁢

𝟏

⁢

(

X

∖

A

)

∥

<

2

⁢

ϵ

.

Let

t

=

ϵ

/

μ

⁢

(

A

)

1

/

p

>

0

,
and

E

m

⁢

n

=

{

f

m

⁢

n

≥

t

}

.
By condition (ii) choose

δ

>

0

so that

∥

f

n

⁢

𝟏

⁢

(

E

)

∥

<

ϵ

whenever

μ

⁢

(

E

)

<

δ

.
By condition (i), take

N

such that if

m

,

n

≥

N

,
then

μ

⁢

(

E

m

⁢

n

)

<

δ

;
it follows immediately that

∥

f

m

⁢

n

⁢

𝟏

⁢

(

E

m

⁢

n

)

∥

<

2

⁢

ϵ

.

Finally,

∥

f

m

⁢

n

⁢

𝟏

⁢

(

A

∖

E

m

⁢

n

)

∥

≤

t

⁢

μ

⁢

(

A

)

1

/

p

=

ϵ

, since

f

m

⁢

n

<

t

on the

complement

of

E

m

⁢

n

.
Hence

∥

f

m

⁢

n

∥

<

5

⁢

ϵ

for

m

,

n

≥

N

.
∎

Remark.

In the statement of the

theorem

, instead
of dealing with Cauchy sequences,
we can directly speak of convergence of

f

n

to

f

in

𝐋

p

and in measure.
This variation of the theorem
is easily proved,
for:

•

a sequence

converges

in

𝐋

p

if and only if it is
Cauchy in

𝐋

p

;

•

a sequence that

converges in measure

is automatically Cauchy in measure;

•

a simple adaptation of the

argument

shows that

f

n

→

f

in

𝐋

p

implies

f

n

→

f

in measure; and

•

the limit in measure is unique.

Title

proof of Vitali convergence theorem

Canonical name

ProofOfVitaliConvergenceTheorem

Date of creation

2013-03-22 17:31:04

Last modified on

2013-03-22 17:31:04

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

5

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 28A20

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