uniformly integrable

uniformly integrable

Let

μ

be a

positive measure

on a

measurable space

.
A

collection

of

functions

{

f

α

}

⊂

𝐋

1

⁢

(

μ

)

is

uniformly integrable

, if for every

ϵ

>

0

, there
exists

δ

>

0

such that

|

∫

E

f

α

⁢

𝑑

μ

|

<

ϵ

whenever

μ

⁢

(

E

)

<

δ

, for any

α

.

(The

absolute value

sign outside of the

integral

above may appear under the integral sign instead without affecting the definition.)

The usefulness of this definition comes from the

Vitali convergence theorem

,
which uses it to characterize the convergence of functions
in

𝐋

1

⁢

(

μ

)

.

Definition in probability theory

In

probability

, a different, and slightly stronger, definition of “uniform integrability”, is more commonly used:

A collection of functions

{

f

α

}

⊂

𝐋

1

⁢

(

μ

)

is

uniformly integrable

, if for every

ϵ

>

0

, there
exists

t

≥

0

such that

∫

[

|

f

α

|

≥

t

]

|

f

α

|

⁢

𝑑

μ

<

ϵ

for every

α

.

Assuming

μ

is a

probability measure

, this definition is

equivalent

to the previous one together with the condition that

∫

|

f

α

|

⁢

𝑑

μ

is uniformly bounded for all

α

.

Properties

1.

If a finite number of collections are uniformly integrable,
then so is their finite union.

2.

A single

f

∈

𝐋

1

⁢

(

μ

)

is always uniformly integrable.

To see this, observe that

f

must be almost everywhere non-infinite.
Thus

f

⋅

1

[

|

f

|

>

k

]

goes to zero a.e. as

k

→

∞

, and it is

bounded

by

|

f

|

.
Then

∫

[

|

f

|

>

k

]

|

f

|

⁢

𝑑

μ

→

0

by the

dominated convergence theorem

.
Choosing

k

big enough so that

∫

[

|

f

|

>

k

]

|

f

|

⁢

𝑑

μ

<

ϵ

, and letting

δ

=

ϵ

/

k

, we have, when

μ

⁢

(

E

)

<

δ

,

∫

E

|

f

|

⁢

𝑑

μ

=

∫

E

∩

[

|

f

|

≤

k

]

|

f

|

⁢

𝑑

μ

+

∫

E

∩

[

|

f

|

>

k

]

|

f

|

⁢

𝑑

μ

≤

k

⁢

μ

⁢

(

E

)

+

ϵ

=

2

⁢

ϵ

.

Examples

1.

If

g

is an integrable function, then
the collection consisting of
all

measurable functions

f

dominated by

g

— that is,

|

f

|

≤

g

—
is uniformly integrable.

2.

If

X

is a

𝐋

1

random variable

on a

probability space

Ω

,
then the set of all of its

conditional expectations

,

{

𝔼

[

X

∣

𝒢

]

:

𝒢

is a

σ

-algebra of

Ω

}

,

is always uniformly integrable.

3.

If there is an

unbounded

increasing function

ϕ

:

[

0

,

∞

)

→

[

0

,

∞

)

such that

∫

|

f

α

|

⁢

ϕ

⁢

(

|

f

α

|

)

⁢

𝑑

μ

is uniformly bounded for all

α

,
then the collection

{

f

α

}

is uniformly integrable.

References

1

Kai Lai Chung.

A Course in Probability Theory

, third ed. Academic Press, 2001.

2

Gerald B. Folland.

Real Analysis: Modern Techniques and Their Applications

, second ed. Wiley-Interscience, 1999.

3

Jeffrey S. Rosenthal.

A First Look at

Rigorous

Probability Theory

.
World Scientific, 2003.

Title

uniformly integrable

Canonical name

UniformlyIntegrable

Date of creation

2013-03-22 15:22:55

Last modified on

2013-03-22 15:22:55

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

23

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 28A20

Synonym

uniform integrability

Synonym

uniform absolute continuity

Related topic

VitaliConvergenceTheorem

Related topic

ConditionalExpectationsAreUniformlyIntegrable

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