preservation of uniform integrability

preservation of uniform integrability

Let

(

Ω

,

ℱ

,

ℙ

)

be a

measure space

. Then, a

uniformly integrable

set

S

of

measurable functions

f

:

Ω

→

ℝ

will remain uniformly integrable if it is enlarged by various

operations

, such as taking convex combinations and

conditional expectations

. The following

theorem

lists some of the operations which preserve uniform integrability.

Theorem

.

Suppose that

S

is a bounded and uniformly integrable subset of

L

1

. Let

S

′

be the smallest set containing

S

such that all of the following conditions are satisfied. Then,

S

′

is also a bounded and uniformly integrable subset of

L

1

.

1.

S

′

is absolutely convex. That is, if

f

,

g

∈

S

′

and

a

,

b

∈

ℝ

are such that

|

a

|

+

|

b

|

≤

1

then

a

⁢

f

+

b

⁢

g

∈

S

′

.

2.

If

f

∈

S

′

and

|

g

|

≤

|

f

|

then

g

∈

S

′

.

3.

S

′

is closed under

convergence in measure

. That is, if

f

n

∈

S

′

converge in measure to

f

, then

f

∈

S

′

.

4.

If

f

∈

S

′

and

𝒢

is a sub-

σ

-algebra of

ℱ

such that

μ

|

𝒢

is

σ

-finite, then the conditional expectation

𝔼

μ

[

f

∣

𝒢

]

is in

S

′

.

To prove this we use the condition that the set

S

is uniformly integrable if and only if there is a convex and symmetric function

Φ

:

ℝ

→

[

0

,

∞

)

such that

Φ

⁢

(

x

)

/

|

x

|

→

∞

as

|

x

|

→

∞

and

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

is bounded over all

f

∈

S

(see

equivalent conditions for uniform integrability

). Suppose that it is bounded by

K

>

0

. Also, by replacing

Φ

by

Φ

⁢

(

x

)

+

|

x

|

if necessary, we may suppose that

Φ

⁢

(

x

)

≥

|

x

|

. Then, let

S

¯

be

S

¯

=

{

f

∈

L

1

:

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

≤

K

}

,

which is a bounded and uniformly integrable subset of

L

1

containing

S

. To prove the result, it just needs to be shown that

S

¯

is closed under each of the operations listed above, as that will imply

S

′

⊆

S

¯

.

First, the convexity and

symmetry

of

Φ

gives

∫

Φ

⁢

(

a

⁢

f

+

b

⁢

g

)

⁢

𝑑

μ

≤

∫

(

|

a

|

⁢

Φ

⁢

(

f

)

+

|

b

|

⁢

Φ

⁢

(

g

)

)

⁢

𝑑

μ

=

|

a

|

⁢

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

+

|

b

|

⁢

∫

Φ

⁢

(

g

)

⁢

𝑑

μ

≤

K

for any

f

,

g

∈

S

¯

and

a

,

b

∈

ℝ

with

|

a

|

+

|

b

|

≤

1

. So,

a

⁢

f

+

b

⁢

g

∈

S

¯

.
Similarly, if

|

g

|

≤

|

f

|

and

f

∈

S

¯

then

Φ

⁢

(

g

)

≤

Φ

⁢

(

f

)

and,

g

∈

S

¯

.

Now suppose that

f

n

∈

S

¯

converge in measure to

f

. Then Fatou’s lemma gives,

∫

Φ

⁢

(

f

)

⁢

𝑑

μ

≤

lim inf

n

→

∞

⁡

∫

Φ

⁢

(

f

n

)

⁢

𝑑

μ

≤

K

so,

f

∈

S

¯

.

Finally suppose that

f

∈

S

¯

and

g

=

𝔼

μ

[

f

∣

𝒢

]

. Using Jensen’s inequality,

∫

Φ

(

g

)

d

μ

≤

∫

𝔼

μ

[

Φ

(

f

)

∣

𝒢

]

d

μ

=

∫

Φ

(

f

)

d

μ

≤

K

,

so

g

∈

S

¯

.

Title

preservation of uniform integrability

Canonical name

PreservationOfUniformIntegrability

Date of creation

2013-03-22 18:40:20

Last modified on

2013-03-22 18:40:20

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A20

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