functional monotone class theorem

functional monotone class theorem

The

monotone class theorem

is a result in

measure theory

which allows statements about particularly simple classes of

functions

to be generalized to arbitrary

measurable

and bounded functions.

Theorem 1

.

Let

(

X

,

A

)

be a

measurable space

and

S

be a

π

-system (

http://planetmath.org/PiSystem

) generating the

σ

-

algebra

(

http://planetmath.org/SigmaAlgebra

)

A

.
Suppose that

H

be a

vector space

of real-valued functions on

X

containing the

constant functions

and satisfying the following,

•

if

f

:

X

→

ℝ

is

bounded

and there is a

sequence

of nonnegative functions

f

n

∈

ℋ

increasing

pointwise

to

f

, then

f

∈

ℋ

.

•

for every set

A

∈

𝒮

the

characteristic function

1

A

is in

ℋ

.

Then,

H

contains every bounded and

measurable function

from

X

to

R

.

That

ℋ

is a vector space just means that it is

closed under

taking

linear combinations

, so

λ

⁢

f

+

μ

⁢

g

∈

ℋ

whenever

f

,

g

∈

ℋ

and

λ

,

μ

∈

ℝ

.

As an example application, consider Fubini’s

theorem

(

http://planetmath.org/FubinisTheorem

), which states that for any two

finite measure spaces

(

X

,

𝒜

,

μ

)

and

(

Y

,

ℬ

,

ν

)

then we may commute the order of integration

∫

∫

f

⁢

(

x

,

y

)

⁢

𝑑

μ

⁢

(

x

)

⁢

𝑑

ν

⁢

(

y

)

=

∫

∫

f

⁢

(

x

,

y

)

⁢

𝑑

ν

⁢

(

y

)

⁢

𝑑

μ

⁢

(

x

)

.

(1)

Here,

f

:

X

×

Y

→

ℝ

is a bounded and

𝒜

⊗

ℬ

-measurable function. The space

ℋ

of functions for which this

identity

holds is easily shown to be linearly closed and, by the

monotone convergence theorem

, is closed under taking

monotone

limits of functions. Furthermore, the

π

-system of sets of the form

A

×

B

for

A

∈

𝒜

,

B

∈

ℬ

generates the

σ

-algebra

𝒜

⊗

ℬ

and

∫

∫

1

A

×

B

⁢

(

x

,

y

)

⁢

𝑑

μ

⁢

(

x

)

⁢

𝑑

ν

⁢

(

y

)

=

μ

⁢

(

A

)

⁢

ν

⁢

(

B

)

=

∫

∫

1

A

×

B

⁢

(

x

,

y

)

⁢

𝑑

ν

⁢

(

y

)

⁢

𝑑

μ

⁢

(

x

)

.

So,

1

A

×

B

∈

ℋ

and the monotone class theorem allows us to conclude that all real valued and bounded

𝒜

⊗

ℬ

-measurable functions are in

ℋ

, and equation (

1

) is satisfied.

An alternative, more general form of the monotone class theorem is as follows. The version of the monotone class theorem given above follows from this by letting

𝒦

be the characteristic functions

1

A

for

A

∈

𝒮

.

Theorem 2

.

Let

X

be a set and

K

be a

collection

of bounded and real valued functions on

X

which is closed under

multiplication

, so that

f

⁢

g

∈

K

for all

f

,

g

∈

K

. Let

A

be the

σ

-algebra on

X

generated by

K

.

Suppose that

H

is a vector space of real valued functions on

X

containing

K

and the constant functions, and satisfying the following

•

if

f

:

X

→

ℝ

is bounded and there is a sequence of nonnegative functions

f

n

∈

ℋ

increasing pointwise to

f

, then

f

∈

ℋ

.

Then,

H

contains every bounded and real valued

A

-measurable function on

X

.

Saying that

𝒜

is the

σ

-algebra generated by

𝒦

means that it is the smallest

σ

-algebra containing the sets

f

-

1

⁢

(

B

)

for

f

∈

𝒦

and

Borel subset

B

of

ℝ

.

For example, letting

𝒦

be as in theorem

2

and

μ

,

ν

be finite

measures

on

(

X

,

𝒜

)

such that

μ

⁢

(

X

)

=

ν

⁢

(

X

)

and

∫

f

⁢

𝑑

μ

=

∫

f

⁢

𝑑

ν

for all

f

∈

𝒦

, then

∫

f

⁢

𝑑

μ

=

∫

f

⁢

𝑑

ν

for all bounded and measurable real valued functions

f

. Therefore,

μ

=

ν

. In particular, a finite measure

μ

on

(

ℝ

,

ℬ

⁢

(

ℝ

)

)

is uniquely determined by its characteristic function

χ

⁢

(

x

)

≡

∫

e

i

⁢

x

⁢

y

⁢

𝑑

μ

⁢

(

y

)

for

x

∈

ℝ

.
Similarly, a finite measure

μ

on a bounded interval is uniquely determined by the

integrals

∫

x

n

⁢

𝑑

μ

⁢

(

x

)

for

n

∈

ℤ

+

.

Title

functional monotone class theorem

Canonical

name

FunctionalMonotoneClassTheorem

Date of creation

2013-03-22 18:38:36

Last modified on

2013-03-22 18:38:36

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A20

Related topic

MonotoneClassTheorem

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