proof of functional monotone class theorem

proof of functional monotone class theorem

We start by proving the following version of the

monotone class theorem

.

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

.

Let

𝒟

consist of the

collection

of subsets

B

of

X

such that the characteristic function

1

B

is in

ℋ

. Then, by the conditions of the

theorem

, the constant function

1

X

is in

V

so that

X

∈

𝒟

, and

𝒮

⊆

𝒟

. For any

A

⊆

B

in

𝒟

then

1

B

∖

A

=

1

B

-

1

A

∈

ℋ

, as

ℋ

is

closed under

linear combinations

, and therefore

B

∖

A

is in

𝒟

.
If

A

n

∈

𝒟

is an increasing sequence, then

1

A

n

∈

ℋ

increases pointwise to

1

⋃

n

A

n

, which is therefore in

ℋ

, and

⋃

n

A

n

∈

𝒟

. It follows that

𝒟

is a

Dynkin system

, and Dynkin’s lemma shows that it contains the

σ

-algebra

𝒜

.

We have shown that

1

A

∈

ℋ

for every

A

∈

𝒜

. Now consider any bounded and measurable function

f

:

X

→

ℝ

taking values in a

finite set

S

⊆

ℝ

. Then,

f

=

∑

s

∈

S

s

⁢

1

f

-

1

⁢

(

{

s

}

)

is in

ℋ

.

We denote the

floor function

by

⌊

⋅

⌋

. That is,

⌊

a

⌋

is defined to be the largest integer less than or equal to the

real number

a

.
Then, for any nonnegative bounded and

measurable

f

:

X

→

ℝ

, the sequence of functions

f

n

⁢

(

x

)

=

2

-

n

⁢

⌊

2

n

⁢

f

⁢

(

x

)

⌋

each take values in a finite set, so are in

ℋ

, and increase pointwise to

f

. So,

f

∈

ℋ

.

Finally, as every measurable and bounded function

f

:

X

→

ℝ

can be written as the

difference

of its positive and negative parts

f

=

f

+

-

f

-

, then

f

∈

ℋ

.

We now extend this result to prove the following more general form of the theorem.

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 bounded 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

.

Let us start by showing that

ℋ

is closed under uniform convergence. That is, if

f

n

is a sequence in

ℋ

and

∥

f

n

-

f

∥

≡

sup

x

⁡

|

f

n

⁢

(

x

)

-

f

⁢

(

x

)

|

converges

to zero, then

f

∈

ℋ

. By passing to a subsequence if necessary, we may assume that

∥

f

n

-

f

m

∥

≤

2

-

n

for all

m

≥

n

. Define

g

n

≡

f

n

-

2

1

-

n

+

2

+

∥

f

∥

. Then

g

n

∈

ℋ

since

ℋ

is a vector space containing the constant functions. Also,

g

n

are nonnegative functions increasing pointwise to

f

+

2

+

∥

f

∥

which must therefore be in

ℋ

, showing that

f

∈

ℋ

as required.

Now let

ℋ

0

consist of linear combinations of constant functions and functions in

𝒦

and

ℋ

¯

0

be its

closure

(

http://planetmath.org/Closure

) under uniform convergence. Then

ℋ

¯

0

⊆

ℋ

since we have just shown that

ℋ

is closed under uniform convergence.
As

𝒦

is already closed under

products

,

ℋ

0

and

ℋ

¯

0

will also be closed under products, so are algebras (

http://planetmath.org/Algebra

). In particular,

p

⁢

(

f

)

∈

ℋ

¯

0

for every

f

∈

ℋ

¯

0

and

polynomial

p

∈

ℝ

⁢

[

X

]

.
Then, for any

continuous function

p

:

ℝ

→

ℝ

, the

Weierstrass approximation theorem

says that there is a sequence of polynomials

p

n

converging uniformly to

p

on

bounded intervals

, so

p

n

⁢

(

f

)

→

p

⁢

(

f

)

uniformly. It follows that

p

⁢

(

f

)

∈

ℋ

¯

0

. In particular, the

minimum

of any two functions

f

,

g

∈

ℋ

¯

0

,

f

∧

g

=

f

-

|

f

-

g

|

and the

maximum

f

∨

g

=

f

+

|

g

-

f

|

will be in

ℋ

¯

0

.

We let

𝒮

consist of the sets

A

⊆

X

such that there is a sequence of nonnegative

f

n

∈

ℋ

¯

0

increasing pointwise to

1

A

. Once it is shown that this is a

π

-system generating the

σ

-algebra

𝒜

, then the result will follow from theorem

1

.

If

f

n

,

g

n

∈

ℋ

¯

0

are nonnegative functions increasing pointwise to

1

A

,

1

B

then

f

n

⁢

g

n

increases pointwise to

1

A

∩

B

, so

A

∩

B

∈

𝒮

and

𝒮

is a

π

-system.

Finally, choose any

f

∈

𝒦

and

a

∈

ℝ

. Then,

f

n

=

(

(

n

⁢

(

f

-

a

)

)

∨

0

)

∧

1

is a sequence of functions in

ℋ

¯

0

increasing pointwise to

1

f

-

1

⁢

(

(

a

,

∞

)

)

. So,

f

-

1

⁢

(

(

a

,

∞

)

)

∈

𝒮

. As

intervals

of the form

(

a

,

∞

)

generate

the Borel

σ

-algebra on

ℝ

, it follows that

𝒮

generates the

σ

-algebra

𝒜

, as required.

Title

proof of functional monotone class theorem

Canonical name

ProofOfFunctionalMonotoneClassTheorem

Date of creation

2013-03-22 18:38:44

Last modified on

2013-03-22 18:38:44

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 28A20

Generated on Fri Feb 9 20:08:21 2018 by

LaTeXML