monotone class theorem

monotone class theorem

Monotone Class

theoremFernando Sanz Gamiz

Theorem

.

Let

F

0

an

algebra

of subsets of

Ω

. Let

M

be the
smallest monotone class such that

F

0

⊂

M

and

σ

⁢

(

F

0

)

be the

sigma algebra

generated by

F

0

. Then

M

=

σ

⁢

(

F

0

)

.

Proof.

It is enough to prove that

ℳ

is an algebra, because an algebra
which is a monotone class is obviously a

σ

-algebra.

Let

ℳ

A

=

{

B

∈

ℳ

|

A

∩

B

,

A

∩

B

∁

⁢

and

⁢

A

∁

∩

B

∈

ℳ

}

. Then is clear that

ℳ

A

is a monotone class and, in
fact,

ℳ

A

=

ℳ

, for if

A

∈

ℱ

0

, then

ℱ

0

⊂

ℳ

A

since

ℱ

0

is a field, hence

ℳ

⊂

ℳ

A

by minimality of

ℳ

;
consequently

ℳ

=

ℳ

A

by definition of

ℳ

A

. But this shows that
for any

B

∈

ℳ

we have

A

∩

B

,

A

∩

B

∁

⁢

and

⁢

A

∁

∩

B

∈

ℳ

for any

A

∈

ℱ

0

, so that

ℱ

0

⊂

ℳ

B

and
again by minimality

ℳ

=

ℳ

B

. But what we have just proved is that

ℳ

is an algebra, for if

A

,

B

∈

ℳ

=

ℳ

A

we have showed that

A

∩

B

,

A

∩

B

∁

⁢

and

⁢

A

∁

∩

B

∈

ℳ

, and, of course,

Ω

∈

ℳ

.
∎

Remark 1

.

One of the main applications of the

Monotone Class Theorem

is that
of showing that certain property is satisfied by all sets in an

σ

-algebra, generally starting by the fact that the field
generating the

σ

-algebra

satisfies

such property and that the
sets that satisfies it constitutes a monotone class.

Example 1

.

Consider an

infinite

sequence

of

independent

random variables

{

X

n

,

n

∈

ℕ

}

. The definition of independence is

P

(

X

1

∈

A

1

,

X

2

∈

A

2

,

…

,

X

n

∈

A

n

)

=

P

(

X

1

∈

A

1

)

P

(

X

2

∈

A

2

)

⋯

P

(

X

n

∈

A

n

)

for any

Borel sets

A

1

,

A

2

,

.

.

,

A

n

and any finite

n

.
Using the Monotone Class Theorem one can show, for example, that any
event in

σ

⁢

(

X

1

,

X

2

,

…

,

X

n

)

is independent of any event in

σ

⁢

(

X

n

+

1

,

X

n

+

2

,

…

)

. For, by independence

P

(

(

X

1

,

X

2

,

…

,

X

n

)

∈

A

,

(

X

n

+

1

,

X

n

+

2

,

…

)

∈

B

)

=

P

(

(

X

1

,

X

2

,

…

,

X

n

)

∈

A

)

P

(

(

X

n

+

1

,

X

n

+

2

,

…

)

∈

B

)

when A and
B are

measurable rectangles

in

ℬ

n

and

ℬ

∞

respectively. Now it is clear that the sets A which
satisfies the above

relation

form a monotone class. So

P

(

(

X

1

,

X

2

,

…

,

X

n

)

∈

A

,

(

X

n

+

1

,

X

n

+

2

,

…

)

∈

B

)

=

P

(

(

X

1

,

X

2

,

…

,

X

n

)

∈

A

)

P

(

(

X

n

+

1

,

X

n

+

2

,

…

)

∈

B

)

for every

A

∈

σ

⁢

(

X

1

,

X

2

,

…

,

X

n

)

and any measurable
rectangle

B

∈

ℬ

∞

. A second application of the

theorem

shows finally that the above relation holds for any

A

∈

σ

⁢

(

X

1

,

X

2

,

…

,

X

n

)

and

B

∈

σ

⁢

(

X

n

+

1

,

X

n

+

2

,

…

)

Title

monotone class theorem

Canonical name

MonotoneClassTheorem

Date of creation

2013-03-22 17:07:34

Last modified on

2013-03-22 17:07:34

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

8

Author

fernsanz (8869)

Entry type

Theorem

Classification

msc 28A05

Related topic

MonotoneClass

Related topic

SigmaAlgebra

Related topic

Algebra

Related topic

FunctionalMonotoneClassTheorem

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