measure on a Boolean algebra

measure on a Boolean algebra

Let

A

be a

Boolean algebra

. A

measure

on

A

is a non-negative extended real-valued function

m

defined on

A

such that

1.

there is an

a

∈

A

such that

m

⁢

(

a

)

is a

real number

(not

∞

),

2.

if

a

∧

b

=

0

, then

m

⁢

(

a

∨

b

)

=

m

⁢

(

a

)

+

m

⁢

(

b

)

.

For example, a

sigma algebra

ℬ

over a set

E

is a Boolean algebra, and a measure (

http://planetmath.org/Measure

)

μ

on the

measurable space

(

ℬ

,

E

)

is a measure on the Boolean algebra

ℬ

.

The following are some of the

elementary

properties

of

m

:

•

m

⁢

(

0

)

=

0

.

By condition 1, suppose

m

⁢

(

a

)

=

r

∈

ℝ

, then

m

⁢

(

a

)

=

m

⁢

(

0

∨

a

)

=

m

⁢

(

0

)

+

m

⁢

(

a

)

, so that

m

⁢

(

0

)

=

0

.

•

m

is non-decreasing:

m

⁢

(

a

)

≤

m

⁢

(

b

)

for

a

≤

b

If

a

≤

b

, then

c

=

b

-

a

and

a

are

disjoint

(

c

∧

a

=

0

) and

b

=

c

∨

a

. So

m

⁢

(

b

)

=

m

⁢

(

c

∨

a

)

=

m

⁢

(

c

)

+

m

⁢

(

a

)

. As a result,

m

⁢

(

a

)

≤

m

⁢

(

b

)

.

•

m

is subadditive:

m

⁢

(

a

∨

b

)

≤

m

⁢

(

a

)

+

m

⁢

(

b

)

.

Since

a

∨

b

=

(

a

-

b

)

∨

b

, and

a

-

b

and

b

are disjoint, we have that

m

⁢

(

a

∨

b

)

=

m

⁢

(

(

a

-

b

)

∨

b

)

=

m

⁢

(

a

-

b

)

+

m

⁢

(

b

)

. Since

a

-

b

≤

a

, the result follows.

From the three properties above, one readily deduces that

I

:=

{

a

∈

A

∣

m

⁢

(

a

)

=

0

}

is a

Boolean ideal

of

A

.

A measure on

A

is called a

two-valued measure

if

m

maps onto the two-element set

{

0

,

1

}

. Because of the

existence

of an

element

a

∈

A

with

m

⁢

(

a

)

=

1

, it follows that

m

⁢

(

1

)

=

1

. Consequently, the set

F

:=

{

a

∈

A

∣

m

⁢

(

a

)

=

1

}

is a Boolean filter. In fact, because

m

is two-valued,

F

is an

ultrafilter

(and correspondingly, the set

{

a

∣

m

⁢

(

a

)

=

0

}

is a maximal ideal).

Conversely

, given an ultrafilter

F

of

A

, the

function

m

:

A

→

{

0

,

1

}

, defined by

m

⁢

(

a

)

=

1

iff

a

∈

F

, is a two-valued measure on

A

. To see this, suppose

a

∧

b

=

0

. Then at least one of them, say

a

, can not be in

F

(or else

0

=

a

∧

b

∈

F

). This means that

m

⁢

(

a

)

=

0

. If

b

∈

F

, then

a

∨

b

∈

F

, so that

m

⁢

(

a

∨

b

)

=

1

=

m

⁢

(

b

)

=

m

⁢

(

b

)

+

m

⁢

(

a

)

. On the other hand, if

b

∉

F

, then

a

′

,

b

′

∈

F

, so

a

′

∧

b

′

∈

F

, or

a

∨

b

∉

F

. This means that

m

⁢

(

a

∨

b

)

=

0

=

m

⁢

(

a

)

+

m

⁢

(

b

)

.

Remark

. A measure (on a Boolean algebra) is sometimes called

finitely additive

to emphasize the defining condition 2 above. In

addition

, this terminology is used when there is a need to contrast a

stronger

form of

additivity

:

countable additivity

. A measure is said to be

countably additive

if whenever

K

is a

countable set

of

pairwise disjoint

elements in

A

such that

⋁

K

exists, then

m

⁢

(

⋁

K

)

=

∑

a

∈

K

m

⁢

(

a

)

.

Title

measure on a Boolean algebra

Canonical

name

MeasureOnABooleanAlgebra

Date of creation

2013-03-22 17:59:16

Last modified on

2013-03-22 17:59:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B99

Related topic

Measure

Defines

measure

Defines

two-valued measure

Defines

finitely additive

Defines

countably additive

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