properties for measure

properties for measure

Theorem

[

1

,

2

,

3

,

4

]

Let

(

E

,

ℬ

,

μ

)

be a

measure space

, i.e.,
let

E

be a set, let

ℬ

be a

σ

-

algebra of sets

in

E

, and let

μ

be a

measure

on

ℬ

.
Then the following properties hold:

1.

Monotonicity

:

If

A

,

B

∈

ℬ

, and

A

⊂

B

, then

μ

⁢

(

A

)

≤

μ

⁢

(

B

)

.

2.

If

A

,

B

in

ℬ

,

A

⊂

B

, and

μ

⁢

(

A

)

<

∞

, then

μ

⁢

(

B

∖

A

)

=

μ

⁢

(

B

)

-

μ

⁢

(

A

)

.

3.

For any

A

,

B

in

ℬ

, we have

μ

⁢

(

A

∪

B

)

+

μ

⁢

(

A

∩

B

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

)

.

4.

Subadditivity:

If

{

A

i

}

i

=

1

∞

is a

collection

of sets from

ℬ

, then

μ

⁢

(

⋃

i

=

1

∞

A

i

)

≤

∑

i

=

1

∞

μ

⁢

(

A

i

)

.

5.

Continuity

from below:

If

{

A

i

}

i

=

1

∞

is a collection of sets from

ℬ

such that

A

i

⊂

A

i

+

1

for all

i

, then

μ

⁢

(

⋃

i

=

1

∞

A

i

)

=

lim

i

→

∞

⁡

μ

⁢

(

A

i

)

.

6.

Continuity from above:

If

{

A

i

}

i

=

1

∞

is a collection of sets from

ℬ

such that

μ

⁢

(

A

1

)

<

∞

, and

A

i

⊃

A

i

+

1

for all

i

, then

μ

⁢

(

⋂

i

=

1

∞

A

i

)

=

lim

i

→

∞

⁡

μ

⁢

(

A

i

)

.

Remarks

In (2), the

assumption

μ

⁢

(

A

)

<

∞

assures
that the right hand side is always well defined, i.e., not of
the form

∞

-

∞

. Without the assumption we can prove that

μ

⁢

(

B

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

∖

A

)

(see below).
In (3), it is tempting to
move the term

μ

⁢

(

A

∩

B

)

to the other side for aesthetic reasons.
However, this is only possible if the term is finite.

Proof.

For (1), suppose

A

⊂

B

. We can then
write

B

as the

disjoint union

B

=

A

∪

(

B

∖

A

)

, whence

μ

⁢

(

B

)

=

μ

⁢

(

A

∪

(

B

∖

A

)

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

∖

A

)

.

Since

μ

⁢

(

B

∖

A

)

≥

0

, the claim follows.
Property (2) follows from the above equation; since

μ

⁢

(

A

)

<

∞

, we can subtract this quantity from both sides.
For property (3), we can write

A

∪

B

=

A

∪

(

B

∖

A

)

, whence

μ

⁢

(

A

∪

B

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

∖

A

)

≤

μ

⁢

(

A

)

+

μ

⁢

(

B

)

.

If

μ

⁢

(

A

∪

B

)

is

infinite

, the last

inequality

must
be

equality

, and either of

μ

⁢

(

A

)

or

μ

⁢

(

B

)

must be infinite.
Together with (1), we obtain that if any of the quantities

μ

⁢

(

A

)

,

μ

⁢

(

B

)

,

μ

⁢

(

A

∩

B

)

or

μ

⁢

(

A

∪

B

)

is infinite,
both sides in the equation are infinite and the claim holds.
We can therefore
without loss of generality assume that all quantities are finite.
From

A

∪

B

=

B

∪

(

A

∖

B

)

, we have

μ

⁢

(

A

∪

B

)

=

μ

⁢

(

B

)

+

μ

⁢

(

A

∖

B

)

and thus

2

⁢

μ

⁢

(

A

∪

B

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

)

+

μ

⁢

(

A

∖

B

)

+

μ

⁢

(

B

∖

A

)

.

For the last two terms we have

μ

⁢

(

A

∖

B

)

+

μ

⁢

(

B

∖

A

)

=

μ

⁢

(

(

A

∖

B

)

∪

(

B

∖

A

)

)

=

μ

⁢

(

(

A

∪

B

)

∖

(

A

∩

B

)

)

=

μ

⁢

(

A

∪

B

)

-

μ

⁢

(

A

∩

B

)

,

where, in the second equality we have used properties for the

symmetric set difference

(

http://planetmath.org/SymmetricDifference

), and
the last equality follows from
property (2). This

completes

the proof of
property (3).
For property (4), let us define the

sequence

{

D

i

}

i

=

1

∞

as

D

1

=

A

1

,

D

i

=

A

i

∖

⋃

k

=

1

i

-

1

A

k

.

Now

D

i

∩

D

j

=

∅

for

i

<

j

, so

{

D

i

}

is a sequence of

disjoint sets

.
Since

∪

i

=

1

∞

D

i

=

∪

i

=

1

∞

A

i

, and since

D

i

⊂

A

i

, we have

μ

⁢

(

⋃

i

=

1

∞

A

i

)

=

μ

⁢

(

⋃

i

=

1

∞

D

i

)

=

∑

i

=

1

∞

μ

⁢

(

D

i

)

≤

∑

i

=

1

∞

μ

⁢

(

A

i

)

,

and property (4) follows.

TODO:

proofs for (5)-(6).

References

1

G.B. Folland,

Real Analysis

: Modern Techniques and Their Applications

, 2nd ed, John Wiley & Sons, Inc., 1999.

2

A. Mukherjea, K. Pothoven,

Real and

Functional analysis

,
Plenum press, 1978.

3

D.L. Cohn,

Measure Theory

, Birkhäuser, 1980.

4

A. Friedman,

Foundations

of Modern

Analysis

,
Dover publications, 1982.

Title

properties for measure

Canonical name

PropertiesForMeasure

Date of creation

2013-03-22 13:45:28

Last modified on

2013-03-22 13:45:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Theorem

Classification

msc 60A10

Classification

msc 28A10

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