properties of the Lebesgue integral of Lebesgue integrable functions

properties of the Lebesgue integral of Lebesgue integrable functions

Theorem

.

Let

(

X

,

B

,

μ

)

be a

measure space

,

f

:

X

→

[

-

∞

,

∞

]

and

g

:

X

→

[

-

∞

,

∞

]

be

Lebesgue integrable

functions

, and

A

,

B

∈

B

. Then the following properties hold:

1.

|

∫

A

f

⁢

𝑑

μ

|

≤

∫

A

|

f

|

⁢

𝑑

μ

2.

If

f

≤

g

, then

∫

A

f

⁢

𝑑

μ

≤

∫

A

g

⁢

𝑑

μ

.

3.

∫

A

f

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

, where

χ

A

denotes the

characteristic function

of

A

4.

If

c

∈

ℝ

, then

∫

A

c

⁢

f

⁢

𝑑

μ

=

c

⁢

∫

A

f

⁢

𝑑

μ

.

5.

If

μ

⁢

(

A

)

=

0

, then

∫

A

f

⁢

𝑑

μ

=

0

.

6.

∫

A

(

f

+

g

)

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

A

g

⁢

𝑑

μ

.

7.

If

A

∩

B

=

∅

, then

∫

A

∪

B

f

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

B

f

⁢

𝑑

μ

.

8.

If

f

=

g

almost everywhere with respect to

μ

, then

∫

A

f

⁢

𝑑

μ

=

∫

A

g

⁢

𝑑

μ

.

Proof.

1.

|

∫

A

f

⁢

𝑑

μ

|

=

|

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

|

by definition

≤

|

∫

A

f

+

⁢

𝑑

μ

|

+

|

∫

A

f

-

⁢

𝑑

μ

|

by the

triangle inequality

=

∫

A

f

+

⁢

𝑑

μ

+

∫

A

f

-

⁢

𝑑

μ

by the

properties of the

Lebesgue integral

of nonnegative

measurable functions

(property 1),

=

∫

A

(

f

+

+

f

-

)

⁢

𝑑

μ

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 7),

=

∫

A

|

f

|

⁢

𝑑

μ

2.

Since

f

≤

g

, the following must hold:

–

f

+

=

max

⁡

{

0

,

f

}

≤

max

⁡

{

0

,

g

}

=

g

+

;

–

-

f

≥

-

g

;

–

f

-

=

max

⁡

{

0

,

-

f

}

≥

max

⁡

{

0

,

-

g

}

=

g

-

.

Thus, by the properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 2),

∫

A

f

+

⁢

𝑑

μ

≤

∫

A

g

+

⁢

𝑑

μ

and

∫

A

f

-

⁢

𝑑

μ

≥

∫

A

g

-

⁢

𝑑

μ

. Therefore,

-

∫

A

f

-

⁢

𝑑

μ

≤

-

∫

A

g

-

⁢

𝑑

μ

. Hence,

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

≤

∫

A

g

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

≤

∫

A

g

+

⁢

𝑑

μ

-

∫

A

g

-

⁢

𝑑

μ

. It follows that

∫

A

f

⁢

𝑑

μ

≤

∫

A

g

⁢

𝑑

μ

.

3.

∫

A

f

⁢

𝑑

μ

=

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

by definition

=

∫

X

χ

A

⁢

f

+

⁢

𝑑

μ

-

∫

X

χ

A

⁢

f

-

⁢

𝑑

μ

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 3),

=

∫

X

(

χ

A

⁢

f

)

+

⁢

𝑑

μ

-

∫

X

(

χ

A

⁢

f

)

-

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

by definition

4.

If

c

≥

0

, then

∫

A

c

⁢

f

⁢

𝑑

μ

=

∫

A

(

c

⁢

f

)

+

⁢

𝑑

μ

-

∫

A

(

c

⁢

f

)

-

⁢

𝑑

μ

by definition

=

∫

A

c

⁢

f

+

⁢

𝑑

μ

-

∫

A

c

⁢

f

-

⁢

𝑑

μ

=

c

⁢

∫

A

f

+

⁢

𝑑

μ

-

c

⁢

∫

A

f

-

⁢

𝑑

μ

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 5)

=

c

⁢

(

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

)

=

c

⁢

∫

A

f

⁢

𝑑

μ

by definition.

If

c

<

0

, then

∫

A

c

⁢

f

⁢

𝑑

μ

=

∫

A

(

c

⁢

f

)

+

⁢

𝑑

μ

-

∫

A

(

c

⁢

f

)

-

⁢

𝑑

μ

by definition

=

∫

A

(

-

c

)

⁢

f

-

⁢

𝑑

μ

-

∫

A

(

-

c

)

⁢

f

+

⁢

𝑑

μ

=

-

c

⁢

∫

A

f

-

⁢

𝑑

μ

+

c

⁢

∫

A

f

+

⁢

𝑑

μ

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 5)

=

c

⁢

(

-

∫

A

f

-

⁢

𝑑

μ

+

∫

A

f

+

⁢

𝑑

μ

)

=

c

⁢

∫

A

f

⁢

𝑑

μ

by definition.

5.

Note that

∫

A

f

+

⁢

𝑑

μ

=

0

and

∫

A

f

-

⁢

𝑑

μ

=

0

by the properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 6). It follows that

∫

A

f

⁢

𝑑

μ

=

0

.

6.

Let

{

s

n

}

be a nondecreasing

sequence

of nonnegative

simple functions

converging

pointwise

to

f

+

+

g

+

and

{

t

n

}

be a nondecreasing sequence of nonnegative simple functions converging pointwise to

f

-

+

g

-

. Note that, for every

n

,

∫

A

s

n

⁢

𝑑

μ

-

∫

A

t

n

⁢

𝑑

μ

=

∫

A

(

s

n

-

t

n

)

⁢

𝑑

μ

.

Since

f

and

g

are

integrable

and

|

f

+

g

|

≤

|

f

|

+

|

g

|

,

f

+

g

is integrable. Thus,

∫

A

f

⁢

𝑑

μ

+

∫

A

g

⁢

𝑑

μ

=

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

+

∫

A

g

+

⁢

𝑑

μ

-

∫

A

g

-

⁢

𝑑

μ

by definition

=

∫

A

f

+

⁢

𝑑

μ

+

∫

A

g

+

⁢

𝑑

μ

-

(

∫

A

f

-

⁢

𝑑

μ

+

∫

A

g

-

⁢

𝑑

μ

)

=

∫

A

(

f

+

+

g

+

)

⁢

𝑑

μ

-

(

∫

A

(

f

-

+

g

-

)

⁢

𝑑

μ

)

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 7)

=

lim

n

→

∞

⁡

∫

A

s

n

⁢

𝑑

μ

-

(

lim

n

→

∞

⁡

∫

A

t

n

⁢

𝑑

μ

)

by Lebesgue’s

monotone convergence theorem

=

lim

n

→

∞

⁡

(

∫

A

s

n

⁢

𝑑

μ

-

∫

A

t

n

⁢

𝑑

μ

)

=

lim

n

→

∞

⁡

∫

A

(

s

n

-

t

n

)

⁢

𝑑

μ

=

∫

A

(

f

+

+

g

+

-

(

f

-

+

g

-

)

)

⁢

𝑑

μ

by Lebesgue’s

dominated convergence theorem

=

∫

A

(

f

+

-

f

-

+

g

+

-

g

-

)

⁢

𝑑

μ

=

∫

A

(

f

+

g

)

⁢

𝑑

μ

by definition.

7.

∫

A

∪

B

f

⁢

𝑑

μ

=

∫

A

∪

B

f

+

⁢

𝑑

μ

-

∫

A

∪

B

f

-

⁢

𝑑

μ

by definition

=

∫

A

f

+

⁢

𝑑

μ

+

∫

B

f

+

⁢

𝑑

μ

-

(

∫

A

f

-

⁢

𝑑

μ

+

∫

B

f

-

⁢

𝑑

μ

)

by the

properties of the Lebesgue integral of nonnegative measurable functions (

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

) (property 8),

=

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

+

∫

B

f

+

⁢

𝑑

μ

-

∫

B

f

-

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

B

f

⁢

𝑑

μ

by definition

8.

Let

E

=

{

x

∈

A

:

f

⁢

(

x

)

=

g

⁢

(

x

)

}

. Since

f

and

g

are measurable functions and

A

∈

𝔅

, it must be the case that

E

∈

𝔅

. Thus,

A

-

E

∈

𝔅

. By

hypothesis

,

μ

⁢

(

A

∖

E

)

=

0

. Note that

E

∩

(

A

∖

E

)

=

∅

and

E

∪

(

A

∖

E

)

=

A

. Thus,

∫

A

f

⁢

𝑑

μ

=

∫

E

f

⁢

𝑑

μ

+

∫

A

∖

E

f

⁢

𝑑

μ

=

∫

E

f

⁢

𝑑

μ

+

0

=

∫

E

g

⁢

𝑑

μ

+

0

=

∫

E

g

⁢

𝑑

μ

+

∫

A

∖

E

g

⁢

𝑑

μ

=

∫

A

g

⁢

𝑑

μ

.

∎

Title

properties of the Lebesgue integral of Lebesgue integrable functions

Canonical name

PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions

Date of creation

2013-03-22 16:14:01

Last modified on

2013-03-22 16:14:01

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

19

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A42

Classification

msc 28A25

Related topic

PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

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