properties of the Lebesgue integral of nonnegative measurable functions

properties of the Lebesgue integral of nonnegative measurable functions

Theorem

.

Let

(

X

,

B

,

μ

)

be a

measure space

,

f

:

X

→

[

0

,

∞

]

and

g

:

X

→

[

0

,

∞

]

be

measurable functions

, and

A

,

B

∈

B

. Then the following properties hold:

1.

∫

A

f

⁢

𝑑

μ

≥

0

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

A

⊆

B

, then

∫

A

f

⁢

𝑑

μ

≤

∫

B

f

⁢

𝑑

μ

.

5.

If

c

≥

0

, then

∫

A

c

⁢

f

⁢

𝑑

μ

=

c

⁢

∫

A

f

⁢

𝑑

μ

.

6.

If

μ

⁢

(

A

)

=

0

, then

∫

A

f

⁢

𝑑

μ

=

0

.

7.

∫

A

(

f

+

g

)

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

A

g

⁢

𝑑

μ

8.

If

A

∩

B

=

∅

, then

∫

A

∪

B

f

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

B

f

⁢

𝑑

μ

.

9.

If

f

=

g

almost everywhere with respect to

μ

, then

∫

A

f

⁢

𝑑

μ

=

∫

A

g

⁢

𝑑

μ

.

Proof.

1.

Let

s

be a

simple function

with

0

≤

s

≤

f

. Let

s

=

∑

k

=

1

n

c

k

⁢

χ

A

k

for

c

k

∈

[

0

,

∞

]

and

A

k

∈

𝔅

. Then

∫

A

s

⁢

𝑑

μ

=

∑

k

=

1

n

c

k

⁢

μ

⁢

(

A

∩

A

k

)

≥

0

. By definition,

∫

A

f

⁢

𝑑

μ

≥

∫

A

s

⁢

𝑑

μ

. It follows that

∫

A

f

⁢

𝑑

μ

≥

0

.

2.

Let

s

be a simple function with

0

≤

s

≤

f

. Since

f

≤

g

,

0

≤

s

≤

g

. By definition,

∫

A

s

⁢

𝑑

μ

≤

∫

A

g

⁢

𝑑

μ

. Since this holds for every simple function

s

with

0

≤

s

≤

f

, it follows that

∫

A

f

⁢

𝑑

μ

≤

∫

A

g

⁢

𝑑

μ

.

3.

Let

s

be a simple function with

0

≤

s

≤

f

. Then

0

≤

χ

A

⁢

s

≤

χ

A

⁢

f

. Let

s

=

∑

k

=

1

n

c

k

⁢

χ

A

k

for

c

k

∈

[

0

,

∞

]

and

A

k

∈

𝔅

. Then

∫

A

s

⁢

𝑑

μ

=

∑

k

=

1

n

c

k

⁢

μ

⁢

(

A

∩

A

k

)

=

∫

X

∑

k

=

1

n

c

k

⁢

χ

A

∩

A

k

⁢

d

⁢

μ

=

∫

X

∑

k

=

1

n

c

k

⁢

χ

A

⁢

χ

A

k

⁢

d

⁢

μ

=

∫

X

χ

A

⁢

∑

k

=

1

n

c

k

⁢

χ

A

k

⁢

d

⁢

μ

=

∫

X

χ

A

⁢

s

⁢

𝑑

μ

≤

∫

X

χ

A

⁢

f

⁢

𝑑

μ

.

Thus,

∫

A

f

⁢

𝑑

μ

≤

∫

X

χ

A

⁢

f

⁢

𝑑

μ

.

Let

t

be a simple function with

0

≤

t

≤

χ

A

⁢

f

. Then

χ

A

⁢

t

=

t

. Thus,

∫

X

t

⁢

𝑑

μ

=

∫

X

χ

A

⁢

t

⁢

𝑑

μ

=

∫

A

t

⁢

𝑑

μ

. Therefore,

∫

X

χ

A

⁢

f

⁢

𝑑

μ

=

∫

A

χ

A

⁢

f

⁢

𝑑

μ

. Since

χ

A

⁢

f

≤

f

,

∫

A

χ

A

⁢

f

⁢

𝑑

μ

≤

∫

A

f

⁢

𝑑

μ

by property 2. Hence,

∫

A

f

⁢

𝑑

μ

≤

∫

X

χ

A

⁢

f

⁢

𝑑

μ

=

∫

A

χ

A

⁢

f

⁢

𝑑

μ

≤

∫

A

f

⁢

𝑑

μ

. It follows that

∫

A

f

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

.

4.

Since

A

⊆

B

,

χ

A

≤

χ

B

. Thus,

χ

A

⁢

f

≤

χ

B

⁢

f

. By property 2,

∫

X

χ

A

⁢

f

⁢

𝑑

μ

≤

∫

X

χ

B

⁢

f

⁢

𝑑

μ

. By property 3,

∫

A

f

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

≤

∫

X

χ

B

⁢

f

⁢

𝑑

μ

=

∫

B

f

⁢

𝑑

μ

.

5.

If

c

=

0

, then

∫

A

c

⁢

f

⁢

𝑑

μ

=

∫

A

0

⁢

𝑑

μ

=

0

=

0

⁢

∫

A

f

⁢

𝑑

μ

=

c

⁢

∫

A

f

⁢

𝑑

μ

.

If

c

>

0

, let

S

=

{

s

:

X

→

[

0

,

∞

]

∣

s

is simple and

s

≤

c

f

}

and

T

=

{

t

:

X

→

[

0

,

∞

]

∣

t

is simple and

t

≤

f

}

. Then

∫

A

c

⁢

f

⁢

𝑑

μ

=

sup

s

∈

S

⁡

∫

A

s

⁢

𝑑

μ

=

sup

s

∈

S

⁡

∫

A

c

⋅

s

c

⁢

𝑑

μ

=

c

⁢

sup

s

∈

S

⁡

∫

A

s

c

⁢

𝑑

μ

=

c

⁢

sup

t

∈

T

⁡

∫

A

t

⁢

𝑑

μ

=

c

⁢

∫

A

f

⁢

𝑑

μ

.

6.

Let

s

be a simple function with

0

≤

s

≤

f

. Let

s

=

∑

k

=

1

n

c

k

⁢

χ

A

k

for

c

k

∈

[

0

,

∞

]

and

A

k

∈

𝔅

. Then

∫

A

s

⁢

𝑑

μ

=

∑

k

=

1

n

c

k

⁢

μ

⁢

(

A

∩

A

k

)

=

∑

k

=

1

n

c

k

⋅

0

=

0

. Thus,

∫

A

f

⁢

𝑑

μ

=

0

.

7.

Let

{

s

n

}

be a nondecreasing

sequence

of nonnegative simple functions converging

pointwise

to

f

and

{

t

n

}

be a nondecreasing sequence of nonnegative simple functions converging pointwise to

g

. Then

{

s

n

+

t

n

}

is a nondecreasing sequence of nonnegative simple functions converging pointwise to

f

+

g

. Note that, for every

n

,

∫

A

(

s

n

+

t

n

)

⁢

𝑑

μ

=

∫

A

s

n

⁢

𝑑

μ

+

∫

A

t

n

⁢

𝑑

μ

. By Lebesgue’s

monotone convergence theorem

,

∫

A

(

f

+

g

)

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

A

g

⁢

𝑑

μ

.

8.

∫

A

∪

B

f

⁢

𝑑

μ

=

∫

X

χ

A

∪

B

⁢

f

⁢

𝑑

μ

=

∫

X

(

χ

A

+

χ

B

-

χ

A

∩

B

)

⁢

f

⁢

𝑑

μ

=

∫

X

(

χ

A

+

χ

B

-

χ

∅

)

⁢

f

⁢

𝑑

μ

=

∫

X

(

χ

A

+

χ

B

-

0

)

⁢

f

⁢

𝑑

μ

=

∫

X

(

χ

A

⁢

f

+

χ

B

⁢

f

)

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

+

∫

X

χ

B

⁢

f

⁢

𝑑

μ

=

∫

A

f

⁢

𝑑

μ

+

∫

B

f

⁢

𝑑

μ

9.

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 nonnegative measurable functions

Canonical name

PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions

Date of creation

2013-03-22 16:13:50

Last modified on

2013-03-22 16:13:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

22

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A42

Classification

msc 28A25

Related topic

PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions

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