Lebesgue integral

Lebesgue integral

The

integral

of a

measurable function

f

:

X

→

ℝ

∪

{

±

∞

}

on a

measure space

(

X

,

𝔅

,

μ

)

is usually written

∫

X

f

⁢

𝑑

μ

,

(1)

although alternative notations such as

∫

X

f

⁢

𝑑

x

or even

∫

f

are commonplace.

It is defined via the following steps:

•

If

f

=

𝟏

A

is the

characteristic function

of a set

A

∈

𝔅

, then set

∫

X

𝟏

A

⁢

𝑑

μ

:=

μ

⁢

(

A

)

.

(2)

•

If

f

is a

simple function

(i.e. if

f

can be written as

f

=

∑

k

=

1

n

c

k

⁢

𝟏

A

k

,

c

k

∈

ℝ

(3)

for some finite

collection

A

k

∈

𝔅

), then define

∫

X

f

⁢

𝑑

μ

:=

∑

k

=

1

n

c

k

⁢

∫

X

𝟏

A

k

⁢

𝑑

μ

=

∑

k

=

1

n

c

k

⁢

μ

⁢

(

A

k

)

.

(4)

•

If

f

is a nonnegative measurable function (possibly attaining the value

∞

at some points), then we define

∫

X

f

⁢

𝑑

μ

:=

sup

⁡

{

∫

X

h

⁢

𝑑

μ

:

h

⁢

is simple and

⁢

h

⁢

(

x

)

≤

f

⁢

(

x

)

⁢

for all

⁢

x

∈

X

}

.

(5)

•

For any measurable function

f

(possibly attaining the values

∞

or

-

∞

at some points), write

f

=

f

+

-

f

-

where

f

+

:=

max

⁡

(

f

,

0

)

and

f

-

:=

max

⁡

(

-

f

,

0

)

,

(6)

so that

|

f

|

=

f

+

+

f

-

, and define the integral of

f

as

∫

X

f

⁢

𝑑

μ

:=

∫

X

f

+

⁢

𝑑

μ

-

∫

X

f

-

⁢

𝑑

μ

,

(7)

provided that

∫

X

f

+

⁢

𝑑

μ

and

∫

X

f

-

⁢

𝑑

μ

are not both

∞

.

If

μ

is

Lebesgue measure

and

X

is any

interval

in

ℝ

n

then the integral is called the

Lebesgue integral

. If the Lebesgue integral of a

function

f

on a set

X

exists and is finite (or, equivalently, if

∫

X

|

f

|

⁢

𝑑

μ

<

∞

), then

f

is said to be

Lebesgue integrable

. The Lebesgue integral equals the

Riemann integral

everywhere the latter is defined; the advantage to the Lebesgue integral is that it is often well defined even when the corresponding Riemann integral is undefined. For example, the Riemann integral of the characteristic function of the

rationals

in

[

0

,

1

]

is undefined, while the Lebesgue integral of this function is simply the

measure

of the rationals in

[

0

,

1

]

, which is 0. Moreover, the conditions under which Lebesgue integrals may be exchanged with each other or with limits or

derivatives

, etc., are far less stringent, making the Lebesgue theory a more convenient tool than the Riemann integral for theoretical purposes.

The

introduction

of the Lebesgue integral was a major advancement in

real analysis

, soon awakening a large interest in the scientific community. In 1916 Edward Burr Van Vleck, in ”Bulletin of the

American Mathematical Society

”, vol. 23, wrote: ”This new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.”

Title

Lebesgue integral

Canonical name

LebesgueIntegral

Date of creation

2013-03-22 12:18:54

Last modified on

2013-03-22 12:18:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

28

Author

djao (24)

Entry type

Definition

Classification

msc 26A42

Classification

msc 28A25

Related topic

Measure

Related topic

LebesgueMeasure

Related topic

RiemannIntegral

Related topic

RiemannMultipleIntegral

Related topic

SimpleFunction

Defines

simple function

Defines

Lebesgue integrable

Defines

integral

Generated on Fri Feb 9 19:53:37 2018 by

LaTeXML