Riemann multiple integral

Riemann multiple integral

We are going to extend the

concept

of

Riemann integral

to

functions

of several variables.

Let

f

:

ℝ

n

→

ℝ

be a bounded function with compact support.
Recalling the definitions of

polyrectangle

and the definitions of upper and

lower Riemann sums

on polyrectangles,
we define

S

*

⁢

(

f

)

:=

inf

⁡

{

S

*

⁢

(

f

,

P

)

:

P

is a polyrectangle,

f

⁢

(

x

)

=

0

for every

x

∈

ℝ

n

∖

∪

P

}

,

S

*

⁢

(

f

)

:=

sup

⁡

{

S

*

⁢

(

f

,

P

)

:

P

is a polyrectangle,

f

⁢

(

x

)

=

0

for every

x

∈

ℝ

n

∖

∪

P

}

.

If

S

*

⁢

(

f

)

=

S

*

⁢

(

f

)

we say that

f

is

Riemann-integrable

on

ℝ

n

and we define the Riemann integral of

f

:

∫

f

⁢

(

x

)

⁢

𝑑

x

:=

S

*

⁢

(

f

)

=

S

*

⁢

(

f

)

.

Clearly one has

S

*

⁢

(

f

,

P

)

≥

S

*

⁢

(

f

,

P

)

. Also one has

S

*

⁢

(

f

,

P

)

≥

S

*

⁢

(

f

,

P

′

)

when

P

and

P

′

are any two polyrectangles containing the

support

of

f

. In fact one can always find a common refinement

P

′′

of both

P

and

P

′

so that

S

*

⁢

(

f

,

P

)

≥

S

*

⁢

(

f

,

P

′′

)

≥

S

*

⁢

(

f

,

P

′′

)

≥

S

*

⁢

(

f

,

P

′

)

. So, to prove that a function is Riemann-integrable it is enough to prove that for every

ϵ

>

0

there exists a polyrectangle

P

such that

S

*

⁢

(

f

,

P

)

-

S

*

⁢

(

f

,

P

)

<

ϵ

.

Next we are going to define the

integral

on more general domains. As a byproduct we also define the

measure

of sets in

ℝ

n

.

Let

D

⊂

ℝ

n

be a bounded set. We say that

D

is

Riemann

measurable

if
the

characteristic function

χ

D

⁢

(

x

)

:=

{

1

if

x

∈

D

0

otherwise

is Riemann measurable on

ℝ

n

(as defined above). Moreover we define the

Peano-Jordan measure

of

D

as

𝐦𝐞𝐚𝐬

⁢

(

D

)

:=

∫

χ

D

⁢

(

x

)

⁢

𝑑

x

.

When

n

=

3

the Peano Jordan measure of

D

is called the

volume

of

D

,
and when

n

=

2

the Peano Jordan measure of

D

is called the

area

of

D

.

Let now

D

⊂

ℝ

n

be a Riemann

measurable set

and let

f

:

D

→

ℝ

be a bounded function. We say that

f

is

Riemann measurable

if the function

f

¯

:

ℝ

n

→

ℝ

f

¯

⁢

(

x

)

:=

{

f

⁢

(

x

)

if

x

∈

D

0

otherwise

is Riemann integrable as defined before. In this case we denote with

∫

D

f

⁢

(

x

)

⁢

𝑑

x

:=

∫

f

¯

⁢

(

x

)

⁢

𝑑

x

the

Riemann integral

of

f

on

D

.

Title

Riemann multiple integral

Canonical name

RiemannMultipleIntegral

Date of creation

2013-03-22 15:03:34

Last modified on

2013-03-22 15:03:34

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

14

Author

paolini (1187)

Entry type

Definition

Classification

msc 26A42

Related topic

Polyrectangle

Related topic

RiemannIntegral

Related topic

Integral2

Related topic

AreaOfPlaneRegion

Related topic

DevelopableSurface

Related topic

VolumeAsIntegral

Related topic

AreaOfPolygon

Related topic

MoscowMathematicalPapyrus

Related topic

IntegralOverPlaneRegion

Defines

Riemann integrable

Defines

Peano Jordan

Defines

measurable

Defines

area

Defines

volume

Defines

Jordan content

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