aarjaneiro/planetmath huggingface dataset:
Lebesgue integral over a subset of the measure space

Lebesgue integral over a subset of the measure space

Let

(

X

,

𝔅

,

μ

)

be a

measure space

and

A

∈

𝔅

.

Let

s

:

X

→

[

0

,

∞

]

be a

simple function

. Then

∫

A

s

⁢

𝑑

μ

is defined as

∫

A

s

⁢

𝑑

μ

:=

∫

X

χ

A

⁢

s

⁢

𝑑

μ

, where

χ

A

denotes the

characteristic function

of

A

.

Let

f

:

X

→

[

0

,

∞

]

be a

measurable function

and

S

=

{

s

:

X

→

[

0

,

∞

]

|

s

is a simple function and

s

≤

f

}

. Then

∫

A

f

⁢

𝑑

μ

is defined as

∫

A

f

⁢

𝑑

μ

:=

sup

s

∈

S

⁡

∫

A

s

⁢

𝑑

μ

.

By the properties of the

Lebesgue integral

of nonnegative measurable functions (property 3), we have that

∫

A

f

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

.

Let

f

:

X

→

[

-

∞

,

∞

]

be a measurable function such that not both of

∫

A

f

+

⁢

𝑑

μ

and

∫

A

f

-

⁢

𝑑

μ

are

infinite

. (Note that

f

+

and

f

-

are defined in the entry Lebesgue integral.) Then

∫

A

f

⁢

𝑑

μ

is defined as

∫

A

f

⁢

𝑑

μ

:=

∫

A

f

+

⁢

𝑑

μ

-

∫

A

f

-

⁢

𝑑

μ

.

By the properties of the Lebesgue integral of

Lebesgue integrable

functions

(property 3), we have that

∫

A

f

⁢

𝑑

μ

=

∫

X

χ

A

⁢

f

⁢

𝑑

μ

.

Title

Lebesgue integral over a subset of the measure space

Canonical name

LebesgueIntegralOverASubsetOfTheMeasureSpace

Date of creation

2013-03-22 16:13:54

Last modified on

2013-03-22 16:13:54

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 26A42

Classification

msc 28A25

Generated on Fri Feb 9 19:53:39 2018 by

LaTeXML
monotone convergence theorem

monotone convergence theorem

Let

X

be a

measure space

, and let

0

≤

f

1

≤

f

2

≤

⋯

be a

monotone increasing

sequence

of nonnegative

measurable functions

. Let

f

:

X

→

ℝ

∪

{

∞

}

be the

function

defined by

f

⁢

(

x

)

=

lim

n

→

∞

⁡

f

n

⁢

(

x

)

.
Then

f

is

measurable

, and

lim

n

→

∞

⁡

∫

X

f

n

=

∫

X

f

.

Remark.

This

theorem

is the first of several theorems which allow us to “exchange integration and limits”. It requires the use of the

Lebesgue integral

: with the

Riemann integral

, we cannot even formulate the theorem, lacking, as we do, the

concept

of “almost everywhere”. For

instance

, the

characteristic function

of the

rational numbers

in

[

0

,

1

]

is not Riemann integrable, despite being the limit of an

increasing sequence

of Riemann integrable functions.

Title

monotone convergence theorem

Canonical name

MonotoneConvergenceTheorem

Date of creation

2013-03-22 12:47:27

Last modified on

2013-03-22 12:47:27

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Theorem

Classification

msc 26A42

Classification

msc 28A20

Synonym

Lebesgue’s monotone convergence theorem

Synonym

Beppo Levi’s theorem

Related topic

DominatedConvergenceTheorem

Related topic

FatousLemma

Generated on Fri Feb 9 19:53:42 2018 by

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polyrectangle

polyrectangle

A

polyrectangle

P

in

ℝ

n

is a finite

collection

P

=

{

R

1

,

…

,

R

N

}

of compact rectangles

R

i

⊂

ℝ

n

with

disjoint

interior

.
A

compact rectangle

R

i

is a

Cartesian product

of

compact

intervals

:

R

i

=

[

a

1

i

,

b

1

i

]

×

⋯

×

[

a

n

i

,

b

n

i

]

where

a

j

i

<

b

j

i

(these are also called

n

-dimensional intervals

).

The union of the compact rectangles of a polyrectangle

P

is denoted by

∪

P

:=

⋃

R

∈

P

R

=

R

1

∪

⋯

∪

R

N

.

It is a compact subset of

ℝ

n

.

We can define the (

n

-dimensional) measure of

∪

P

in a way.
If

R

=

[

a

1

,

b

1

]

×

⋯

×

[

a

n

,

b

n

]

is a

rectangle

we define the measure of

R

as

meas

⁢

(

R

)

:=

(

b

1

-

a

1

)

⁢

⋯

⁢

(

b

n

-

a

n

)

and define the measure of the polyrectangle

P

as:

meas

⁢

(

P

)

:=

∑

R

∈

P

meas

⁢

(

R

)

.

Moreover if we are given a

bounded function

f

:

∪

P

→

ℝ

we can define the

upper

and

lower Riemann sums

of

f

over

∪

P

by

S

*

⁢

(

f

,

P

)

:=

∑

R

∈

P

meas

⁢

(

R

)

⁢

sup

x

∈

R

⁡

f

⁢

(

x

)

,

S

*

⁢

(

f

,

P

)

:=

∑

R

∈

P

meas

⁢

(

R

)

⁢

inf

x

∈

R

⁡

f

⁢

(

x

)

.

Polyrectangles are then used to define the

Peano Jordan

measure of subsets of

ℝ

n

and to define Riemann multiple integrals.
To achieve this, it is useful to introduce the so called

refinements

. The family of rectangles

R

i

which appear in the definition

LABEL:defpoly

are called a

partition

of

∪

P

¯

in rectangles.
It is clear that the set

∪

P

can be represented by different polyrectangles. For example any rectangle

R

can be split in

2

n

smaller rectangles by dividing in two parts each of the

n

intervals defining

R

.
We claim that given two polyrectangles

P

and

Q

there exists a polyrectangle

S

such that

(

∪

P

)

∪

(

∪

Q

)

⊂

∪

S

and such that given any rectangle

R

in

P

or

Q

,

R

is the union of rectangles in

S

.

Title

polyrectangle

Canonical name

Polyrectangle

Date of creation

2013-03-22 15:03:31

Last modified on

2013-03-22 15:03:31

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

23

Author

paolini (1187)

Entry type

Definition

Classification

msc 26A42

Related topic

RiemannMultipleIntegral

Defines

Riemann sums on polyrectangles

Defines

compact rectangle

Generated on Fri Feb 9 19:53:44 2018 by

LaTeXML
proof of continuous functions are Riemann integrable

proof of continuous functions are Riemann integrable

Recall the

definition

of

Riemann integral

. To prove that

f

is

integrable

we have to prove that

lim

δ

→

0

+

⁡

S

*

⁢

(

δ

)

-

S

*

⁢

(

δ

)

=

0

.
Since

S

*

⁢

(

δ

)

is

decreasing

and

S

*

⁢

(

δ

)

is increasing it is enough to show that given

ϵ

>

0

there exists

δ

>

0

such that

S

*

⁢

(

δ

)

-

S

*

⁢

(

δ

)

<

ϵ

.

So let

ϵ

>

0

be fixed.

By

Heine-Cantor Theorem

f

is

uniformly continuous

i.e.

∃

δ

>

0

⁢

|

x

-

y

|

<

δ

⇒

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

<

ϵ

b

-

a

.

Let now

P

be any

partition

of

[

a

,

b

]

in

C

⁢

(

δ

)

i.e. a partition

{

x

0

=

a

,

x

1

,

…

,

x

N

=

b

}

such that

x

i

+

1

-

x

i

<

δ

. In any small

interval

[

x

i

,

x

i

+

1

]

the

function

f

(being

continuous

) has a

maximum

M

i

and

minimum

m

i

. Since

f

is uniformly continuous and

x

i

+

1

-

x

i

<

δ

we have

M

i

-

m

i

<

ϵ

/

(

b

-

a

)

. So the

difference

between upper and

lower Riemann sums

is

∑

i

M

i

⁢

(

x

i

+

1

-

x

i

)

-

∑

i

m

i

⁢

(

x

i

+

1

-

x

i

)

≤

ϵ

b

-

a

⁢

∑

i

(

x

i

+

1

-

x

i

)

=

ϵ

.

This being true for every partition

P

in

C

⁢

(

δ

)

we conclude that

S

*

⁢

(

δ

)

-

S

*

⁢

(

δ

)

<

ϵ

.

Title

proof of continuous functions are Riemann integrable

Canonical

name

ProofOfContinuousFunctionsAreRiemannIntegrable

Date of creation

2013-03-22 13:45:34

Last modified on

2013-03-22 13:45:34

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Proof

Classification

msc 26A42

Generated on Fri Feb 9 19:53:46 2018 by

LaTeXML
properties of Riemann–Stieltjes integral

properties of Riemann–Stieltjes integral

Denote by

R

⁢

(

g

)

the set of bounded

real functions

which are

http://planetmath.org/node/3187

Riemann–Stieltjes

integrable

with respect to a given

monotonically nondecreasing

function

g

on  a given

interval

.

The

http://planetmath.org/node/3187

Riemann–

Stieltjes integral

is a generalisation of the

Riemann integral

, and both have properties; N.B. however the items 5, 7 and 9.

1.

If

f

1

,

f

2

∈

R

⁢

(

g

)

on

[

a

,

b

]

,  then also

f

1

+

f

2

,

c

⁢

f

1

∈

R

⁢

(

g

)

on

[

a

,

b

]

and

∫

a

b

(

f

1

+

f

2

)

⁢

𝑑

g

=

∫

a

b

f

1

⁢

𝑑

g

+

∫

a

b

f

2

⁢

𝑑

g

,

∫

a

b

c

⁢

f

1

⁢

𝑑

g

=

c

⁢

∫

a

b

f

1

⁢

𝑑

g

.

2.

If

f

1

,

f

2

∈

R

⁢

(

g

)

on

[

a

,

b

]

, then also

f

1

⁢

f

2

∈

R

⁢

(

g

)

on

[

a

,

b

]

.

3.

If

f

1

,

f

2

∈

R

⁢

(

g

)

on

[

a

,

b

]

and

inf

x

∈

[

a

,

b

]

⁡

|

f

2

⁢

(

x

)

|

>

0

,  then also

f

1

f

2

∈

R

⁢

(

g

)

on

[

a

,

b

]

.

4.

If

f

1

,

f

2

∈

R

⁢

(

g

)

and

f

1

≤

f

2

on

[

a

,

b

]

,  then

∫

a

b

f

1

⁢

𝑑

g

≤

∫

a

b

f

2

⁢

𝑑

g

.

5.

If

f

∈

R

⁢

(

g

)

on

[

a

,

b

]

,  and

V

g

is the

total variation

of

g

on

[

a

,

b

]

,  then

|

∫

a

b

f

⁢

𝑑

g

|

≤

sup

x

∈

[

a

,

b

]

⁡

f

⁢

(

x

)

⋅

V

g

.

6.

If

f

∈

R

⁢

(

g

)

on

[

a

,

b

]

,  then also

|

f

|

∈

R

⁢

(

g

)

on

[

a

,

b

]

and

|

∫

a

b

f

⁢

𝑑

g

|

≤

∫

a

b

|

f

|

⁢

𝑑

g

.

7.

If

f

∈

R

⁢

(

g

)

and

m

≤

f

⁢

(

x

)

≤

M

on

[

a

,

b

]

,  then

m

⁢

[

g

⁢

(

b

)

-

g

⁢

(

a

)

]

≤

∫

a

b

f

⁢

𝑑

g

≤

M

⁢

[

g

⁢

(

b

)

-

g

⁢

(

a

)

]

.

8.

If

f

∈

R

⁢

(

g

)

on

[

a

,

b

]

and on

[

b

,

c

]

,  then also

f

∈

R

⁢

(

g

)

on

[

a

,

c

]

and

∫

a

c

f

⁢

𝑑

g

=

∫

a

b

f

⁢

𝑑

g

+

∫

b

c

f

⁢

𝑑

g

.

9.

If

f

∈

R

⁢

(

g

1

)

,

R

⁢

(

g

2

)

on

[

a

,

b

]

,  then

f

∈

R

⁢

(

g

1

+

g

2

)

on the same interval and

∫

a

b

f

⁢

d

⁢

(

g

1

+

g

2

)

=

∫

a

b

f

⁢

𝑑

g

1

+

∫

a

b

f

⁢

𝑑

g

2

.

10.

If

f

∈

R

⁢

(

g

)

on

[

a

,

b

]

,  then

g

∈

R

⁢

(

f

)

on the same interval and one can integrate by parts:

∫

a

b

f

⁢

𝑑

g

=

f

⁢

(

b

)

⁢

g

⁢

(

b

)

-

f

⁢

(

a

)

⁢

g

⁢

(

a

)

-

∫

a

b

g

⁢

𝑑

f

.

Title

properties of Riemann–Stieltjes integral

Canonical name

PropertiesOfRiemannStieltjesIntegral

Date of creation

2013-03-22 18:54:59

Last modified on

2013-03-22 18:54:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A42

Synonym

properties of Riemann-Stieltjes integral

Related topic

ProductAndQuotientOfFunctionsSum

Related topic

FactsAboutRiemannStieltjesIntegral

Generated on Fri Feb 9 19:53:52 2018 by

LaTeXML
properties of vector-valued functions

properties of vector-valued functions

If

F

=

(

f

1

,

…

,

f

n

)

and

G

=

(

g

1

,

…

,

g

n

)

are vector-valued and

u

a real-valued function of the real variable

t

, one defines the

vector-valued functions

F

+

G

and

u

⁢

F

componentwise as

F

+

G

:=

(

f

1

+

g

1

,

…

,

f

n

+

g

n

)

,

u

⁢

F

:=

(

u

⁢

f

1

,

…

,

u

⁢

f

n

)

and the real valued

dot product

as

F

⋅

G

:=

f

1

⁢

g

1

+

…

+

f

n

⁢

g

n

.

If

n

=

3

,  one my define also the vector-valued

cross product

function as

F

×

G

:=

(

|

f

2

f

3

g

2

g

3

|

,

|

f

3

f

1

g

3

g

1

|

,

|

f

1

f

2

g

1

g

2

|

)

.

It’s not hard to verify, that if

F

,

G

and

u

are

differentiable

on an

interval

, so are also

F

+

G

,

u

⁢

F

and

F

⋅

G

, and the formulae

(

F

+

G

)

′

=

F

′

+

G

′

,

(

u

⁢

F

)

′

=

u

′

⁢

F

+

u

⁢

F

′

,

(

F

⋅

G

)

′

=

F

′

⋅

G

+

F

⋅

G

′

are valid, in

ℝ

3

additionally

(

F

×

G

)

′

=

F

′

×

G

+

F

×

G

′

.

Likewise one can verify the following theorems.

Theorem 1.

If

u

is

continuous

in the point

t

and

F

in the point

u

⁢

(

t

)

, then

H

=

F

∘

u

:=

(

f

1

∘

u

,

…

,

f

n

∘

u

)

is continuous in the point

t

.  If

u

is differentiable in the point

t

and

F

in the point

u

⁢

(

t

)

, then the composite function

H

is differentiable in

t

and the

chain rule

H

′

⁢

(

t

)

=

F

′

⁢

(

u

⁢

(

t

)

)

⁢

u

′

⁢

(

t

)

is in .

Theorem 2.

If

F

and

G

are integrable on

[

a

,

b

]

,  so is also

c

1

⁢

F

+

c

2

⁢

G

, where

c

1

,

c

2

are real constants, and

∫

a

b

(

c

1

⁢

F

+

c

2

⁢

G

)

⁢

𝑑

t

=

c

1

⁢

∫

a

b

F

⁢

𝑑

t

+

c

2

⁢

∫

a

b

G

⁢

𝑑

t

.

Theorem 3.

Suppose that

F

is continuous on the interval

I

and

c

∈

I

.  Then the vector-valued function

t

↦

∫

c

t

F

⁢

(

τ

)

⁢

𝑑

τ

:=

G

⁢

(

t

)

∀

t

∈

I

is differentiable on

I

and satisfies

G

′

=

F

.

Theorem 4.

Suppose that

F

is continuous on the interval

[

a

,

b

]

and

G

is an arbitrary function such that

G

′

=

F

on this interval.  Then

∫

a

b

F

⁢

(

t

)

⁢

𝑑

t

=

G

⁢

(

b

)

-

G

⁢

(

a

)

.

Theorem 2 may be generalised to

Theorem 5.

If

F

is integrable on

[

a

,

b

]

and

C

=

(

c

1

,

…

,

c

n

)

is an arbitrary vector of

ℝ

n

, then dot product

C

⋅

F

is integrable on this interval and

∫

a

b

C

⋅

F

⁢

(

t

)

⁢

𝑑

t

=

C

⋅

∫

a

b

F

⁢

(

t

)

⁢

𝑑

t

.

Title

properties of vector-valued functions

Canonical name

PropertiesOfVectorvaluedFunctions

Date of creation

2013-03-22 19:02:42

Last modified on

2013-03-22 19:02:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A42

Classification

msc 26A36

Classification

msc 26A24

Related topic

ProductAndQuotientOfFunctionsSum

Generated on Fri Feb 9 19:54:05 2018 by

LaTeXML
Riemann-Stieltjes integral

Riemann-Stieltjes integral

Let

f

and

α

be

bounded

, real-valued functions defined upon a closed finite

interval

I

=

[

a

,

b

]

of

ℝ

(

a

≠

b

)

,

P

=

{

x

0

,

…

,

x

n

}

a

partition

of

I

, and

t

i

a point of the

subinterval

[

x

i

-

1

,

x

i

]

. A sum of the form

S

⁢

(

P

,

f

,

α

)

=

∑

i

=

1

n

f

⁢

(

t

i

)

⁢

(

α

⁢

(

x

i

)

-

α

⁢

(

x

i

-

1

)

)

is called a

Riemann-Stieltjes sum

of

f

with respect to

α

.

f

is said to be

Riemann Stieltjes

integrable

with respect to

α

on

I

if there exists

A

∈

ℝ

such that given any

ϵ

>

0

there exists a partition

P

ϵ

of

I

for which, for all

P

finer than

P

ϵ

and for every choice of points

t

i

, we have

|

S

⁢

(

P

,

f

,

α

)

-

A

|

<

ϵ

If such an

A

exists, then it is unique and is known as the

Riemann-Stieltjes integral of

f

with respect to

α

.

f

is known as the

integrand

and

α

the

integrator

. The

integral

is denoted by

∫

a

b

f

⁢

𝑑

α

or

∫

a

b

f

⁢

(

x

)

⁢

𝑑

α

⁢

(

x

)

Title

Riemann-Stieltjes integral

Canonical name

RiemannStieltjesIntegral

Date of creation

2013-03-22 12:51:13

Last modified on

2013-03-22 12:51:13

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26A42

Related topic

RiemannSum

Related topic

IntegralSign

Defines

Riemann-Stieltjes sum

Defines

integrator

Generated on Fri Feb 9 19:54:10 2018 by

LaTeXML
Riemann sum

Riemann sum

Let

I

=

[

a

,

b

]

be a

closed interval

,

f

:

I

→

ℝ

be

bounded

on

I

,

n

∈

ℕ

, and

P

=

{

[

x

0

,

x

1

)

,

[

x

1

,

x

2

)

,

…

⁢

[

x

n

-

1

,

x

n

]

}

be a

partition

of

I

. The

Riemann sum

of

f

over

I

with respect to the partition

P

is defined as

S

=

∑

j

=

1

n

f

⁢

(

c

j

)

⁢

(

x

j

-

x

j

-

1

)

where

c

j

∈

[

x

j

-

1

,

x

j

]

is chosen arbitrary.

If

c

j

=

x

j

-

1

for all

j

, then

S

is called a

left Riemann sum

.

If

c

j

=

x

j

for all

j

, then

S

is called a

Riemann sum

.

Equivalently, the Riemann sum can be defined as

S

=

∑

j

=

1

n

b

j

⁢

(

x

j

-

x

j

-

1

)

where

b

j

∈

{

f

⁢

(

x

)

:

x

∈

[

x

j

-

1

,

x

j

]

}

is chosen arbitrarily.

If

b

j

=

sup

x

∈

[

x

j

-

1

,

x

j

]

⁡

f

⁢

(

x

)

, then

S

is called an

upper Riemann sum

.

If

b

j

=

inf

x

∈

[

x

j

-

1

,

x

j

]

⁡

f

⁢

(

x

)

, then

S

is called a

lower Riemann sum

.

For some examples of Riemann sums, see the entry examples of estimating a

Riemann integral

.

Title

Riemann sum

Canonical name

RiemannSum

Date of creation

2013-03-22 11:49:17

Last modified on

2013-03-22 11:49:17

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 26A42

Related topic

RiemannIntegral

Related topic

RiemannStieltjesIntegral

Related topic

LeftHandRule

Related topic

RightHandRule

Related topic

MidpointRule

Defines

left Riemann sum

Defines

right Riemann sum

Defines

upper Riemann sum

Defines

lower Riemann sum

Generated on Fri Feb 9 19:54:12 2018 by

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substitution notation

substitution notation

The following are two commonly used

substitution notations

for calculating

definite integrals

with the

antiderivative

:

•

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

[

F

⁢

(

x

)

]

a

b

•

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

F

⁢

(

x

)

|

a

b

Here, the right hand the

difference

F

⁢

(

b

)

-

F

⁢

(

a

)

.  For example, one has

∫

1

2

1

x

⁢

𝑑

x

=

[

ln

⁡

x

]

1

2

.

In Finland (only?) the corresponding notation is

∫

1

2

1

x

⁢

𝑑

x

=

/

1

2

⁡

ln

⁡

x

which may be somewhat better;  it is read in same manner as the definite integral notation, “

sijoitus 1:stä 2:een ln x

” (literally: “

substitution from 1 to 2  ln x

”).  The position of the substitution symbol in front of the function to be substituted is perhaps more natural in the sense that the symbol has an operator (as e.g. the

summing

symbol).  One of benefits of the Finnish notation is that one can comfortably clarify in it which is the

variable

to be substituted (as in the sum notation), e.g. in the case

∫

0

π

sin

⁡

t

⁢

x

⁢

d

⁢

t

=

-

1

x

⁢

/

t

=

0

π

⁡

cos

⁡

t

⁢

x

.

The notation

/

a

b

⁡

F

⁢

(

x

)

:=

F

⁢

(

b

)

-

F

⁢

(

a

)

is extended also to such cases as

/

a

∞

⁡

F

⁢

(

x

)

:=

lim

b

→

∞

⁡

/

a

b

⁡

F

⁢

(

x

)

.

Formulae

•

/

a

b

⁡

F

⁢

(

x

)

=

-

/

b

a

⁡

F

⁢

(

x

)

•

/

a

b

⁡

k

⁢

F

⁢

(

x

)

=

k

⁢

/

a

b

⁡

F

⁢

(

x

)

•

/

a

b

⁡

[

F

1

⁢

(

x

)

+

…

+

F

n

⁢

(

x

)

]

=

/

a

b

⁡

F

1

⁢

(

x

)

+

…

+

/

a

b

⁡

F

n

⁢

(

x

)

•

∫

a

b

u

⁢

(

x

)

⁢

v

′

⁢

(

x

)

⁢

𝑑

x

=

/

a

b

⁡

u

⁢

(

x

)

⁢

v

⁢

(

x

)

-

∫

a

b

u

′

⁢

(

x

)

⁢

v

⁢

(

x

)

⁢

𝑑

x

Note.

There are in Finland also some other “national”, unofficial mathematical notations used in universities, e.g.

-

⁣

∋

⁣

-

which means ‘such that’.  For example, one may write

∀

x

∈

ℤ

⁢

∃

y

∈

ℤ

-

∋

-

x

+

y

=

0

.

Title

substitution notation

Canonical

name

SubstitutionNotation

Date of creation

2013-03-22 15:08:12

Last modified on

2013-03-22 15:08:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A42

Related topic

HermitePolynomials

Related topic

AreaUnderGaussianCurve

Related topic

SineIntegralInInfinity

Related topic

Tractrix

Related topic

FourierSineAndCosineSeries

Related topic

ProofOfClosedDifferentialFormsOnASimpleConnectedDomain

Related topic

TaylorSeriesOfArcusTangent

Related topic

PerimeterOfAstroid

Related topic

PotentialOfHollowBall

Related topic

VolumeOfEllipsoid

Related topic

Arc

Generated on Fri Feb 9 19:54:14 2018 by

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support of integrable function is σ-finite

support of integrable function is

σ

-finite

Theroem -

Let

(

X

,

ℬ

,

μ

)

be a

measure space

and

f

:

X

→

ℂ

a

measurable function

. If

f

is integrable, then the

support

of

f

is

σ

-finite (

http://planetmath.org/SigmaFinite

).

It follows easily from this result that any

function

in an

L

p

-space (

http://planetmath.org/LpSpace

), with

1

≤

p

<

∞

, must have

σ

-

finite support

.

:

Let

A

0

:=

[

1

,

∞

[

, and for each

n

∈

ℕ

let

A

n

:=

[

1

n

+

1

,

1

n

[

. Since

f

is integrable, we must necessarily have

μ

⁢

(

|

f

|

-

1

⁢

(

A

n

)

)

<

∞

for each

n

∈

ℕ

∪

{

0

}

, because

μ

⁢

(

|

f

|

-

1

⁢

(

A

n

)

)

⋅

1

n

+

1

≤

∫

|

f

|

-

1

⁢

(

A

n

)

|

f

|

⁢

𝑑

μ

≤

∫

X

|

f

|

⁢

𝑑

μ

<

∞

.

Since

f

and

|

f

|

have the same support, and the the support of the latter is

supp

⁢

|

f

|

=

⋃

n

=

0

∞

|

f

|

-

1

⁢

(

A

n

)

, it follows that the support of

f

is

σ

-finite.

□

Title

support of integrable function is

σ

-finite

Canonical name

SupportOfIntegrableFunctionIssigmafinite

Date of creation

2013-03-22 18:38:47

Last modified on

2013-03-22 18:38:47

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 26A42

Classification

msc 28A25

Related topic

SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable

Defines

L

p

functions have

σ

-finite support

Generated on Fri Feb 9 19:54:17 2018 by

LaTeXML
Torricelli’s trumpet

Torricelli’s trumpet

Torricelli’s trumpet

is a fictional infinitely long

solid of revolution

formed when the closed domain

A

:=

{

(

x

,

y

)

∈

ℝ

2

⋮

x

≥

1

,

0

≤

y

≤

1

x

}

rotates about the

x

-axis. It has a finite volume,

π

volume
, but the area of its surface is infinite; in
fact even the area of

A

is infinite, i.e., the

improper integral

∫

1

∞

1

x

⁢

𝑑

x

is not

convergent

.

Torricelli’s trumpet is surprising since it can be filled by a finite
amount of paint, but this paint can never suffice for painting its
surface, no matter how a coat of paint is
used!

Title

Torricelli’s trumpet

Canonical

name

TorricellisTrumpet

Date of creation

2013-03-22 17:17:53

Last modified on

2013-03-22 17:17:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A42

Classification

msc 26A36

Classification

msc 57M20

Classification

msc 51M04

Synonym

Gabriel’s horn

Generated on Fri Feb 9 19:54:27 2018 by

LaTeXML
two improper integrals

two improper integrals

Let us consider first the improper integral

I

⁢

(

k

)

:=

∫

0

∞

1

-

cos

⁡

k

⁢

x

x

2

⁢

𝑑

x

.

The

derivative

I

′

⁢

(

k

)

may be formed by differentiating under the

integral sign

(

http://

planetmath

.org/DifferentiationUnderIntegralSign

):

I

′

⁢

(

k

)

=

∫

0

∞

(

∂

∂

⁡

k

⁢

1

-

cos

⁡

k

⁢

x

x

2

)

⁢

𝑑

x

=

∫

0

∞

sin

⁡

k

⁢

x

x

⁢

𝑑

x

=

∫

0

∞

sin

⁡

t

t

⁢

𝑑

t

Here, the last form has been gotten by the substitution (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

k

⁢

x

=

t

.
But since by the parent entry (

http://planetmath.org/SineIntegralInInfinity

) we have

∫

0

∞

sin

⁡

t

t

⁢

𝑑

t

=

π

2

and since

I

⁢

(

0

)

=

0

, we can write

I

⁢

(

k

)

=

∫

0

k

π

2

⁢

𝑑

k

=

π

⁢

k

2

.

Thus we have evaluated the

integral

I

⁢

(

k

)

:

∫

0

∞

1

-

cos

⁡

k

⁢

x

x

2

⁢

𝑑

x

=

π

⁢

k

2

.

(1)

The

formula

(1) gives

I

⁢

(

1

)

=

∫

0

∞

1

-

cos

⁡

x

x

2

⁢

𝑑

x

=

π

2

.

We use here the

consequence

formula

1

-

cos

⁡

x

=

2

⁢

sin

2

⁡

x

2

of the

double angle formula

cos

⁡

2

⁢

α

=

1

-

2

⁢

sin

2

⁡

α

,  obtaining

π

2

=

2

⁢

∫

0

∞

sin

2

⁡

x

2

x

2

⁢

𝑑

x

=

∫

0

∞

sin

2

⁡

u

u

2

⁢

𝑑

u

,

where the substitution

x

2

=

u

has produced the last form.  Accordingly, we can write as result the formula

∫

0

∞

(

sin

⁡

x

x

)

2

⁢

𝑑

x

=

π

2

.

(2)

Title

two improper integrals

Canonical

name

TwoImproperIntegrals

Date of creation

2013-03-22 18:43:11

Last modified on

2013-03-22 18:43:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Example

Classification

msc 26A42

Classification

msc 26A06

Classification

msc 26A03

Generated on Fri Feb 9 19:54:29 2018 by

LaTeXML