aarjaneiro/planetmath huggingface dataset:
Douady rabbit

Douady rabbit

The

Douady rabbit

is a

Julia set

(

http://planetmath.org/SetDeJulia

) produced by

c

=

-

1

8

+

3

4

⁢

i

.

As the

Mandelbrot set

indicates, the

real part

can be varied to as much as

-

.2

while the

imaginary part

can be varied to as much as

.75

⁢

i

and still produce a connected set.

Title

Douady rabbit

Canonical name

DouadyRabbit

Date of creation

2013-03-22 17:15:51

Last modified on

2013-03-22 17:15:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 28A80

Generated on Fri Feb 9 20:10:42 2018 by

LaTeXML
fractal

fractal

There are several ways of defining a

fractal

, and a reader will need to reference their source to see which definition is being used.

Perhaps the simplest definition is to define a

fractal

to be a subset of

ℝ

n

with

Hausdorff dimension

greater than its

Lebesgue covering dimension

. It is worth noting that typically (but not always), fractals have non-integer Hausdorff dimension. See, for example, the

Koch snowflake

and the

Mandelbrot set

(named after

Benoit Mandelbrot

, who also coined the term “fractal” for these objects).

A looser definition of a

fractal

is a “self-similar object”. That is, a subset or

ℝ

n

which can be covered by copies of itself using a set of (usually two or more)

transformation

mappings

. Another way to say this would be “an object with a discrete approximate

scaling

symmetry

”.

See also the discussion near the end of the entry Hausdorff dimension (

http://

planetmath

.org/HausdorffDimension

).

Title

fractal

Canonical name

Fractal

Date of creation

2013-03-22 12:41:51

Last modified on

2013-03-22 12:41:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

17

Author

mathcam (2727)

Entry type

Definition

Classification

msc 28A80

Related topic

MengerSponge

Generated on Fri Feb 9 20:10:44 2018 by

LaTeXML
Hausdorff dimension

Hausdorff dimension

Let

Θ

be a

bounded

subset of

ℝ

n

let

N

Θ

⁢

(

ϵ

)

be the minimum number of balls of radius

ϵ

required to cover

Θ

. Then define the

Hausdorff dimension

d

H

of

Θ

to be

d

H

⁢

(

Θ

)

:=

-

lim

ϵ

→

0

⁡

log

⁡

N

Θ

⁢

(

ϵ

)

log

⁡

ϵ

.

Hausdorff dimension is easy to calculate for simple objects like the

Sierpinski gasket

or a

Koch curve

. Each of these may be covered with a

collection

of scaled-down copies of itself. In fact, in the case of the Sierpinski gasket, one can take the

individual

triangles

in each approximation as balls in the covering. At stage

n

, there are

3

n

triangles of radius

1

2

n

, and so the Hausdorff dimension of the Sierpinski triangle is at most

-

n

⁢

log

⁡

3

n

⁢

log

⁡

1

/

2

=

log

⁡

3

log

⁡

2

, and it can be shown that it is equal to

log

⁡

3

log

⁡

2

.

From some notes from Koro

This

definition

can be extended to a general

metric space

X

with distance function

d

.

Define the

diameter

|

C

|

of a bounded subset

C

of

X

to be

sup

x

,

y

∈

C

⁡

d

⁢

(

x

,

y

)

.

Define a

r

-cover

of

X

to be a collection of subsets

C

i

of

X

indexed by

some

countable set

I

, such that

|

C

i

|

<

r

and

X

=

∪

i

∈

I

C

i

.

We also define the

function

H

r

D

⁢

(

X

)

=

inf

⁡

∑

i

∈

I

|

C

i

|

D

where the

infimum

is over all

countable

r

-covers of

X

.
The

Hausdorff dimension

of

X

may then be defined as

d

H

⁢

(

X

)

=

inf

⁡

{

D

∣

lim

r

→

0

⁡

H

r

D

⁢

(

X

)

=

0

}

.

When

X

is a subset of

ℝ

n

with any norm-induced metric, then this definition reduces to that given above.

Title

Hausdorff dimension

Canonical

name

HausdorffDimension

Date of creation

2013-05-18 23:14:26

Last modified on

2013-05-18 23:14:26

Owner

Mathprof (13753)

Last modified by

unlord (1)

Numerical id

16

Author

Mathprof (1)

Entry type

Definition

Classification

msc 28A80

Related topic

Dimension3

Related topic

HausdorffMeasure

Defines

countable r-cover

Defines

diameter

Generated on Fri Feb 9 20:10:46 2018 by

LaTeXML
Julia set

Julia set

Let

U

be an open subset of the

complex plane

and let

f

:

U

→

U

be

analytic

. Denote the

n

-th

iterate

of

f

by

f

n

, i.e.

f

1

=

f

and

f

n

+

1

=

f

∘

f

n

. Then the

Julia set

of

f

is the
subset

J

of

U

characterized by the following

property

: if

z

∈

J

then the

restriction

of

{

f

n

∣

n

∈

ℕ

}

to any

neighborhood

of

z

is not a

normal family

.

It can also be shown that the Julia set of

f

is the closure of the set of

repelling periodic points

of

f

. (Repelling periodic point means that, for
some

n

, we have

f

n

⁢

(

z

)

=

z

and

|

f

′

⁢

(

z

)

|

>

1

.)

A simple example is afforded by the map

f

⁢

(

z

)

=

z

2

; in this case, the Julia
set is the

unit circle

. In general, however, things are much more complicated
and the Julia set is a fractal.

From the

definition

, it follows that the Julia set is closed under

f

and
its

inverse

—

f

⁢

(

J

)

=

J

and

f

-

1

⁢

(

J

)

=

J

. Topologically, Julia sets
are perfect and have empty interior.

Title

Julia set

Canonical

name

JuliaSet

Date of creation

2013-03-22 17:15:26

Last modified on

2013-03-22 17:15:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 28A80

Generated on Fri Feb 9 20:10:49 2018 by

LaTeXML
Mandelbrot set

Mandelbrot set

The

Mandelbrot set

is a

fractal

in the

complex plane

defined as the collection of

c

∈

ℂ

for which the orbit of

0

under the quadratic

dynamical system

f

c

⁢

(

z

)

:

z

↦

z

2

+

c

is

bounded

. That is, to test whether

c

is in the Mandelbrot set, one checks whether the sequence

(

f

c

⁢

(

0

)

,

f

c

⁢

(

f

c

⁢

(

0

)

)

,

f

c

⁢

(

f

c

⁢

(

f

c

⁢

(

0

)

)

)

,

…

)

tends to

infinity

or stays in a bounded region. The point

c

is in the set if the sequence does not tend to infinity.

There are deep connections between points in the Mandelbrot set and the corresponding

Julia sets

(

http://planetmath.org/SetDeJulia

)

J

⁢

(

f

c

)

. In particular,

c

is in the Mandelbrot set if and only if the corresponding to

c

is

connected

.

The Mandelbrot set is perhaps the most famous fractal. At every level, it has smaller copies of the overall shape that are connected to the main shape. The higher the resolution, the easier it is to see that the entire Mandelbrot set is simply connected, with no holes. Discovered by Benoît Mandelbrot, the first print-out of it was in black and white.

Most of the Mandelbrot set lies within the unit disk of the complex plane. The second largest ‘circle’ (to the left of the main ‘circle’ in the standard illustration) is centered at

-

1

+

0

⁢

i

. Points with

real part

outside the range

-

2

<

a

<

1

and

imaginary part

(

1

<

|

b

|

)

i

will certainly tend to infinity.

Nowadays, most images of the Mandelbrot set include colors, with the colors for escaping values of

c

being assigned in some correspondence to how many iterations it took to determine that

c

escaped.

References

1

H. Lauwerier, translated by Sophia Gill-Hoffstädt.

Fractals: Endlessly Repeated Geometric Figures

Princeton: Princeton University Press (1991): 124 - 151

Title

Mandelbrot set

Canonical name

MandelbrotSet

Date of creation

2013-03-22 16:57:52

Last modified on

2013-03-22 16:57:52

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

11

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 28A80

Generated on Fri Feb 9 20:10:51 2018 by

LaTeXML
Menger sponge

Menger sponge

PMlinkescapeworddivisions

A

Sierpinski carpet

is the set of all points

(

x

,

y

)

such that

x

or

y

is in
the

Cantor set

. An

equivalent

and perhaps simpler

definition

is:

Let

S

0

be a unit square. Let

S

n

+

1

be

S

n

, with each square divided
into ninths, by being divided into thirds horizontally and vertically, and the central resulting square removed, and the other resulting squares treated separately in further

divisions

. The limit as

n

→

∞

of

S

n

is a Sierpinski carpet. An approximation is shown in figure

1

.

Figure 1:

An approximation of the Sierpinski carpet

The

Menger sponge

is a

fractal

embedded in 3-dimensional space. It can be seen as a 3-d

generalization

of the Sierpinski carpet, which is itself a 2-dimensional generalization of the Cantor set. The Menger sponge is almost always represented as being constructed from Cantor sets using the “middle third” rule.

The Menger sponge consists of all points

(

x

,

y

,

z

)

such that

(

x

,

y

)

,

(

y

,

z

)

, and

(

x

,

z

)

are all in Sierpinski carpets. Each “face” is a Sierpinski carpet.

Similarily to the Sierpinski carpet the Menger sponge can be constructed in the following way:

Start with a unit cube and split it into 27 smaller cubes of equal size. Remove the central cube and the ones joining a face with it. Then start over with the remaining smaller cubes.

Figure 2:

An iteration of the Menger sponge – created in Blender 2.35.
(The

\PMlinktofile

Blender filemenger.blend for this picture.)

Title

Menger sponge

Canonical

name

MengerSponge

Date of creation

2013-03-22 14:27:05

Last modified on

2013-03-22 14:27:05

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

23

Author

mathwizard (128)

Entry type

Definition

Classification

msc 28A80

Related topic

Fractal

Related topic

CantorSet

Related topic

SierpinskiGasket

Defines

Sierpinski carpet

Generated on Fri Feb 9 20:10:53 2018 by

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Peano curve

Peano curve

A

Peano curve

or

space-filling curve

is a

continuous mapping

of a

closed interval

onto a square.

Such

mappings

, introduced by Peano in 1890, played an
important role in the

development

of

topology

as a

counterexample

to the naive ideas of

dimension

— while it is
true that one cannot map a space onto a space of higher dimension using a

smooth map

, this is no longer true if one only requires

continuity

as opposed to
smoothness. The Peano curve and

similar

examples led to a rethinking of the

foundations

of topology and

analysis

, and underscored the importance of formulating
topological notions in a

rigorous

fashion.

However, still, a space-filling curve cannot ever be

one-to-one

;
otherwise invariance of dimension would be violated.

Many space-filling curves may be obtained as the limit of a

sequence

,

⟨

h

n

∣

n

∈

ℕ

⟩

, of

continuous functions

h

n

:

[

0

,

1

]

→

[

0

,

1

]

×

[

0

,

1

]

. Figure

1

(

\PMlinktofile

source codehilbert.cc), showing the ranges of the first few approximations to Hilbert’s space-filling curve, illustrates a common case in which each successive approximation is obtained by applying a

recursive

procedure to its

predecessor

.

Figure 1:

The ranges of the first six approximations to Hilbert’s space-filling curve

Title

Peano curve

Canonical name

PeanoCurve

Date of creation

2013-03-22 16:32:29

Last modified on

2013-03-22 16:32:29

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 28A80

Synonym

space-filling curve

Synonym

space filling curve

Generated on Fri Feb 9 20:10:54 2018 by

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proof of existence and unicity of self-similar fractals

proof of existence and unicity of self-similar fractals

We consider the space

𝒦

⁢

(

X

)

=

{

K

⊂

X

:

K

⁢

compact

⁢

and

⁢

non

⁢

empty

}

endowed with the

Hausdorff distance

δ

.
Since

Hausdorff metric inherits completeness

, being

X

complete

,

(

𝒦

⁢

(

X

)

,

δ

)

is complete too. We then consider the

mapping

T

:

𝒦

⁢

(

X

)

→

𝒦

⁢

(

X

)

defined by

T

⁢

(

A

)

=

⋃

i

=

1

N

T

i

⁢

(

A

)

.

We claim that

T

is a

contraction

. In fact, recalling that

δ

⁢

(

A

1

∪

A

2

,

B

1

∪

B

2

)

≤

max

⁡

{

δ

⁢

(

A

1

,

B

1

)

,

δ

⁢

(

A

2

,

B

2

)

}

while

δ

⁢

(

T

i

⁢

(

A

)

,

T

i

⁢

(

B

)

)

≤

λ

i

⁢

δ

⁢

(

A

,

B

)

if

T

i

is

λ

i

-

Lipschitz

, we have

δ

⁢

(

T

⁢

(

A

)

,

T

⁢

(

B

)

)

=

δ

⁢

(

⋃

i

T

i

⁢

(

A

)

,

⋃

i

T

i

⁢

(

B

)

)

≤

max

i

⁡

δ

⁢

(

T

i

⁢

(

A

)

,

T

i

⁢

(

B

)

)

≤

max

i

⁡

λ

i

⁢

δ

⁢

(

A

,

B

)

=

λ

⁢

δ

⁢

(

A

,

B

)

with

λ

=

max

i

⁡

λ

i

<

1

.

So

T

is a contraction on the complete metric space

𝒦

⁢

(

X

)

and hence,
by

Banach Fixed Point Theorem

, there exists one and only one

K

∈

𝒦

⁢

(

X

)

such that

T

⁢

(

K

)

=

K

.

Title

proof of existence and unicity of self-similar fractals

Canonical name

ProofOfExistenceAndUnicityOfSelfsimilarFractals

Date of creation

2013-03-22 16:05:30

Last modified on

2013-03-22 16:05:30

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Proof

Classification

msc 28A80

Generated on Fri Feb 9 20:10:57 2018 by

LaTeXML
quadratic Julia set

quadratic Julia set

For each

complex number

c

, there is an associated

quadratic map

f

c

:

ℂ

→

ℂ

defined by

f

c

⁢

(

z

)

=

z

2

+

c

.
Since

polynomials

are

analytic

, it follows that

f

c

has a

Julia set

J

⁢

(

f

c

)

, which we call the

quadratic Julia set

associated to

c

and denote by

J

c

.

The

function

can also be viewed as having

ℝ

2

as its domain
and codomain. If

c

=

a

+

i

⁢

b

, then

f

c

⁢

(

x

,

y

)

=

(

x

2

-

y

2

+

a

,

2

⁢

x

⁢

y

+

b

)

.

The characterization of the Julia set

J

c

as all points

z

for which the
collection of iterates

{

f

n

:

n

∈

ℕ

}

is not a

normal
family

can be roughly interpreted as saying that the Julia set
includes only those points exhibiting chaotic behavior. In
particular, points whose orbit under

f

goes to

infinity

are omitted
from

J

c

, as well as points whose orbit converges to a point.

Sometimes for aesthetic purposes a Julia set is displayed with points
of the latter type included. However, the chaoticity of the true Julia
set can be exploited to plot an approximation very quickly. Given a
single point

z

known to be in the quadratic Julia set

J

c

, its

inverses

under

f

, that is, the square roots of

z

-

c

, are also in

J

c

. Moreover,
by the chaoticity condition the “backwards orbit” of

z

(selecting just one
square root at each step) will be distributed fairly evenly
over

J

c

, so this gives a computationally inexpensive method to plot
Julia sets.

Before the advent of computers, the French mathematician Gaston Julia
proved under what conditions a Julia set is

connected

or not
connected. After computers became available, it became possible to
make pictures displaying some of the complexity of these Julia sets,
and the

Mandelbrot set

, a kind of index into connected quadratic Julia
sets, was discovered.

In the same way that some people see recognizable shapes in clouds,
some people see recognizable shapes in Julia sets, and some of them
have been named accordingly. To give two examples: the

San Marco
dragon

at

-

3

4

+

0

⁢

i

and the

Douady rabbit

at

-

1

8

+

745

1000

⁢

i

(the

coordinates

can be varied by small values and
still give very

similar

shapes).

Julia sets can be generalized to other iterated

holomorphic functions

on the

complex plane

or in a 3-dimensional space.

References

1

H. Lauwerier, translated by Sophia Gill-Hoffstädt.

Fractals

: Endlessly Repeated Geometric Figures

Princeton: Princeton University Press (1991): 124 - 151

Title

quadratic Julia set

Canonical name

QuadraticJuliaSet

Date of creation

2013-03-22 17:14:40

Last modified on

2013-03-22 17:14:40

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 28A80

Generated on Fri Feb 9 20:10:58 2018 by

LaTeXML
San Marco dragon

San Marco dragon

The

San Marco dragon

is a

Julia set

(

http://planetmath.org/SetDeJulia

) produced by

c

=

-

3

4

+

0

⁢

i

.

Like other on the horizontal center line of the

Mandelbrot set

, the San Marco dragon is symmetrical around its horizontal axis, but this particular set reminded Benoît Mandelbrot of St. Mark’s cathedral in Venice (and its

reflection

in the canal) more than the others.

References

1

H. Lauwerier, translated by Sophia Gill-Hoffstädt.

Fractals

: Endlessly Repeated Geometric Figures

Princeton: Princeton University Press (1991): 144

Title

San Marco dragon

Canonical name

SanMarcoDragon

Date of creation

2013-03-22 17:15:54

Last modified on

2013-03-22 17:15:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 28A80

Generated on Fri Feb 9 20:11:00 2018 by

LaTeXML
self-similar fractals

self-similar fractals

Let

(

X

,

d

)

be a

metric space

and let

T

1

,

…

,

T

N

be a finite number of

contractions

on

X

i.e. each

T

i

:

X

→

X

enjoys the

property

d

⁢

(

T

i

⁢

(

x

)

,

T

i

⁢

(

y

)

)

≤

λ

i

⁢

d

⁢

(

x

,

y

)

(

T

i

is

λ

i

-Lipschitz) for some

λ

i

<

1

.

Given a set

A

⊂

X

we can define

T

⁢

(

A

)

=

⋃

i

=

1

N

T

i

⁢

(

A

)

.

Definition 1

.

A set

K

such that

T

⁢

(

K

)

=

K

(invariant set) is called a

self-similar fractal

with respect to the contractions

{

T

1

,

…

,

T

N

}

.

The most famous example of self-similar fractal is the

Cantor set

.
This is constructed in

X

=

ℝ

with the usual

Euclidean metric

structure

, by
choosing

N

=

2

contractions:

T

1

⁢

(

x

)

=

x

/

3

,

T

2

⁢

(

x

)

=

1

-

(

1

-

x

)

/

3

.

A more interesting example is the

Koch curve

in

X

=

ℝ

2

. In this case we choose

N

=

4

similitudes with factor

1

/

3

.

By choosing other appropriate

transformations

one can obtain the beautiful example of the

Barnsley Fern

, which shows how the

fractal

geometry

can successfully describe nature.

An important result is given by the following

Theorem

.

Theorem 1

.

Let

X

be a

complete metric space

and let

T

1

,

…

,

T

N

:

X

→

X

be a given set of contractions.
Then there exists one and only one non empty

compact set

K

⊂

X

such that

T

⁢

(

K

)

=

K

.

Notice that the

empty set

always

satisfies

the

relation

T

⁢

(

∅

)

=

∅

and hence is not an interesting case. On the other hand, if at least one of the

T

i

is

surjective

(as happens in the examples above), then the whole set

X

satisfies

T

⁢

(

X

)

=

X

.

Title

self-similar fractals

Canonical

name

SelfsimilarFractals

Date of creation

2013-03-22 16:05:12

Last modified on

2013-03-22 16:05:12

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

11

Author

paolini (1187)

Entry type

Definition

Classification

msc 28A80

Generated on Fri Feb 9 20:11:02 2018 by

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Sierpiński gasket

Sierpiński gasket

Let

S

0

be a triangular area, and define

S

n

+

1

to be obtained from

S

n

by replacing each triangular area in

S

n

with three

similar

and similarly
oriented triangular areas each intersecting with each of the other two at exactly one vertex, each one half the linear scale of the original in size.
The limiting set as

n

→

∞

(alternately the

intersection

of all these sets) is a

S

ierpiński gasket, also known as a

S

ierpiński

triangle

.

Figure 1:

Sierpiński gasket stage 0, a single triangle, and at stage 1, three triangles

Figure 2:

Stage 2, nine triangles, and stage

n

,

3

n

triangles

Title

Sierpiński gasket

Canonical

name

SierpinskiGasket

Date of creation

2013-05-18 22:53:46

Last modified on

2013-05-18 22:53:46

Owner

mathwizard (128)

Last modified by

unlord (1)

Numerical id

41

Author

mathwizard (1)

Entry type

Definition

Classification

msc 28A80

Synonym

Sierpinski triangle

Synonym

Sierpinski gasket

Synonym

Sierpiński triangle

Related topic

MengerSponge

Generated on Fri Feb 9 20:11:04 2018 by

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Banach-Tarski paradox

Banach-Tarski paradox

The

3

-dimensional ball can be split in a finite number of pieces which can be pasted together to give two balls of the same volume as the first!

Let us formulate the

theorem

formally.
We say that a set

A

⊂

ℝ

n

is

decomposable

in

N

pieces

A

1

,

…

,

A

N

if there exist some

isometries

θ

1

,

…

,

θ

N

of

ℝ

n

such that

A

=

θ

1

⁢

(

A

1

)

∪

…

∪

θ

N

⁢

(

A

N

)

while

θ

1

⁢

(

A

1

)

,

…

,

θ

N

⁢

(

A

N

)

are all

disjoint

.

We then say that two sets

A

,

B

⊂

ℝ

n

are

equi-decomposable

if both

A

and

B

are decomposable in the same pieces

A

1

,

…

,

A

N

.

Theorem 1

(Banach-Tarski).

The

unit ball

B

3

⊂

R

3

is equi-decomposable to the union of two disjoint unit balls.

1

Comments

The actual number of pieces needed for this

decomposition

is not so large. Say that ten pieces are enough.

Also it is not important that the set considered is a ball. Every two set with non empty

interior

are equi-decomposable in

ℝ

3

. Also the

ambient space

can be chosen larger. The theorem is true in all

ℝ

n

with

n

≥

3

but it is not true in

ℝ

2

nor in

ℝ

.

Where is the

paradox

? We are saying that a piece of (say) gold can be cut and pasted to obtain two pieces equal to the previous one. And we may divide these two pieces in the same way to obtain four pieces and so on…

We believe that this is not possible since the weight of the piece of gold does not change when I cut it.

A

consequence

of this theorem is, in fact, that it is not possible to define the volume for all subsets of the

3

-dimensional space. In particular the volume cannot be computed for some of the pieces in which the unit ball is decomposed (some of them are not

measurable

).

The

existence

of non-measurable sets is proved more simply and in all

dimension

by Vitali Theorem. However

Banach-Tarski paradox

says something more. It says that it is not possible to define a measure on all the subsets of

ℝ

3

even if we drop the

countable additivity

and replace it with a

finite additivity

:

μ

⁢

(

A

∪

B

)

=

μ

⁢

(

A

)

+

μ

⁢

(

B

)

∀

A

,

B

⁢

disjoint

.

Another point to be noticed is that the proof needs the

axiom of choice

(

http://planetmath.org/AxiomOfChoice

). So some of the pieces in which the ball is divided are not constructable.

See

http://www.math.metu.edu.tr/ berkman/choice/

http://www.math.metu.edu.tr/ berkman/choice/
for more details.

Title

Banach-Tarski paradox

Canonical name

BanachTarskiParadox

Date of creation

2013-03-22 13:45:40

Last modified on

2013-03-22 13:45:40

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

12

Author

paolini (1187)

Entry type

Theorem

Classification

msc 28A99

Classification

msc 03B99

Related topic

PsuedoparadoxInMeasureTheory

Related topic

HausdorffParadox

Related topic

ProofOfHausdorffParadox

Related topic

DehnsTheorem

Defines

decomposable

Defines

equi-decomposable

Generated on Fri Feb 9 20:11:06 2018 by

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Vitali’s Theorem

Vitali’s Theorem

There exists a set

V

⊂

[

0

,

1

]

which is not

Lebesgue measurable

.
Notice that this result requires the

Axiom of Choice

.

Title

Vitali’s

Theorem

Canonical name

VitalisTheorem

Date of creation

2013-03-22 13:45:47

Last modified on

2013-03-22 13:45:47

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Theorem

Classification

msc 28Axx

Synonym

existence of non measurable sets

Related topic

Integral2

Related topic

LebesgueMeasure

Related topic

PsuedoparadoxInMeasureTheory

Related topic

ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets

Generated on Fri Feb 9 20:11:08 2018 by

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