aarjaneiro/planetmath huggingface dataset:
example of function not Lebesgue Measurable with measurable level sets

example of function not Lebesgue Measurable with measurable level sets

Consider

V

as in Vitali’s

theorem

. Define the

function

f

:

[

0

,

1

]

→

[

0

,

+

∞

[

by:

f

⁢

(

x

)

=

{

x

if

⁢

x

∉

V

2

+

x

if

⁢

x

∈

V

The

level sets

of

f

will either be the

empty set

, or a singleton and thus

measurable

.

f

-

1

⁢

(

{

x

}

)

=

{

{

x

}

if

⁢

0

≤

x

≤

1

∧

x

∉

V

{

2

-

x

}

if

⁢

2

≤

x

≤

3

∧

x

-

2

∈

V

{

}

otherwise

f

is not a

measurable function

since

f

-

1

(

[

2

,

+

∞

[

)

=

V

and

V

is not a

measurable set

.

Title

example of function not

Lebesgue Measurable

with measurable level sets

Canonical name

ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets

Date of creation

2013-03-22 15:51:22

Last modified on

2013-03-22 15:51:22

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Example

Classification

msc 28B15

Related topic

measurableFunctions

Related topic

VitalisTheorem

Related topic

MeasurableFunctions

Generated on Fri Feb 9 20:11:12 2018 by

LaTeXML
fundamental lemma of calculus of variations

fundamental lemma of calculus of variations

The idea in the

calculus of variations

is to study

stationary points

of functionals.
To derive a

differential equation

for such stationary
points, the following theorem is needed, and hence
named thereafter. It is also used in distribution theory
to recover traditional

calculus

from distributional calculus.

Theorem 1

.

Suppose

f

:

U

→

C

is a

locally integrable function

on an
open subset

U

⊂

R

n

,
and suppose that

∫

U

f

⁢

ϕ

⁢

𝑑

x

=

0

for all

smooth functions with compact support

ϕ

∈

C

0

∞

⁢

(

U

)

.
Then

f

=

0

almost everywhere.

By linearity of the

integral

, it is easy to see that one only needs to
prove the claim for real

f

. If

f

is

continuous

, this can be seen
by purely geometrical

arguments

. A full proof
based on the

Lebesgue differentiation theorem

is given
in

[

1

]

. Another proof is given in

[

2

]

.

References

1

L. Hörmander,

The

Analysis

of Linear Partial Differential

Operators

I,
(Distribution theory and Fourier Analysis)

, 2nd ed, Springer-Verlag, 1990.

2

S. Lang,

Analysis II

,
Addison-Wesley Publishing Company Inc., 1969.

Title

fundamental lemma of calculus of variations

Canonical name

FundamentalLemmaOfCalculusOfVariations

Date of creation

2013-03-22 15:02:04

Last modified on

2013-03-22 15:02:04

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Theorem

Classification

msc 28B15

Synonym

fundamental theorem of the calculus of variations

Related topic

CalculusOfVariations

Generated on Fri Feb 9 20:11:14 2018 by

LaTeXML
locally integrable function

locally integrable function

Definition

Suppose that

U

is an

open set

in

ℝ

n

, and

f

:

U

→

ℂ

is a

Lebesgue measurable

function

.
If the

Lebesgue integral

∫

K

|

f

|

⁢

𝑑

x

is finite for all

compact subsets

K

in

U

, then

f

is

locally integrable

. The set of all
such functions is denoted by

L

loc

1

⁢

(

U

)

.

Example

1.

L

1

⁢

(

U

)

⊂

L

loc

1

⁢

(

U

)

, where

L

1

⁢

(

U

)

is the set of (globally) integrable functions.

2.

Continuous functions

are locally integrable.

3.

The function

f

⁢

(

x

)

=

1

/

x

for

x

≠

0

and

f

⁢

(

0

)

=

0

is not locally integrable.

Title

locally integrable function

Canonical name

LocallyIntegrableFunction

Date of creation

2013-03-22 13:44:19

Last modified on

2013-03-22 13:44:19

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 28B15

Generated on Fri Feb 9 20:11:16 2018 by

LaTeXML
L^p-space

L

p

-space

Definition

Let

(

X

,

𝔅

,

μ

)

be a

measure space

. Let

0

<

p

<

∞

. The

L

p

-norm

of a

function

f

:

X

→

ℂ

is defined as

||

f

||

p

:=

(

∫

X

|

f

|

p

⁢

𝑑

μ

)

1

p

(1)

when the

integral

exists. The set of functions with finite

L

p

-

norm forms

a

vector space

V

with the usual

pointwise addition

and

scalar
multiplication

of functions. In particular, the set of functions with zero

L

p

-norm form a

linear subspace

of

V

, which for this article
will be called

K

. We are then interested in the quotient space

V

/

K

, which consists of complex functions on

X

with finite

L

p

-norm,
identified up to equivalence almost everywhere. This quotient space is the complex

L

p

-space on

X

.

Theorem

If

1

≤

p

<

∞

, the vector space

V

/

K

is

complete

with respect to the

L

p

norm.

The space

L

∞

.

The space

L

∞

is somewhat special, and may be defined without explicit reference to an integral. First, the

L

∞

-norm of

f

is
defined to be the

essential supremum

of

|

f

|

:

||

f

||

∞

:=

ess

⁢

sup

⁢

|

f

|

=

inf

⁡

{

a

∈

ℝ

:

μ

⁢

(

{

x

:

|

f

⁢

(

x

)

|

>

a

}

)

=

0

}

(2)

However, if

μ

is the trivial measure, then essential supremum of every

measurable function

is defined to be 0.

The definitions of

V

,

K

, and

L

∞

then proceed as above, and again we have that

L

∞

is complete. Functions in

L

∞

are also called

essentially bounded

.

Example

Let

X

=

[

0

,

1

]

and

f

⁢

(

x

)

=

1

x

. Then

f

∈

L

1

⁢

(

X

)

but

f

∉

L

2

⁢

(

X

)

.

Title

L

p

-space

Canonical name

Lpspace

Date of creation

2013-03-22 12:21:32

Last modified on

2013-03-22 12:21:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

28

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 28B15

Synonym

L

p

space

Synonym

essentially bounded function

Related topic

MeasureSpace

Related topic

Norm

Related topic

EssentialSupremum

Related topic

Measure

Related topic

FeynmannPathIntegral

Related topic

AmenableGroup

Related topic

VectorPnorm

Related topic

VectorNorm

Related topic

SobolevInequality

Related topic

L2SpacesAreHilbertSpaces

Related topic

RieszFischerTheorem

Related topic

BoundedLinearFunctionalsOnLpmu

Related topic

ConvolutionsOfComplexFunctionsOnLocallyCompactG

Defines

p

-integrable function

Defines

L

∞

Defines

essentially bounded

Defines

L

p

-norm

Generated on Fri Feb 9 20:11:18 2018 by

LaTeXML
fractional calculus

fractional calculus

The idea of

calculus

in fractional order is nearly as old as its integer counterpart. In a letter dated September 30th 1650, l’Hôpital posed the question of the meaning of

d

α

⁢

f

d

⁢

x

α

if

α

=

1

2

to Leibniz.
There are different approaches to define calculus of fractional order. The following approaches are the most common and we can prove that they are

equivalent

(1) Riemann-Liouville approach of

fractional integration

(2) Grunwald-Letnikov approach of

fractional differentiation

Title

fractional calculus

Canonical name

FractionalCalculus

Date of creation

2013-03-22 16:18:27

Last modified on

2013-03-22 16:18:27

Owner

bchui (10427)

Last modified by

bchui (10427)

Numerical id

14

Author

bchui (10427)

Entry type

Definition

Classification

msc 28B20

Synonym

fractional calculus

Defines

fractional calculus

Generated on Fri Feb 9 20:11:19 2018 by

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Haar integral

Haar integral

Let

Γ

be a

locally compact topological group

and

𝒞

be the

algebra

of all

continuous

real-valued functions on

Γ

with compact support. In

addition

we define

𝒞

+

to be the set of non-negative functions that belong to

𝒞

. The

Haar integral

is a real linear map

I

of

𝒞

into the field of the

real number

for

Γ

if it satisfies:

•

I

is not the zero map

•

I

only takes non-negative values on

𝒞

+

•

I

has the following

property

I

⁢

(

γ

⋅

f

)

=

I

⁢

(

f

)

for all elements

f

of

𝒞

and all element

γ

of

Γ

.

The

Haar integral

may be denoted in the following way

(there are also other ways)

:

∫

γ

∈

Γ

f

⁢

(

γ

)

or

∫

Γ

f

or

∫

Γ

f

⁢

𝑑

γ

or

I

⁢

(

f

)

The following are necessary and sufficient conditions for the existence of
a unique Haar integral:
There is a real-valued function

I

+

1.

(Linearity)

.

I

+

⁢

(

λ

⁢

f

+

μ

⁢

g

)

=

λ

⁢

I

+

⁢

(

f

)

+

μ

⁢

I

+

⁢

(

g

)

where

f

,

g

∈

𝒞

+

and

λ

,

μ

∈

ℝ

+

.

2.

(Positivity)

. If

f

⁢

(

γ

)

≥

0

for all

γ

∈

Γ

then

I

+

⁢

(

f

⁢

(

γ

)

)

≥

0

.

3.

(Translation-Invariance)

.

I

⁢

(

f

⁢

(

δ

⁢

γ

)

)

=

I

⁢

(

f

⁢

(

γ

)

)

for any fixed

δ

∈

Γ

and every

f

in

𝒞

+

.

An additional property is if

Γ

is a compact group then the Haar integral has right translation-invariance:

∫

γ

∈

Γ

f

⁢

(

γ

⁢

δ

)

=

∫

γ

∈

Γ

f

⁢

(

γ

)

for any fixed

δ

∈

Γ

.
In addition we can define

normalized Haar integral

to be

∫

Γ

1

=

1

since

Γ

is

compact

, it implies that

∫

Γ

1

is finite.

(The proof for existence and uniqueness of the Haar integral is presented in

[

HG

]

on page 9.)

(the information of this entry is in part quoted and paraphrased from

[

GSS

]

)

References

GSS

Golubsitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

HG

Gochschild, G.: The Structure of

Lie Groups

. Holden-Day, San Francisco, 1965.

Title

Haar integral

Canonical

name

HaarIntegral

Date of creation

2013-03-22 13:39:56

Last modified on

2013-03-22 13:39:56

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 28C05

Defines

normalized Haar integral

Generated on Fri Feb 9 20:11:22 2018 by

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Haar measure

Haar measure

1

Definition of Haar measures

Let

G

be a

locally compact topological group

, and denote by

ℬ

the

sigma algebra

generated by the closed

compact subsets

of

G

. A

left Haar measure

on

G

is a

measure

μ

on

ℬ

which is:

1.

outer regular

on all sets

B

∈

ℬ

2.

inner regular on all open sets

U

∈

ℬ

3.

finite on all compact sets

K

∈

ℬ

4.

invariant under left

translation

:

μ

⁢

(

g

⁢

B

)

=

μ

⁢

(

B

)

for all sets

B

∈

ℬ

5.

nontrivial:

μ

⁢

(

B

)

>

0

for all non-empty open sets

B

∈

ℬ

.

A

right Haar measure

on

G

is defined similarly, except with left translation invariance replaced by right translation invariance (

μ

⁢

(

B

⁢

g

)

=

μ

⁢

(

B

)

for all sets

B

∈

ℬ

). A

bi-invariant Haar measure

is a Haar measure that is both left invariant and right invariant.

2

Existence of Haar measures

For any discrete

topological group

G

, the

counting measure

on

G

is a bi-invariant Haar measure. More generally, every locally compact topological group

G

has a left Haar measure

μ

, which is unique up to scalar multiples. In addition,

G

also admits a right Haar measure, and for an

abelian group

G

the left and right Haar measures are always equal. The Haar measure plays an important role in the

development

of Fourier analysis and

representation theory

on

locally compact groups

such as

Lie groups

and

profinite groups

.

Title

Haar measure

Canonical name

HaarMeasure

Date of creation

2013-03-22 12:40:55

Last modified on

2013-03-22 12:40:55

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 28C10

Defines

left Haar measure

Defines

right Haar measure

Defines

bi-invariant Haar measure

Generated on Fri Feb 9 20:11:30 2018 by

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regular measure

regular measure

Definition 0.1

.

A

regular measure

μ

R

on a

topological space

X

is a

measure

on

X

such that
for each

A

∈

ℬ

⁢

(

X

)

, with

μ

R

⁢

(

A

)

<

∞

), and each

ε

>

0

there exist a

compact subset

K

of

X

and an

open subset

G

of

X

with

K

⊂

A

⊂

G

,
such that for all sets

A

′

∈

ℬ

⁢

(

X

)

with

A

′

⊂

G

-

K

, one has

μ

R

⁢

(

A

′

)

<

ε

.

Title

regular measure

Canonical name

RegularMeasure

Date of creation

2013-03-22 18:23:07

Last modified on

2013-03-22 18:23:07

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

5

Author

bci1 (20947)

Entry type

Definition

Classification

msc 28C15

Classification

msc 28A12

Classification

msc 28A10

Related topic

OuterMeasure

Generated on Fri Feb 9 20:11:32 2018 by

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essential supremum

essential supremum

Essential supremum of a function

Let

(

Ω

,

ℱ

,

μ

)

be a

measure space

and let

f

be a

Borel measurable function

from

Ω

to the

extended real numbers

ℝ

¯

. The

essential supremum

of

f

is the smallest number

a

∈

ℝ

¯

for which

f

only exceeds

a

on a set of

measure zero

. This allows us to generalize the

maximum

of a

function

in a useful way.

More formally, we define

ess

⁢

sup

⁡

f

as follows. Let

a

∈

ℝ

, and define

M

a

=

{

x

:

f

⁢

(

x

)

>

a

}

,

the subset of

X

where

f

⁢

(

x

)

is

greater

than

a

. Then let

A

0

=

{

a

∈

ℝ

:

μ

⁢

(

M

a

)

=

0

}

,

the set of real numbers for which

M

a

has measure zero. The essential supremum of

f

is

ess

⁢

sup

⁡

f

:=

inf

⁡

A

0

.

The

supremum

is taken in the set of extended real numbers so,

ess

⁢

sup

⁡

f

=

∞

if

A

0

=

∅

and

ess

⁢

sup

⁡

f

=

-

∞

if

A

0

=

ℝ

.

Essential supremum of a collection of functions

Let

(

Ω

,

ℱ

,

μ

)

be a measure space, and

𝒮

be a

collection

of

measurable functions

f

:

Ω

→

ℝ

¯

. The Borel

σ

-algebra on

ℝ

¯

is used.

If

𝒮

is

countable

then we can define the

pointwise

supremum of the functions in

𝒮

, which will itself be

measurable

. However, if

𝒮

is

uncountable

then this is often not useful, and does not even have to be measurable. Instead, the

essential supremum

can be used.

The essential supremum of

𝒮

, written as

ess

⁢

sup

⁡

𝒮

, if it exists, is a measurable function

f

:

Ω

→

ℝ

¯

satisfying the following.

•

f

≥

g

,

μ

-almost everywhere (

http://planetmath.org/AlmostSurely

), for any

g

∈

𝒮

.

•

if

g

:

Ω

→

ℝ

¯

is measurable and

g

≥

h

(

μ

-a.e.) for every

h

∈

𝒮

, then

g

≥

f

(

μ

-a.e.).

Similarly, the

essential infimum

,

ess

⁢

inf

⁡

𝒮

is defined by replacing the

inequalities

‘

≥

’ by ‘

≤

’ in the above definition.

Note that if

f

is the essential supremum and

g

:

Ω

→

ℝ

¯

is equal to

f

μ

-almost everywhere, then

g

is also an essential supremum.

Conversely

, if

f

,

g

are both essential supremums then, from the above definition,

f

≤

g

and

g

≤

f

, so

f

=

g

(

μ

-a.e.). So, the essential supremum (and the essential infimum), if it exists, is only defined almost everywhere.

It can be shown that, for a

σ

-finite

measure

μ

, the essential supremum and essential infimum always exist (

http://planetmath.org/ExistenceOfTheEssentialSupremum

). Furthermore, they are always equal to the supremum or

infimum

of some countable subset of

𝒮

.

Title

essential supremum

Canonical name

EssentialSupremum

Date of creation

2013-03-22 12:21:29

Last modified on

2013-03-22 12:21:29

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

9

Author

gel (22282)

Entry type

Definition

Classification

msc 28C20

Synonym

ess-sup

Synonym

ess sup

Related topic

Supremum

Related topic

LpSpace

Related topic

ExistenceOfTheEssentialSupremum

Defines

essential infimum

Defines

ess-inf

Defines

ess inf

Generated on Fri Feb 9 20:11:34 2018 by

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ergodic

ergodic

\PMlinkescapephrase

invariant

1

Ergodicity

Definition -

Let

(

X

,

𝔅

,

μ

)

be a

probability space

and

T

:

X

⟶

X

a

measure-preserving transformation

. We say that

T

is

ergodic

if all the subsets

A

∈

𝔅

such that

T

-

1

⁢

(

A

)

=

A

have

measure

0

or

1

.

In other words, if

A

∈

𝔅

is invariant (

http://planetmath.org/InvariantByAMeasurePreservingTransformation

) by

T

then

μ

⁢

(

A

)

=

0

or

μ

⁢

(

A

)

=

1

.

1.0.1

Motivation

Suppose

(

X

,

𝔅

,

μ

)

is a probability space and

T

:

X

⟶

X

is a measure-preserving transformation. If

A

∈

𝔅

is an invariant (

http://planetmath.org/InvariantByAMeasurePreservingTransformation

) measurable subset, with

0

<

μ

⁢

(

A

)

<

1

,
then

X

∖

A

is also invariant and

0

<

μ

⁢

(

X

∖

A

)

<

1

. Thus, in this situation, we can study the

transformation

T

by studying the two simpler transformations

T

|

A

and

T

|

X

∖

A

in the spaces

A

and

X

∖

A

, respectively.

The transformation

T

is ergodic precisely when

T

cannot be decomposed into simpler transformations. Thus,

ergodic transformations

are the

measure-preserving transformations, in the sense described above.

Remark:

When the invariant

A

∈

𝔅

has measure

μ

⁢

(

A

)

=

0

we can ignore it (as usual in

measure theory

), as its presence does not affect

T

significantly. Thus, the study of

T

is not simplified when restricting to

X

∖

A

.

2

Examples

•

The

identity transformation

in a probability space

(

X

,

𝔅

,

μ

)

is ergodic if (and only if) all

measurable sets

have measure

0

or

1

.

•

Let

𝕋

be the

unit circle

in

ℂ

, endowed with the arc length

Lebesgue measure

(or

Haar measure

). The transformation of

𝕋

given by

S

⁢

(

x

)

=

a

⁢

x

, where

a

∈

𝕋

, is ergodic if and only if

a

is not a root of unity.

Title

ergodic

Canonical name

Ergodic

Date of creation

2013-03-22 12:19:38

Last modified on

2013-03-22 12:19:38

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Definition

Classification

msc 28D05

Classification

msc 37A25

Synonym

ergodicity

Synonym

ergodic transformation

Synonym

ergodic map

Related topic

Measure

Related topic

ErgodicTheorem

Related topic

MeasurePreserving

Generated on Fri Feb 9 20:11:36 2018 by

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entropy of a partition

entropy of a partition

Let

(

X

,

ℬ

,

μ

)

be a

probability space

. A

measurable partition

of

X

is a

partition

such that each of its
elements is a

measurable set

(i.e. an element of

ℬ

).

Given a finite measurable partition

𝒫

,
its

is

H

μ

⁢

(

𝒫

)

=

∑

P

∈

𝒫

-

μ

⁢

(

P

)

⁢

log

⁡

μ

⁢

(

P

)

,

where we assume

0

⁢

log

⁡

0

=

0

for convenience.

Remarks.

1.

Entropy

can be interpreted as a

measure

of the

a priori

uncertainity about the
outcome of the measurement an experiment, assuming that we are measuring it through the given partition
(i.e., we are going to be told in which atom of the partition the result is).
Thus, the finer a partition is, the higher the resulting entropy. In particular, the trivial
partition

{

X

}

has entropy

0

, since there is only one possible outcome, so there is no
uncertainity at all. On the other hand, the measurement gives no information at all
about the “real” outcome of the experiment, which reflects the complementary intepretation of entropy:
as the information gained from the measurement.
This is because of the intuitive fact that more
uncertainity about the outcome of the measurement means that more information will be obtained from
knowing it about the “real” outcome.

2.

Equally intuitive is the fact that among all measurable partitions of

X

into

n

atoms, the maximum
possible
entropy is attained at those in which the atoms are equally likely (i.e., all atoms have equal
measure

1

/

n

).
This can be proved by means of standard calculus, and
a direct computation shows that the maximum value is

log

⁡

n

.

3.

Since the definition of entropy involves only the measure of atoms of the given partition, two
partitions which are equal modulo

measure zero

have the same entropy.

4.

There is a natural correspondence between finite measurable partitions and finite sub-

σ

-algebras of

ℬ

. For this reason,
to each finite sub-

σ

-algebra

𝒫

we can define its entropy by

H

μ

⁢

(

𝒫

)

where

𝒫

is the (unique) partition which generates

𝒫

. For short, we denote this entropy by

H

μ

⁢

(

𝒫

)

.

Title

entropy of a partition

Canonical name

EntropyOfAPartition

Date of creation

2013-03-22 14:31:56

Last modified on

2013-03-22 14:31:56

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 28D20

Classification

msc 37A35

Related topic

HartleyFunction

Defines

entropy of a

σ

-algebra

Defines

entropy of a

sigma-algebra

Defines

measurable partition

Generated on Fri Feb 9 20:11:40 2018 by

LaTeXML
metric entropy

metric entropy

Let

(

X

,

ℬ

,

μ

)

be a

probability space

, and

T

:

X

→

X

a

measure-preserving transformation

.
The

entropy

of

T

with respect to a finite

measurable partition

𝒫

is

h

μ

⁢

(

T

,

𝒫

)

=

lim

n

→

∞

⁡

H

μ

⁢

(

⋁

k

=

0

n

-

1

T

-

k

⁢

𝒫

)

,

where

H

μ

is the entropy of a

partition

and

∨

denotes the join of partitions.
The above limit always exists, although it can be

+

∞

.
The entropy of

T

is then defined as

h

μ

⁢

(

T

)

=

sup

𝒫

⁡

h

μ

⁢

(

T

,

𝒫

)

,

with the supremum taken over all finite measurable partitions.
Sometimes

h

μ

⁢

(

T

)

is called the metric or measure theoretic entropy of

T

, to differentiate it from

topological entropy

.

Remarks.

1.

There is a natural correspondence between finite measurable partitions and finite
sub-

σ

-

algebras

of

ℬ

. Each finite sub-

σ

-algebra is

generated by

a unique partition, and clearly each finite partition generates a finite

σ

-algebra.
Because of this, sometimes

h

μ

⁢

(

T

,

𝒫

)

is called the entropy of

T

with respect to
the

σ

-algebra

𝒫

generated by

𝒫

, and denoted by

h

μ

⁢

(

T

,

𝒫

)

.
This simplifies the notation in some instances.

Title

metric entropy

Canonical name

MetricEntropy

Date of creation

2013-03-22 14:31:59

Last modified on

2013-03-22 14:31:59

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 28D20

Classification

msc 37A35

Synonym

entropy

Synonym

measure theoretic entropy

Generated on Fri Feb 9 20:11:42 2018 by

LaTeXML
proof that a finite collection of sets will not suffice

proof that a finite collection of sets will not suffice

Suppose that you cut

[

0

,

1

]

into

A

0

,

…

,

A

n

.
Displacing the parts is simply translating them; you can suppose
that you leave

A

0

in place and

translate

all the others to the
right. Let

ϵ

be the smallest

translation

length : if after
translation the union contains

[

0

,

1

]

, necessarily

[

0

,

ϵ

]

⊂

A

0

. A

contradiction

ensues.

Title

proof that a finite

collection

of sets will not suffice

Canonical name

ProofThatAFiniteCollectionOfSetsWillNotSuffice

Date of creation

2013-03-22 14:38:46

Last modified on

2013-03-22 14:38:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 28E99

Generated on Fri Feb 9 20:11:48 2018 by

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