aarjaneiro/planetmath huggingface dataset:
sequence of sets convergence

sequence of sets convergence

Let

{

A

n

}

n

=

1

∞

be a sequence of sets,
and

A

a set.

The sequence

{

A

n

}

n

=

1

∞

is said to

from below

to

A

, (shortly,

A

n

↑

A

or

A

n

↗

A

), iff

1)

A

n

⊆

A

n

+

1

∀

n

≥

1

2)

A

=

⋃

n

=

1

∞

A

n

The sequence

{

A

n

}

n

=

1

∞

is said to

from above

to

A

, (shortly,

A

n

↓

A

or

A

n

↘

A

), iff

1)

A

n

+

1

⊆

A

n

∀

n

≥

1

2)

A

=

⋂

n

=

1

∞

A

n

In both cases the less accurate notation

A

=

lim

n

⟶

∞

⁡

A

n

is also used.

Title

sequence of sets convergence

Canonical

name

SequenceOfSetsConvergence

Date of creation

2013-03-22 16:15:12

Last modified on

2013-03-22 16:15:12

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

8

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 28A05

Generated on Fri Feb 9 20:05:52 2018 by

LaTeXML
sigma–ring of sets

sigma–ring of sets

A

σ

-

ring of sets

is a nonempty

collection

𝒮

of sets such that

•

if

A

∈

𝒮

and

B

∈

𝒮

then

A

-

B

∈

𝒮

and

•

if

A

i

∈

𝒮

for

i

=

1

,

2

⁢

…

,

then

∪

i

=

1

∞

A

i

∈

𝒮

A

σ

-ring is also

closed under

countable

intersections

since

∩

i

=

1

∞

A

i

=

A

-

∪

i

=

1

∞

(

A

-

A

i

)

where

A

=

∪

i

=

1

∞

A

i

.

σ

-rings are used in

measure theory

.

Title

sigma–ring of sets

Canonical

name

SigmaringOfSets

Date of creation

2013-03-22 17:04:34

Last modified on

2013-03-22 17:04:34

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 28A05

Generated on Fri Feb 9 20:05:53 2018 by

LaTeXML
Souslin scheme

Souslin scheme

A

Souslin scheme

is a method of representing and defining

analytic sets

on a

paved space

(

X

,

ℱ

)

.
Let

𝒮

be the

collection

of

finite sequences

of positive integers. That is

𝒮

is the

disjoint union

of

ℕ

n

for

n

=

1

,

2

,

…

.

A Souslin scheme on

ℱ

is a collection

(

A

s

)

s

∈

𝒮

of sets in

ℱ

.
If

𝒩

=

ℕ

ℕ

is

Baire space

then, for any

s

∈

𝒩

and

n

∈

ℕ

, we write

s

|

n

≡

(

s

1

,

…

,

s

n

)

for the

restriction

of

s

to

{

1

,

…

,

n

}

. So,

s

|

n

∈

ℕ

n

.

The

result

of the Souslin scheme

(

A

s

)

is defined to be

A

=

⋃

s

∈

𝒩

⋂

n

=

1

∞

A

s

|

n

.

The set

𝒮

can be partially ordered as follows. Say that

s

≤

t

if

s

∈

ℕ

r

and

t

∈

ℕ

s

for

r

≤

s

, and

s

k

=

t

k

for

k

=

1

,

…

,

r

.
The scheme

(

A

s

)

is said to be

regular

if

A

s

⊇

A

t

for all

s

≤

t

.

It can be shown that the result of a Souslin scheme is

ℱ

-

analytic

and,

conversely

, any analytic set is the result of some scheme (see

equivalent definitions of analytic sets

).

References

1

Jean Bourgain,

A stabilization property and its applications in the theory of

sections

. Séminaire Choquet. Initiation à l’analyse, 17 no. 1 (1977).

Title

Souslin scheme

Canonical name

SouslinScheme

Date of creation

2013-03-22 18:48:30

Last modified on

2013-03-22 18:48:30

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 28A05

Synonym

Suslin scheme

Defines

regular scheme

Defines

result of a scheme

Generated on Fri Feb 9 20:05:56 2018 by

LaTeXML
sums of compact pavings are compact

sums of compact pavings are compact

Suppose that

(

K

i

,

𝒦

i

)

is a

paved space

for each

i

in an

index set

I

. The

direct sum

, or

disjoint union

(

http://planetmath.org/DisjointUnion

),

∑

i

∈

I

K

i

is the union of the

disjoint sets

K

i

×

{

i

}

. The direct sum of the paving

𝒦

i

is defined as

∑

i

∈

I

𝒦

i

=

{

∑

i

∈

I

S

i

:

S

i

∈

𝒦

i

∪

{

∅

}

⁢

is empty for all but finitely many

⁢

i

}

.

Theorem

.

Let

(

K

i

,

K

i

)

be

compact

paved spaces for

i

∈

I

. Then,

∑

i

K

i

is a compact paving on

∑

i

K

i

.

The paving

𝒦

′

consisting of subsets of

∑

i

𝒦

i

of the form

∑

i

S

i

where

S

i

=

∅

for all but a single

i

∈

I

is easily shown to be compact.
Indeed, if

𝒦

′′

⊆

𝒦

′

satisfies

the

finite intersection property

then there is an

i

∈

I

such that

S

⊆

K

i

×

{

i

}

for every

S

∈

𝒦

′′

.

Compactness

of

𝒦

i

gives

⋂

𝒦

′′

≠

∅

.

Then, as

∑

i

𝒦

i

consists of finite unions of sets in

𝒦

′

, it is a compact paving (see compact pavings are

closed subsets

of a

compact space

).

Title

sums of compact pavings are compact

Canonical name

SumsOfCompactPavingsAreCompact

Date of creation

2013-03-22 18:45:15

Last modified on

2013-03-22 18:45:15

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Synonym

disjoint unions of compact pavings are compact

Related topic

ProductsOfCompactPavingsAreCompact

Defines

direct sum of pavings

Defines

disjoint union of pavings

Generated on Fri Feb 9 20:05:58 2018 by

LaTeXML
tail event

tail event

Tail eventFernando Sanz Gamiz

Definition

.

Let

Ω

be a set and

ℱ

a

sigma algebra

of subsets
of

Ω

. Given the

random variables

{

X

n

,

n

∈

ℕ

}

, defined
on the

measurable space

(

Ω

,

ℱ

)

, the

tail
events

are the events of the

tail

σ

-algebra

ℱ

∞

=

⋂

n

=

1

∞

σ

⁢

(

X

n

,

X

n

+

1

,

⋯

)

where

σ

⁢

(

X

n

,

X

n

+

1

,

⋯

)

is the

σ

-algebra induced by

(

X

n

,

X

n

+

1

,

⋯

)

.

Remark 1

.

One can intuitively think of tail events as those events whose
ocurrence or not is not affected by altering any finite number of
random variables in the

sequence

. Some examples are

{

lim

sup

X

n

<

c

}

,

∑

X

n

converges

,

lim

X

n

exists

Remark 2

.

One of the most important

theorems

in probability theory due to
Kolomogorv, is the

Kolmogorov zero-one law

which states that, in the case of

independent

random variables, the

probability

of any tail event is 0 or 1 (provided there is a

probability measure

defined on

(

Ω

,

ℱ

)

)

Title

tail event

Canonical name

TailEvent

Date of creation

2013-03-22 17:07:18

Last modified on

2013-03-22 17:07:18

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

9

Author

fernsanz (8869)

Entry type

Definition

Classification

msc 28A05

Related topic

SigmaAlgebra

Related topic

KolmogorovZeroOneLaw

Defines

tail sigma algebra

Generated on Fri Feb 9 20:06:01 2018 by

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Argand diagram

Argand diagram

An

Argand diagram

is the graphical representation of

complex numbers

written in polar coordinates. For example, if

z

∈

ℂ

is a complex number, then

z

can be written as

r

⁢

e

i

⁢

θ

, where

r

is the length of

z

considered as a vector

(

x

,

y

)

∈

ℝ

2

, with

z

=

x

+

y

⁢

i

, and

θ

is the value such that

tan

⁡

θ

=

y

x

, and can be interpreted as the angle

z

makes with the

x

-axis. The Argand diagram of

z

is thus

Argand is the name of Jean-Robert Argand, the Frenchman who is credited with the geometric

interpretation

of the
complex numbers

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Argand.html

[Biography]

Title

Argand diagram

Canonical name

ArgandDiagram

Date of creation

2013-03-22 11:57:25

Last modified on

2013-03-22 11:57:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 28A10

Classification

msc 30-00

Related topic

Complex

Generated on Fri Feb 9 20:06:10 2018 by

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Hahn-Kolmogorov theorem

Hahn-Kolmogorov theorem

Let

𝒜

0

be an

algebra

of subsets of a set

X

. If a

finitely additive

measure

μ

0

:

𝒜

→

ℝ

∪

{

∞

}

satisfies

μ

0

⁢

(

⋃

n

=

1

∞

A

n

)

=

∑

n

=

1

∞

μ

0

⁢

(

A

n

)

for any

disjoint

family

{

A

n

:

n

∈

ℕ

}

of elements of

𝒜

0

such that

∪

n

=

0

∞

A

n

∈

𝒜

0

, then

μ

0

extends to a measure defined on the

σ

-algebra

𝒜

generated by

𝒜

0

; i.e. there exists a measure

μ

:

𝒜

→

ℝ

∪

{

∞

}

such that its

restriction

(

http://planetmath.org/RestrictionOfAFunction

) to

𝒜

0

coincides with

μ

0

.

If

μ

0

is

σ

-finite (

http://planetmath.org/SigmaFinite

), then the

extension

is unique.

Title

Hahn-Kolmogorov theorem

Canonical name

HahnKolmogorovTheorem

Date of creation

2013-03-22 14:03:10

Last modified on

2013-03-22 14:03:10

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 28A10

Synonym

Hahn-Kolmogorov extension theorem

Synonym

Kolmogorov extension theorem

Generated on Fri Feb 9 20:06:12 2018 by

LaTeXML
σ-finite

σ

-finite

A

measure space

(

Ω

,

ℬ

,

μ

)

is a

finite measure space

if

μ

⁢

(

Ω

)

<

∞

; it is

σ

-

finite

if the total space is the union of a finite or

countable

family of sets of finite

measure

, i.e. if there exists a

countable set

ℱ

⊂

ℬ

such that

μ

⁢

(

A

)

<

∞

for each

A

∈

ℱ

, and

Ω

=

⋃

A

∈

ℱ

A

.

In this case we also say that

μ

is a

σ

-finite measure.
If

μ

is not

σ

-finite, we say that it is

σ

-

infinite

.

Examples.

Any finite measure space is

σ

-finite. A more interesting example is the

Lebesgue measure

μ

in

ℝ

n

: it is

σ

-finite but not finite. In fact

ℝ

n

=

⋃

k

∈

ℕ

[

-

k

,

k

]

n

(

[

-

k

,

k

]

n

is a cube with center at

0

and side length

2

⁢

k

, and its measure is

(

2

⁢

k

)

n

), but

μ

⁢

(

ℝ

n

)

=

∞

.

Title

σ

-finite

Canonical name

sigmafinite

Date of creation

2013-03-22 12:29:48

Last modified on

2013-03-22 12:29:48

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

13

Author

Koro (127)

Entry type

Definition

Classification

msc 28A10

Synonym

σ

finite

Synonym

sigma-finite

Synonym

sigma finite

Related topic

Measure

Related topic

MeasureSpace

Related topic

AlternativeDefinitionOfSigmaFiniteMeasure

Related topic

AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure

Defines

σ

-infinite

Defines

sigma-infinite

Defines

finite measure space

Generated on Fri Feb 9 20:06:14 2018 by

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absolutely continuous

absolutely continuous

Let

μ

and

ν

be

signed measures

or

complex measures

on the same

measurable space

(

Ω

,

𝒮

)

. We say that

ν

is

absolutely continuous

with respect to

μ

if, for each

A

∈

𝒮

such that

|

μ

|

⁢

(

A

)

=

0

,
it holds that

ν

⁢

(

A

)

=

0

. This is usually denoted by

ν

≪

μ

.

Remarks.

If

μ

and

ν

are signed measures and

(

ν

+

,

ν

-

)

is the

Jordan decomposition

of

ν

, the following are

equivalent

:

1.

ν

≪

μ

;

2.

ν

+

≪

μ

and

ν

-

≪

μ

;

3.

|

ν

|

≪

|

μ

|

.

If

ν

is a finite signed or complex measure and

ν

≪

μ

, the following useful property holds: for each

ε

>

0

, there is a

δ

>

0

such that

|

ν

|

⁢

(

E

)

<

ε

whenever

|

μ

|

⁢

(

E

)

<

δ

.

Title

absolutely continuous

Canonical name

AbsolutelyContinuous

Date of creation

2013-03-22 13:26:12

Last modified on

2013-03-22 13:26:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 28A12

Related topic

RadonNikodymTheorem

Related topic

AbsolutelyContinuousFunction2

Defines

absolute continuity

Generated on Fri Feb 9 20:06:16 2018 by

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any σ-finite measure is equivalent to a probability measure

any

σ

-finite measure is equivalent to a probability measure

The following

theorem

states that for any

σ

-finite (

http://planetmath.org/SigmaFinite

)

measure

μ

, there is an

equivalent

probability measure

ℙ

— that is, the sets

A

satisfying

μ

⁢

(

A

)

=

0

are the same as those satisfying

ℙ

⁢

(

A

)

=

0

.
This result allows statements about probability measures to be generalized to arbitrary

σ

-finite measures.

Theorem

.

Any nonzero

σ

-finite measure

μ

on a

measurable space

(

X

,

A

)

is equivalent to a probability measure

P

on

(

X

,

A

)

. In particular, there is a positive

measurable function

f

:

X

→

(

0

,

∞

)

satisfying

∫

f

⁢

𝑑

μ

=

1

, and

P

⁢

(

A

)

=

∫

A

f

⁢

𝑑

μ

for all

A

∈

A

.

Proof.

Let

A

1

,

A

2

,

…

be a

sequence

in

𝒜

such that

μ

⁢

(

A

k

)

<

∞

and

⋃

k

A

k

=

X

. Then it is easily verified that

g

≡

∑

k

=

1

∞

2

-

k

⁢

1

A

k

1

+

μ

⁢

(

A

k

)

satisfies

1

≥

g

>

0

and

∫

g

⁢

𝑑

μ

<

∞

. So, setting

f

=

g

/

∫

g

⁢

𝑑

μ

, we have

∫

f

⁢

𝑑

μ

=

1

and therefore

ℙ

⁢

(

A

)

≡

∫

A

f

⁢

𝑑

μ

is a probability measure equivalent to

μ

.
∎

Title

any

σ

-finite measure is equivalent to a probability measure

Canonical name

AnysigmafiniteMeasureIsEquivalentToAProbabilityMeasure

Date of creation

2013-03-22 18:33:44

Last modified on

2013-03-22 18:33:44

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Classification

msc 28A10

Related topic

SigmaFinite

Related topic

Measure

Generated on Fri Feb 9 20:06:18 2018 by

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Bolyai-Gerwien theorem

Bolyai-Gerwien theorem

Theorem 1

(Bolyai-Gerwien Theorem).

Given two

polygons

with the same area, it is possible to cut up one polygon into a finite number of smaller polygonal pieces and from those pieces assemble into the other polygon.

To cast the

theorem

in more mathematically

rigorous

terms, let

P

be a polygon in a plane (

Euclidean plane

). A

dissection

or

polygonal decomposition

of

P

is a

finite set

of polygons

P

1

,

…

,

P

n

such that

P

=

P

1

∪

⋯

∪

P

n

and

P

i

∩

P

j

is either a

line segment

, a point, or the

empty set

. Two polygons

P

and

Q

are said to be

equidecomposable

(or

scissor-equivalent

) if there is a dissection

{

P

i

∣

i

=

1

,

…

,

n

}

of

P

and a finite set of

rigid motions

{

m

i

∣

i

=

1

,

…

,

n

}

such that

{

m

i

⁢

(

P

i

)

∣

i

=

1

,

…

,

n

}

is a dissection of

Q

. From this definition, it is easy to see that equidecomposability is an

equivalence relation

on the set of all polygons in a given plane. Furthermore, if two polygons are equidecomposable, they have the same area, and the

converse

of which is the statement of Bolyai-Gerwien Theorem.

One possible proof is to show that any polygon can be broken down into

triangles

. These triangles can be reassembled into

rectangles

, which can then be joined into one big rectangle. This big rectangle can then be reassembled into a square.

Remarks

. Farkas Bolyai conjectured this in the 1790s and William Wallace proved it in 1808. Unaware of this, Paul Gerwien proved it again in 1833, and then Bolyai, unaware of both earlier results, gave another proof in 1835.

David Hilbert wondered if the same could be done for any pair of

polyhedra

of equal volume (see the third of Hilbert’s problems). Dehn proved that this could not be done shortly after (Dehn’s Theorem). However, under some suitable conditions, Hadwiger proved that equidecomposability is still possible.

Title

Bolyai-Gerwien theorem

Canonical name

BolyaiGerwienTheorem

Date of creation

2013-03-22 16:32:19

Last modified on

2013-03-22 16:32:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 28A12

Synonym

Wallace-Bolyai-Gerwien theorem

Synonym

Bolyai-Gerwin theorem

Synonym

polygonal decomposition

Defines

dissection

Defines

equidecomposable

Generated on Fri Feb 9 20:06:21 2018 by

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Carathéodory’s extension theorem

Carathéodory’s extension theorem

In

measure theory

, Carathéodory’s extension theorem is an important result used in the construction of

measures

, such as the

Lebesgue measure

on the real number line. The result states that a

countably additive

(

http://planetmath.org/Additive

)

set function

on an

algebra of sets

can be extended to a measure on the

σ

-algebra (

http://planetmath.org/SigmaAlgebra

)

generated by

that algebra.

Theorem

(Carathéodory).

Let

X

be a set,

A

be an algebra on

X

, and

A

≡

σ

⁢

(

A

)

be the

σ

-algebra generated by

A

. If

μ

0

:

A

→

R

+

∪

{

∞

}

is a countably additive map then there exists a measure

μ

on

(

X

,

A

)

such that

μ

=

μ

0

on

A

.

References

1

David Williams,

Probability with martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

2

Olav Kallenberg,

Foundations

of modern probability

, Second edition. Probability and its Applications. Springer-Verlag, 2002.

Title

Carathéodory’s extension theorem

Canonical name

CaratheodorysExtensionTheorem

Date of creation

2013-03-22 18:33:00

Last modified on

2013-03-22 18:33:00

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

18

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Related topic

Measure

Related topic

OuterMeasure2

Related topic

LebesgueMeasure

Related topic

CaratheodorysLemma

Related topic

ExistenceOfTheLebesgueMeasure

Generated on Fri Feb 9 20:06:22 2018 by

LaTeXML
Carathéodory’s lemma

Carathéodory’s lemma

In

measure theory

, Carathéodory’s lemma is used for constructing

measures

and, for example, can be applied to the construction of the

Lebesgue measure

and is used in the proof of Carathéodory’s extension theorem.
The idea is that to define a measure on a

measurable space

(

X

,

𝒜

)

we would first construct an

outer measure

(

http://planetmath.org/OuterMeasure2

), which is a

set function

defined on the

power set

of

X

. Then, this outer measure is restricted to

𝒜

and Carathéodory’s lemma is applied to show that this

restriction

does in fact result in a measure. For an example of this procedure, see the proof of Carathéodory’s extension theorem.

Given an outer measure

μ

on a set

X

, the result first defines a

collection

of subsets of

X

— the

μ

-

measurable sets

.
A subset

S

⊆

X

is called

μ

-

measurable

(or Carathéodory measurable with respect to

μ

) if the equality

μ

⁢

(

E

)

=

μ

⁢

(

E

∩

S

)

+

μ

⁢

(

E

∩

S

c

)

holds for every

E

⊆

X

. Then, Caratheodory’s lemma says that a measure is obtained by restricting

μ

to the

μ

-measurable sets.

Lemma

(Carathéodory).

Let

μ

be an outer measure on a set

X

, and

A

be the class of

μ

-measurable sets. Then

A

is a

σ

-algebra (

http://planetmath.org/SigmaAlgebra

) and the restriction of

μ

to

A

is a measure.

It should be noted that for any outer measure

μ

and sets

S

,

E

⊆

X

, subadditivity of

μ

implies that the

inequality

μ

⁢

(

E

)

≤

μ

⁢

(

E

∩

S

)

+

μ

⁢

(

E

∩

S

c

)

is always satisfied. So, only the reverse inequality is required and consequently

S

is

μ

-measurable if and only if

μ

⁢

(

E

)

≥

μ

⁢

(

E

∩

S

)

+

μ

⁢

(

E

∩

S

c

)

for every

E

⊆

X

.

References

1

David Williams,

Probability

with

martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

2

Olav Kallenberg,

Foundations

of modern probability

, Second edition. Probability and its Applications. Springer-Verlag, 2002.

Title

Carathéodory’s lemma

Canonical name

CaratheodorysLemma

Date of creation

2013-03-22 18:33:03

Last modified on

2013-03-22 18:33:03

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

19

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Related topic

CaratheodorysExtensionTheorem

Related topic

OuterMeasure2

Related topic

LebesgueOuterMeasure

Related topic

ConstructionOfOuterMeasures

Related topic

ProofOfCaratheodorysLemma

Related topic

ProofOfCaratheodorysExtensionTheorem

Defines

Carathéodory measurable

Generated on Fri Feb 9 20:06:24 2018 by

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Choquet capacity

Choquet capacity

A

Choquet capacity

, or just

capacity

, on a set

X

is a kind of

set function

,

mapping

the

power set

𝒫

⁢

(

X

)

to the

real numbers

.

Definition

.

Let

F

be a

collection

of subsets of

X

. Then, an

F

-capacity is an

increasing

set function

I

:

𝒫

⁢

(

X

)

→

ℝ

+

satisfying the following.

1.

If

(

A

n

)

n

∈

ℕ

is an

increasing sequence

of subsets of

X

then

I

⁢

(

A

n

)

→

I

⁢

(

⋃

m

A

m

)

as

n

→

∞

.

2.

If

(

A

n

)

n

∈

ℕ

is a

decreasing sequence

of subsets of

X

such that

A

n

∈

ℱ

for each

n

, then

I

⁢

(

A

n

)

→

I

⁢

(

⋂

m

A

m

)

as

n

→

∞

.

The condition that

I

is increasing means that

I

⁢

(

A

)

≤

I

⁢

(

B

)

whenever

A

⊆

B

.
Note that capacities differ from the

concepts

of

measures

and

outer measures

, as no

additivity

or subadditivity conditions are imposed. However, for any finite measure, there is a corresponding capacity (

http://planetmath.org/CapacityGeneratedByAMeasure

). An important application to the theory of measures and

analytic sets

is given by the

capacitability theorem

.

The

(

F

,

I

)

-capacitable

sets are defined as follows. Recall that

ℱ

δ

denotes the collection of

countable

intersections of sets

in the paving

ℱ

.

Definition

.

Let

I

be an

F

-capacity on a set

X

. Then a subset

A

⊆

X

is

(

ℱ

,

I

)

-capacitable

if, for each

ϵ

>

0

, there exists a

B

∈

F

δ

such that

B

⊆

A

and

I

⁢

(

B

)

≥

I

⁢

(

A

)

-

ϵ

.

Alternatively, such sets are called

I

-capacitable or, simply, capacitable.

Title

Choquet capacity

Canonical name

ChoquetCapacity

Date of creation

2013-03-22 18:47:26

Last modified on

2013-03-22 18:47:26

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 28A12

Classification

msc 28A05

Synonym

capacity

Defines

capacitable

Generated on Fri Feb 9 20:06:26 2018 by

LaTeXML
complete measure

complete measure

A

measure space

(

X

,

𝒮

,

μ

)

is said to be

complete

if every subset of a set of

measure

0

is

measurable

(and consequently, has measure

0

); i.e. if for all

E

∈

𝒮

such that

μ

⁢

(

E

)

=

0

and for all

S

⊂

E

we have

μ

⁢

(

S

)

=

0

.

If a measure space is not complete, there exists a

completion

(

http://planetmath.org/CompletionOfAMeasureSpace

) of it, which is a complete measure space

(

X

,

𝒮

¯

,

μ

¯

)

such that

𝒮

⊂

𝒮

¯

and

μ

¯

|

𝒮

=

μ

, where

𝒮

¯

is the smallest

σ

-algebra containing both

𝒮

and all subsets of elements of zero measure of

𝒮

.

Title

complete measure

Canonical name

CompleteMeasure

Date of creation

2013-03-22 14:06:56

Last modified on

2013-03-22 14:06:56

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 28A12

Related topic

UniversallyMeasurable

Defines

completion

Defines

complete

Generated on Fri Feb 9 20:06:28 2018 by

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