aarjaneiro/planetmath huggingface dataset:
Simpson’s rule

Simpson’s rule

Simpson’s rule

is a method of (approximate) numerical definite integration (or

quadrature

). Simpson’s rule is based on a parabolic model of the

function

to be integrated (in contrast to the trapezoidal model of the

trapezoidal rule

). Thus, a

minimum

of three points and three

function values

are required. Here we take three equidistant points:

x

0

⁢

x

2

the

interval

endpoints

,

x

1

=

(

x

0

+

x

2

)

/

2

the

midpoint

, and let

h

=

|

b

-

a

|

/

2

the

distance

between each. The

definite integral

is then approximated by:

∫

x

0

x

2

f

⁢

(

x

)

⁢

𝑑

x

≈

I

=

h

3

⁢

(

f

⁢

(

x

0

)

+

4

⁢

f

⁢

(

x

1

)

+

f

⁢

(

x

2

)

)

We can extend this to

greater

precision

by breaking our target domain into

n

equal-length fragments. The quadrature is then the weighted sum of the above formula for every pair of

adjacent

regions, which works out for even

n

to

I

=

h

3

⁢

(

f

⁢

(

x

0

)

+

4

⁢

f

⁢

(

x

1

)

+

2

⁢

f

⁢

(

x

2

)

+

4

⁢

f

⁢

(

x

3

)

+

⋯

+

4

⁢

f

⁢

(

x

n

-

3

)

+

2

⁢

f

⁢

(

x

n

-

2

)

+

4

⁢

f

⁢

(

x

n

-

1

)

+

f

⁢

(

x

n

)

)

Title

Simpson’s rule

Canonical name

SimpsonsRule

Date of creation

2013-03-22 13:40:12

Last modified on

2013-03-22 13:40:12

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Theorem

Classification

msc 28-00

Classification

msc 26A06

Classification

msc 41A55

Classification

msc 65D32

Related topic

LagrangeInterpolationFormula

Related topic

NewtonAndCotesFormulas

Related topic

Prismatoid

Generated on Fri Feb 9 20:04:41 2018 by

LaTeXML
an integrable function that does not tend to zero

an integrable function that does not tend to zero

In this entry, we give an example of a

function

f

such that

f

is

Lebesgue integrable

on

[

0

,

∞

)

but

f

⁢

(

x

)

does not tend to zero as

x

→

∞

.

First of all, let

g

n

be the function

sin

⁡

(

2

n

⁢

x

)

⁢

χ

[

0

,

π

2

n

]

, where

χ

I

denotes the

characteristic function

of the

interval

I

. In other words,

χ

takes the value

1

on

I

and 0 everywhere else.

Let

μ

denote

Lebesgue measure

. An easy

computation

shows

∫

ℝ

g

n

⁢

𝑑

μ

=

2

1

-

n

,

(1)

and

g

n

⁢

(

π

2

n

+

1

)

=

1

. Let

h

n

⁢

(

x

)

=

g

n

⁢

(

x

-

n

⁢

π

)

, so

h

n

is just a “shifted” version of

g

n

. Note that

h

n

⁢

(

n

⁢

π

+

π

2

n

+

1

)

=

1

.

(2)

We now construct our function

f

by defining

f

=

∑

r

=

0

∞

h

r

. There are no convergence problems with this sum since for a given

x

∈

ℝ

, at most one

h

r

takes a non-zero value at

x

. Also

f

⁢

(

x

)

does not tend to 0 as

x

→

∞

as there are arbitrarily large values of

x

for which

f

takes the value

1

, by (

2

).

All that is left is to show that

f

is Lebesgue integrable. To do this rigorously, we apply the

monotone convergence theorem

(MCT) with

f

n

=

∑

r

=

0

n

h

r

. We must check the hypotheses of the MCT. Clearly

f

n

→

f

as

n

→

∞

, and the

sequence

(

f

n

)

is

monotone increasing

,

positive

, and

integrable

. Furthermore, each

f

n

is

continuous

and zero except on a compact interval, so is integrable. Finally, from (

1

) we see that

∫

ℝ

f

n

⁢

𝑑

μ

≤

4

for all

n

. Therefore, the MCT applies and

f

is integrable.

Title

an

integrable function that does not tend to zero

Canonical name

AnIntegrableFunctionThatDoesNotTendToZero

Date of creation

2013-03-22 16:56:09

Last modified on

2013-03-22 16:56:09

Owner

silverfish (6603)

Last modified by

silverfish (6603)

Numerical id

9

Author

silverfish (6603)

Entry type

Example

Classification

msc 28-01

Generated on Fri Feb 9 20:04:43 2018 by

LaTeXML
example of a non Riemann integrable function

example of a non Riemann integrable function

Let

[

a

,

b

]

be any

closed interval

and
consider the Dirichlet’s function

f

:

[

a

,

b

]

→

ℝ

f

⁢

(

x

)

=

{

1

if

x

is rational

0

otherwise

.

Then

f

is not

Riemann integrable

. In fact given any

interval

[

x

1

,

x

2

]

⊂

[

a

,

b

]

with

x

1

<

x

2

one has

sup

[

x

1

,

x

2

]

⁡

f

⁢

(

x

)

=

1

,

inf

[

x

1

,

x

2

]

⁡

f

⁢

(

x

)

=

0

because every interval contains both

rational and irrational

points.
So all

upper Riemann sums

are equal to

1

and all lower Riemann sums are equal to

0

.

Title

example of a non Riemann integrable function

Canonical name

ExampleOfANonRiemannIntegrableFunction

Date of creation

2013-03-22 15:03:28

Last modified on

2013-03-22 15:03:28

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Example

Classification

msc 28-XX

Classification

msc 26-XX

Generated on Fri Feb 9 20:04:45 2018 by

LaTeXML
a Lebesgue measurable but non-Borel set

a Lebesgue measurable but non-Borel set

We give an example of a subset of the

real numbers

which is

Lebesgue measurable

, but not

Borel measurable

.

Let

S

consist of the set of all

irrational

real numbers with

continued fraction

of the form

x

=

a

0

+

1

a

1

+

1

a

2

+

1

⋱

such that there exists an

infinite

sequence

0

<

i

1

<

i

2

<

⋯

where each

a

i

k

divides

a

i

k

+

1

.
It can be shown that this set is Lebesgue measurable, but not Borel measurable.

In fact, it can be shown that

S

is an

analytic set

(

http://planetmath.org/AnalyticSet2

), meaning that it is the image of a

continuous function

f

:

X

→

ℝ

for some

Polish space

X

and, consequently,

S

is a universally measurable set.

This example is due to Lusin (1927).

Title

a

Lebesgue measurable but non-Borel set

Canonical name

ALebesgueMeasurableButNonBorelSet

Date of creation

2013-03-22 18:37:01

Last modified on

2013-03-22 18:37:01

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Classification

msc 28A20

Generated on Fri Feb 9 20:04:47 2018 by

LaTeXML
analytic sets define a closure operator

analytic sets define a closure operator

For a paving

ℱ

on a set

X

, we denote the

collection

of all

ℱ

-

analytic sets

(

http://planetmath.org/AnalyticSet2

) by

a

⁢

(

ℱ

)

.
Then,

ℱ

↦

a

⁢

(

ℱ

)

is a

closure operator

on the subsets of

X

.
That is,

1.

ℱ

⊆

a

⁢

(

ℱ

)

.

2.

If

ℱ

⊆

𝒢

then

a

⁢

(

ℱ

)

⊆

a

⁢

(

𝒢

)

.

3.

a

⁢

(

a

⁢

(

ℱ

)

)

=

a

⁢

(

ℱ

)

.

For example, if

𝒢

is a collection of

ℱ

-analytic sets then

𝒢

⊆

a

⁢

(

ℱ

)

gives

a

⁢

(

𝒢

)

⊆

a

⁢

(

a

⁢

(

ℱ

)

)

=

a

⁢

(

ℱ

)

and so all

𝒢

-analytic sets are also

ℱ

-

analytic

. In particular, for a

metric space

, the analytic sets are the same regardless of whether they are defined with respect to the collection of open, closed or

Borel sets

.

Properties

1

and

2

follow directly from the definition of analytic sets. We just need to prove

3

. So, for any

A

∈

a

⁢

(

a

⁢

(

ℱ

)

)

we show that

A

∈

a

⁢

(

ℱ

)

. First, there is a

compact

paved space

(

http://planetmath.org/PavedSpace

)

(

K

,

𝒦

)

and

S

∈

(

a

⁢

(

ℱ

)

×

𝒦

)

σ

⁢

δ

such that

A

is equal to the projection

π

X

⁢

(

S

)

.
Write

S

=

⋂

m

=

1

∞

⋃

n

=

1

∞

A

m

,

n

×

B

m

,

n

for

A

m

,

n

∈

a

⁢

(

ℱ

)

and

B

m

,

n

∈

𝒦

. It is clear that

A

m

,

n

×

B

m

,

n

is

ℱ

×

𝒦

-analytic and, as

countable unions and intersections of analytic sets are analytic

,

S

is also

ℱ

×

𝒦

-analytic. Finally, since

projections of analytic sets are analytic

,

A

=

π

X

⁢

(

S

)

must be

ℱ

-analytic as required.

Title

analytic sets define a closure operator

Canonical name

AnalyticSetsDefineAClosureOperator

Date of creation

2013-03-22 18:46:30

Last modified on

2013-03-22 18:46:30

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Generated on Fri Feb 9 20:04:51 2018 by

LaTeXML
atom (measure theory)

atom (measure theory)

Let

(

X

,

ℬ

,

μ

)

be a

measure space

. A set

A

∈

ℬ

is called an

atom

if

A

has

positive measure

and contains no measurable subsets

B

⊂

A

such that

0

<

μ

⁢

(

B

)

<

μ

⁢

(

A

)

.

An

equivalent

definition

can be:

A

has positive measure and for every measurable subset

B

⊂

A

, either

μ

⁢

(

B

)

=

0

or

μ

⁢

(

A

-

B

)

=

0

.

Title

atom (

measure theory

)

Canonical

name

AtommeasureTheory

Date of creation

2013-03-22 17:38:31

Last modified on

2013-03-22 17:38:31

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Definition

Classification

msc 28A05

Synonym

atom

Generated on Fri Feb 9 20:04:52 2018 by

LaTeXML
Borel σ-algebra

Borel

σ

-algebra

For any

topological space

X

, the

Borel sigma algebra

of

X

is the

σ

–algebra

ℬ

generated by

the

open sets

of

X

. In other words, the Borel sigma algebra is equal to the

intersection

of all sigma algebras

𝒜

of

X

having the property that every open set of

X

is an

element of

𝒜

.

An element of

ℬ

is called a

Borel subset

of

X

, or a

Borel set

.

Title

Borel

σ

-algebra

Canonical name

Borelsigmaalgebra

Date of creation

2013-03-22 12:00:31

Last modified on

2013-03-22 12:00:31

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 28A05

Synonym

Borel

σ

algebra

Synonym

Borel sigma algebra

Related topic

SigmaAlgebra

Related topic

OuterRegular

Related topic

LebesgueMeasure

Defines

Borel subset

Defines

Borel set

Generated on Fri Feb 9 20:04:54 2018 by

LaTeXML
capacity generated by a measure

capacity generated by a measure

Any finite

measure

(

http://planetmath.org/SigmaFinite

) can be extended to a

set function

on the

power set

of the underlying space. As the following result states, this will be a

Choquet capacity

.

Theorem

.

Let

(

X

,

F

,

μ

)

be a

finite measure space

. Then,

μ

*

:

𝒫

⁢

(

X

)

→

ℝ

+

,

μ

*

⁢

(

S

)

=

inf

⁡

{

μ

⁢

(

A

)

:

A

∈

ℱ

,

A

⊇

S

}

is an

F

-capacity. Furthermore, a subset

S

⊆

X

is

(

F

,

μ

*

)

-capacitable if and only if it is in the

completion

(

http://planetmath.org/CompleteMeasure

) of

F

with respect to

μ

.

Note that, as well as being a capacity,

μ

*

is also an

outer measure

(see here (

http://planetmath.org/ConstructionOfOuterMeasures

)), which does not require the finiteness of

μ

.
Clearly,

μ

*

⁢

(

A

)

=

μ

⁢

(

A

)

for all

A

∈

ℱ

, so

μ

*

is an

extension

of

μ

to the power set of

X

, and is referred to as the

outer measure generated by

μ

.

Recall that a subset

S

⊆

X

is in the completion of

ℱ

with respect to

μ

if and only if there are sets

A

,

B

∈

ℱ

with

A

⊆

S

⊆

B

and

μ

⁢

(

B

∖

A

)

=

0

which, by the above

theorem

, is

equivalent

to the capacitability of

S

.

Title

capacity

generated by

a measure

Canonical name

CapacityGeneratedByAMeasure

Date of creation

2013-03-22 18:47:35

Last modified on

2013-03-22 18:47:35

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Classification

msc 28A12

Synonym

outer measure generated by a measure

Defines

outer measure generated by

Generated on Fri Feb 9 20:04:56 2018 by

LaTeXML
Choquet’s capacitability theorem

Choquet’s capacitability theorem

Choquet’s

capacitability theorem

states that

analytic sets

(

http://planetmath.org/AnalyticSet2

) are

capacitable

.

Theorem

(Choquet).

Let

F

be a paving that is

closed under

finite unions and finite

intersections

. If

I

is an

F

-capacity, then all

F

-analytic sets are

(

F

,

I

)

-capacitable.

A useful

consequence

of this result for applicatons to

measure theory

is the

universal

measurability of analytic sets

(

http://planetmath.org/MeasurabilityOfAnalyticSets

).

Title

Choquet’s capacitability theorem

Canonical name

ChoquetsCapacitabilityTheorem

Date of creation

2013-03-22 18:47:49

Last modified on

2013-03-22 18:47:49

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Classification

msc 28A12

Synonym

capacitability theorem

Generated on Fri Feb 9 20:04:57 2018 by

LaTeXML
compact pavings are closed subsets of a compact space

compact pavings are closed subsets of a compact space

Recall that a paving

𝒦

is

compact

if every subcollection satisfying the

finite intersection property

has nonempty

intersection

. In particular, a

topological space

is compact (

http://planetmath.org/Compact

) if and only if its

collection

of

closed subsets

forms a compact paving. Compact paved spaces can therefore be constructed by taking closed subsets of a compact topological space. In fact, all compact pavings arise in this way, as we now show.

Given any compact paving

𝒦

the following result says that the collection

𝒦

′

of all intersections of finite unions of sets in

𝒦

is also compact.

Theorem 1

.

Suppose that

(

K

,

K

)

is a compact paved space. Let

K

′

be the smallest collection of subsets of

X

such that

K

⊆

K

′

and which is

closed under

arbitrary intersections and finite unions.
Then,

K

′

is a compact paving.

In particular,

𝒯

≡

{

K

∖

C

:

C

∈

𝒦

′

}

∪

{

∅

,

K

}

is closed under arbitrary unions and finite intersections, and hence is a

topology

on

K

.
The collection of

closed sets

defined with respect to this topology is

𝒦

′

∪

{

∅

,

K

}

which, by

Theorem

1

, is a compact paving. So, the following is obtained.

Corollary

.

A paving

(

K

,

K

)

is compact if and only if there exists a topology on

K

with respect to which

K

are closed sets and

K

is compact.

Title

compact pavings are closed subsets of a

compact space

Canonical name

CompactPavingsAreClosedSubsetsOfACompactSpace

Date of creation

2013-03-22 18:45:01

Last modified on

2013-03-22 18:45:01

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Generated on Fri Feb 9 20:04:59 2018 by

LaTeXML
countable unions and intersections of analytic sets are analytic

countable unions and intersections of analytic sets are analytic

A property of

analytic sets

(

http://planetmath.org/AnalyticSet2

) which makes them particularly suited to applications in

measure theory

is that, in common with

σ

-

algebras

(

http://planetmath.org/SigmaAlgebra

), they are

closed under

countable

unions and

intersections

.

Theorem 1

.

Let

(

X

,

F

)

be a

paved space

and

(

A

n

)

n

∈

N

be a

sequence

of

F

-analytic sets. Then,

⋃

n

A

n

and

⋂

n

A

n

are

F

-

analytic

.

A

consequence

of this is that

measurable sets

are analytic, as follows.

Corollary

.

Let

F

be a nonempty paving on a set

X

such that the

complement

(

http://planetmath.org/Complement

) of any

S

∈

F

is a union of countably many sets in

F

.

Then, every set

A

in the

σ

-algebra

generated by

F

is

F

-analytic.

For example, every

closed subset

of a

metric space

X

is a union of countably many

open sets

. Therefore, the corollary shows that all

Borel sets

are analytic with respect to the open subsets of

X

.

That the corollary does indeed follow from

Theorem

1

is a simple application of the

monotone class theorem

.
First, as the

collection

a

⁢

(

ℱ

)

of

ℱ

-analytic sets is closed under countable unions and finite intersections, it will contain all finite unions of finite

intersections of sets

in

ℱ

and their complements, which is an algebra (

http://planetmath.org/RingOfSets

). Then, Theorem

1

says that

a

⁢

(

ℱ

)

is closed under taking limits of increasing and decreasing sequences of sets. So, by the monotone class theorem, it contains the

σ

-algebra generated by

ℱ

.

Title

countable unions and intersections of analytic sets are analytic

Canonical name

CountableUnionsAndIntersectionsOfAnalyticSetsAreAnalytic

Date of creation

2013-03-22 18:45:18

Last modified on

2013-03-22 18:45:18

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Generated on Fri Feb 9 20:05:01 2018 by

LaTeXML
direct images of analytic sets are analytic

direct images of analytic sets are analytic

For

measurable spaces

(

X

,

ℱ

)

and

(

Y

,

𝒢

)

, consider a

measurable function

f

:

X

→

Y

. By definition, the

inverse image

f

-

1

⁢

(

A

)

will be in

ℱ

whenever

A

is in

𝒢

.
However, the situation is more complicated for

direct images

(

http://planetmath.org/DirectImage

), which in general do not preserve measurability. However, as stated by the following

theorem

, the class of

analytic

subsets of

Polish spaces

is

closed under

direct images.

Theorem

.

Let

f

:

X

→

Y

be a Borel measurable function between Polish spaces

X

and

Y

. Then, the direct image

f

⁢

(

A

)

is analytic whenever

A

is an analytic subset of

X

.

Title

direct images of analytic sets are analytic

Canonical name

DirectImagesOfAnalyticSetsAreAnalytic

Date of creation

2013-03-22 18:46:33

Last modified on

2013-03-22 18:46:33

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Generated on Fri Feb 9 20:05:03 2018 by

LaTeXML
equivalent definitions of analytic sets

equivalent definitions of analytic sets

For a

paved space

(

X

,

ℱ

)

the

ℱ

-

analytic

(

http://planetmath.org/AnalyticSet2

) sets can be defined as the projections (

http://planetmath.org/GeneralizedCartesianProduct

) of sets in

(

ℱ

×

𝒦

)

σ

⁢

δ

onto

X

, for

compact

paved spaces

(

K

,

𝒦

)

. There are, however, many other

equivalent

definitions, some of which we list here.

In conditions

2

and

3

of the following

theorem

,

Baire space

𝒩

=

ℕ

ℕ

is the

collection

of

sequences

of

natural numbers

together with the

product topology

.
In conditions

5

and

6

,

Y

can be any

uncountable

Polish space

. For example, we may take

Y

=

ℝ

with the

standard topology

.

Theorem

.

Let

(

X

,

F

)

be a paved space such that

F

contains the

empty set

, and

A

be a subset of

X

. The following are equivalent.

1.

A

is

ℱ

-analytic.

2.

There is a

closed subset

S

of

𝒩

and

θ

:

ℕ

2

→

ℱ

such that

A

=

⋃

s

∈

S

⋂

n

=

1

∞

θ

⁢

(

n

,

s

n

)

.

3.

There is a closed subset

S

of

𝒩

and

θ

:

ℕ

→

ℱ

such that

A

=

⋃

s

∈

S

⋂

n

=

1

∞

θ

⁢

(

s

n

)

.

4.

A

is the result of a

Souslin scheme

on

ℱ

.

5.

A

is the projection of a set in

(

ℱ

×

𝒢

)

σ

⁢

δ

onto

X

, where

𝒢

is the collection of closed subsets of

Y

.

6.

A

is the projection of a set in

(

ℱ

×

𝒦

)

σ

⁢

δ

onto

X

, where

𝒦

is the collection of compact subsets of

Y

.

For subsets of a

measurable space

, the following result gives a simple condition to be analytic. Again, the space

Y

can be any uncountable Polish space, and its Borel

σ

-algebra is denoted by

ℬ

. In particular, this result shows that a subset of the

real numbers

is analytic if and only if it is the projection of a

Borel set

from

ℝ

2

.

Theorem

.

Let

(

X

,

F

)

be a measurable space. For a subset

A

of

X

the following are equivalent.

1.

A

is

ℱ

-analytic.

2.

A

is the projection of an

ℱ

⊗

ℬ

-measurable subset of

X

×

Y

onto

X

.

We finally state some equivalent definitions of analytic subsets of a Polish space. Again,

𝒩

denotes Baire space and

Y

is any uncountable Polish space.

Theorem

.

For a nonempty subset

A

of a Polish space

X

the following are equivalent.

1.

A

is

ℱ

-analytic (

http://planetmath.org/AnalyticSet2

).

2.

A

is the projection of a closed subset of

X

×

𝒩

onto

X

.

3.

A

is the projection of a Borel subset of

X

×

Y

onto

X

.

4.

A

is the image (

http://planetmath.org/DirectImage

) of a

continuous function

f

:

Z

→

X

for some Polish space

Z

.

5.

A

is the image of a continuous function

f

:

𝒩

→

X

.

6.

A

is the image of a

Borel measurable function

f

:

Y

→

X

.

Title

equivalent definitions of analytic sets

Canonical name

EquivalentDefinitionsOfAnalyticSets

Date of creation

2013-03-22 18:48:28

Last modified on

2013-03-22 18:48:28

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

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

LaTeXML
finite fields of sets

finite fields of sets

If

S

is a

finite set

then any field of subsets of

S

(see “

field of
sets

” in the entry on rings of sets) can be described as the set of unions
of subsets of a

partition

of

S

.

Note that, if

P

is a partition of

S

and

A

,

B

⊂

P

, we
have

⋃

A

¯

=

⋃

(

P

∖

A

)

(

⋃

A

)

∪

(

⋃

B

)

=

⋃

(

A

∪

B

)

(

⋃

A

)

∩

(

⋃

B

)

=

⋃

(

A

∩

B

)

so

{

⋃

X

∣

X

⊂

P

}

is a field of sets.

Now assume that

ℱ

is a field of subsets of a finite set

S

. Let us define the set of “prime elements” of

ℱ

as
follows:

P

=

{

X

∈

(

ℱ

∖

∅

)

∣

(

Y

⊂

X

)

∧

(

Y

∈

ℱ

)

⇒

(

Y

=

∅

∨

Y

=

X

)

}

The choice of terminology “prime element” is meant to be a suggestive
mnemonic of how the only divisors of a prime number are 1 and the
number itself.

We claim that

P

is a partition. To justify this claim, we need to
show that elements of

P

are

pairwise disjoint

and that

⋃

P

=

S

.

Suppose that

A

and

B

are prime elements. Since, by definition,

A

∈

ℱ

and

B

∈

ℱ

and

ℱ

is a field
of sets,

A

∩

B

∈

ℱ

. Since

A

∩

B

⊂

A

, we
must either have

A

∩

B

=

∅

or

A

∩

B

=

A

. In the
former case,

A

and

B

are

disjoint

, whilst in the latter case

A

=

B

.

Suppose that

x

is any element of

S

. Then we claim that the set

X

defined as

X

=

⋃

{

Y

∈

ℱ

∣

x

∈

Y

}

is a prime element of

ℱ

. To begin, note that, since

ℱ

is finite, a forteriori any subset of

ℱ

is
finite and, since fields of sets are assumed to be

closed under

intersection

, it follows that the intersection of a susbet of

ℱ

is an element of

ℱ

, in particular

X

∈

ℱ

.

Suppose that

Z

⊂

X

and

Z

∈

ℱ

. If

x

∉

Z

,
then

x

∈

Z

¯

. Since

ℱ

is a field of sets,

Z

¯

∈

ℱ

. Hence, by the construction of

X

, it
is the case that

X

⊂

Z

¯

, hence

X

∩

Z

=

∅

. Together with

Z

⊂

X

, this implies

Z

=

∅

. If

x

∈

Z

, then, by construction,

X

⊂

Z

,
which implies

X

=

Z

.

Thus, we see that

X

is a prime set. Since

x

was arbitrarily
chosen, this means that every element of

S

is

contained

in a prime
element of

ℱ

, so the union of all prime elements is

S

itself. Together with the previously shown fact that prime elements
are pairwise disjoint, this shows that the prime elements for a
partition of

S

.

Let

A

be an arbitrary element of

ℱ

. Since

P

⊂

ℱ

, it is the case that

(

∀

X

∈

P

)

A

∩

X

∈

ℱ

. Since

P

is a partition of

S

,

A

=

⋃

{

A

∩

X

∣

X

∈

P

}

so every element of

ℱ

can be expressed as a union of
elements of

P

.

Title

finite fields of sets

Canonical name

FiniteFieldsOfSets

Date of creation

2013-03-22 15:47:49

Last modified on

2013-03-22 15:47:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 28A05

Classification

msc 03E20

Generated on Fri Feb 9 20:05:07 2018 by

LaTeXML