Partition regularityWikipedia

Contents

1 Almost disjoint sets 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Other meanings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Delta-ring 32.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Disjoint sets 43.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Dynkin system 84.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Dynkin’s π-λ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Family of sets 105.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Field of sets 12

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6.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 126.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.1.2 Separative and compact fields of sets: towards Stone duality . . . . . . . . . . . . . . . . . 12

6.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.2 Topological fields of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.3 Preorder fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2.4 Complex algebras and fields of sets on relational structures . . . . . . . . . . . . . . . . . . 14

6.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Finite character 167.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Finite intersection property 178.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Greedoid 199.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Partition of a set 2210.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.4 Refinement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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10.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 Partition regularity 3011.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

12 Pi system 3212.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2 Relationship to λ-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12.2.1 The π-λ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.3 π-Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

12.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13 Ring of sets 3613.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Sigma-algebra 3814.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

14.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.1.2 Limits of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.1.3 Sub σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14.2 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2.2 Dynkin’s π-λ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2.3 Combining σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2.4 σ-algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2.5 Relation to σ-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.4 σ-algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4.1 σ-algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4.2 σ-algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4.3 Borel and Lebesgue σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.4.4 Product σ-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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14.4.5 σ-algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.4.6 σ-algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 4414.4.7 σ-algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

15 Sigma-ideal 4615.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

16 Sigma-ring 4716.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Almost disjoint sets

In mathematics, two sets are almost disjoint [1][2] if their intersection is small in some sense; different definitions of“small” will result in different definitions of “almost disjoint”.

1.1 Definition

Themost common choice is to take “small” to mean finite. In this case, two sets are almost disjoint if their intersectionis finite, i.e. if

|A ∩B| < ∞.

(Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1,2] are almost disjoint, because their intersection is the finite set 1. However, the unit interval [0, 1] and the set ofrational numbers Q are not almost disjoint, because their intersection is infinite.This definition extends to any collection of sets. A collection of sets is pairwise almost disjoint ormutually almostdisjoint if any two distinct sets in the collection are almost disjoint. Often the prefix “pairwise” is dropped, and apairwise almost disjoint collection is simply called “almost disjoint”.Formally, let I be an index set, and for each i in I, let Ai be a set. Then the collection of sets Ai : i in I is almostdisjoint if for any i and j in I,

Ai = Aj ⇒ |Ai ∩Aj | < ∞.

For example, the collection of all lines through the origin in R2 is almost disjoint, because any two of them only meetat the origin. If Ai is an almost disjoint collection consisting of more than one set, then clearly its intersection isfinite:

∩i∈I

Ai < ∞.

However, the converse is not true—the intersection of the collection

1, 2, 3, . . ., 2, 3, 4, . . ., 3, 4, 5, . . ., . . .

is empty, but the collection is not almost disjoint; in fact, the intersection of any two distinct sets in this collection isinfinite.

1

2 CHAPTER 1. ALMOST DISJOINT SETS

1.2 Other meanings

Sometimes “almost disjoint” is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative definitions of “almost disjoint” that are sometimes used (similar definitions apply to infinitecollections):

• Let κ be any cardinal number. Then two sets A and B are almost disjoint if the cardinality of their intersectionis less than κ, i.e. if

|A ∩B| < κ.

The case of κ = 1 is simply the definition of disjoint sets; the case of

κ = ℵ0

is simply the definition of almost disjoint given above, where the intersection of A and B is finite.

• Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if theirintersection is a null-set, i.e. if

m(A ∩B) = 0.

• Let X be a topological space. Then two subsets A and B of X are almost disjoint if their intersection is meagrein X.

1.3 References[1] Kunen, K. (1980), “Set Theory; an introduction to independence proofs”, North Holland, p. 47

[2] Jech, R. (2006) “Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118

Chapter 2

Delta-ring

In mathematics, a nonempty collection of setsR is called a δ-ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A ∪B ∈ R if A,B ∈ R

2. A−B ∈ R if A,B ∈ R

3.∩∞

n=1 An ∈ R if An ∈ R for all n ∈ N

If only the first two properties are satisfied, then R is a ring but not a δ-ring. Every σ-ring is a δ-ring, but not everyδ-ring is a σ-ring.δ-rings can be used instead of σ-fields in the development of measure theory if one does not wish to allow sets ofinfinite measure.

2.1 See also• Ring of sets

• Sigma field

• Sigma ring

2.2 References• Cortzen, Allan. “Delta-Ring.” From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.http://mathworld.wolfram.com/Delta-Ring.html

3

Chapter 3

Disjoint sets

This article is about the mathematical concept. For the data structure, see Disjoint-set data structure.In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are

A BTwo disjoint sets.

sets whose intersection is the empty set.[1] For example, 1, 2, 3 and 4, 5, 6 are disjoint sets, while 1, 2, 3 and3, 4, 5 are not.

3.1 Generalizations

This definition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint ormutuallydisjoint if every two different sets in the family are disjoint.[1] For example, the collection of sets 1, 2, 3, ... is pairwise disjoint.Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two infinite setswhose intersection is a finite set may be said to be almost disjoint.[2]

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance,two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, ina metric space, positively separated sets are sets separated by a nonzero distance.[3]

4

3.2. EXAMPLES 5

A

BC

A pairwise disjoint family of sets

3.2 Examples

• The set of the drum and the guitar is disjoint to the set of the card and the book

• A pairwise disjoint family of sets

• A non pairwise disjoint family of sets

3.3 Intersections

Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections.

6 CHAPTER 3. DISJOINT SETS

Two sets A and B are disjoint if and only if their intersection A∩B is the empty set.[1] It follows from this definitionthat every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.[4]

A family F of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.[1] If the familycontains more than one set, this implies that the intersection of the whole family is also empty. However, a familyof only one set is pairwise disjoint, regardless of whether that set is empty, and may have a non-empty intersection.Additionally, a family of sets may have an empty intersection without being pairwise disjoint.[5] For instance, thethree sets 1, 2, 2, 3, 1, 3 have an empty intersection but are not pairwise disjoint. In fact, there are no twodisjoint sets in this collection. Also the empty family of sets is pairwise disjoint.[6]

A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that arepairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closedintervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it mustbe pairwise disjoint.[7]

3.4 Disjoint unions and partitions

A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X.[8] Every partition canequivalently be described by an equivalence relation, a binary relation that describes whether two elements belongto the same set in the partition.[8] Disjoint-set data structures[9] and partition refinement[10] are two techniques incomputer science for efficiently maintaining partitions of a set subject to, respectively, union operations that mergetwo sets or refinement operations that split one set into two.A disjoint union may mean one of two things. Most simply, it may mean the union of sets that are disjoint.[11] Butif two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make themdisjoint before forming the union of the modified sets.[12] For instance two sets may be made disjoint by replacingeach element by an ordered pair of the element and a binary value indicating whether it belongs to the first or secondset.[13] For families of more than two sets, one may similarly replace each element by an ordered pair of the elementand the index of the set that contains it.[14]

3.5 See also• Hyperplane separation theorem for disjoint convex sets

• Mutually exclusive events

• Relatively prime, numbers with disjoint sets of prime divisors

• Set packing, the problem of finding the largest disjoint subfamily of a family of sets

3.6 References[1] Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 15, ISBN 9780387900926.

[2] Halbeisen, Lorenz J. (2011), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer monographs inmathematics, Springer, p. 184, ISBN 9781447121732.

[3] Copson, Edward Thomas (1988),Metric Spaces, Cambridge Tracts in Mathematics 57, Cambridge University Press, p. 62,ISBN 9780521357326.

[4] Oberste-Vorth, Ralph W.; Mouzakitis, Aristides; Lawrence, Bonita A. (2012), Bridge to Abstract Mathematics, MAAtextbooks, Mathematical Association of America, p. 59, ISBN 9780883857793.

[5] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2010), A Transition to Advanced Mathematics, Cengage Learning,p. 95, ISBN 9780495562023.

[6] See answers to the question ″Is the empty family of sets pairwise disjoint?″

[7] Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-bridge University Press, p. 82, ISBN 9780521337038.

3.7. EXTERNAL LINKS 7

[8] Halmos (1960), p. 28.

[9] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), “Chapter 21: Data structures forDisjoint Sets”, Introduction to Algorithms (Second ed.), MIT Press, pp. 498–524, ISBN 0-262-03293-7.

[10] Paige, Robert; Tarjan, Robert E. (1987), “Three partition refinement algorithms”, SIAM Journal on Computing 16 (6):973–989, doi:10.1137/0216062, MR 917035.

[11] Ferland, Kevin (2008), Discrete Mathematics: An Introduction to Proofs and Combinatorics, Cengage Learning, p. 45,ISBN 9780618415380.

[12] Arbib, Michael A.; Kfoury, A. J.; Moll, Robert N. (1981), A Basis for Theoretical Computer Science, The AKM series inTheoretical Computer Science: Texts and monographs in computer science, Springer-Verlag, p. 9, ISBN 9783540905738.

[13] Monin, Jean François; Hinchey,Michael Gerard (2003),Understanding FormalMethods, Springer, p. 21, ISBN9781852332471.

[14] Lee, John M. (2010), Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 (2nd ed.), Springer, p.64, ISBN 9781441979407.

3.7 External links• Weisstein, Eric W., “Disjoint Sets”, MathWorld.

Chapter 4

Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set Ω satisfying a setof axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himselfused this term) or d-system.[1] These set families have applications in measure theory and probability.The primary relevance of λ-systems are their use in applications of the π-λ theorem.

4.1 Definitions

Let Ω be a nonempty set, and let D be a collection of subsets of Ω (i.e., D is a subset of the power set of Ω). ThenD is a Dynkin system if

1. Ω ∈ D ,

2. if A, B ∈ D and A ⊆ B, then B \ A ∈ D ,

3. if A1, A2, A3, ... is a sequence of subsets in D and An ⊆ An₊₁ for all n ≥ 1, then∪∞

n=1 An ∈ D .

Equivalently, D is a Dynkin system if

1. Ω ∈ D ,

2. if A ∈ D, then Ac ∈ D,

3. if A1, A2, A3, ... is a sequence of subsets in D such that Ai ∩ Aj = Ø for all i ≠ j, then∪∞

n=1 An ∈ D .

The second definition is generally preferred as it usually is easier to check.An important fact is that a Dynkin systemwhich is also a π-system (i.e., closed under finite intersection) is a σ-algebra.This can be verified by noting that condition 3 and closure under finite intersection implies closure under countableunions.Given any collection J of subsets of Ω , there exists a unique Dynkin system denotedDJ which is minimal withrespect to containing J . That is, if D is any Dynkin system containing J , then DJ ⊆ D . DJ is called theDynkin system generated by J . Note D∅ = ∅,Ω . For another example, let Ω = 1, 2, 3, 4 and J = 1 ;then DJ = ∅, 1, 2, 3, 4,Ω .

4.2 Dynkin’s π-λ theorem

If P is a π-system andD is a Dynkin system with P ⊆ D , then σP ⊆ D . In other words, the σ-algebra generatedby P is contained in D .One application of Dynkin’s π-λ theorem is the uniqueness of a measure that evaluates the length of an interval(known as the Lebesgue measure):

8

4.3. NOTES 9

Let (Ω, B, λ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let μ be another measure on Ωsatisfying μ[(a,b)] = b − a, and letD be the family of sets S such that μ[S] = λ[S]. Let I = (a,b),[a,b),(a,b],[a,b] : 0 <a ≤ b < 1 , and observe that I is closed under finite intersections, that I ⊂ D, and that B is the σ-algebra generated byI. It may be shown that D satisfies the above conditions for a Dynkin-system. From Dynkin’s π-λ Theorem it followsthat D in fact includes all of B, which is equivalent to showing that the Lebesgue measure is unique on B.Additional applications are in the article on π-systems.

4.3 Notes[1] Charalambos Aliprantis, Kim C. Border (2006). Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer.

Retrieved August 23, 2010.

4.4 References• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN0-387-22833-0.

• Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2.

• David Williams (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6.

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 5

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term “collection” is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

5.1 Examples• The power set P(S) is a family of sets over S.

• The k-subsets S(k) of a set S form a family of sets.

• Let S = a,b,c,1,2, an example of a family of sets over S (in the multiset sense) is given by F = A1, A2, A3,A4 where A1 = a,b,c, A2 = 1,2, A3 = 1,2 and A4 = a,b,1.

• The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

5.2 Special types of set family• A Sperner family is a family of sets in which none of the sets contains any of the others. Sperner’s theorembounds the maximum size of a Sperner family.

• A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Helly’s theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

5.3 Properties• Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members.

• Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe).

• Hall’s marriage theorem, due to Philip Hall gives necessary and sufficient conditions for a finite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

5.4 Related concepts

Certain types of objects from other areas ofmathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

10

5.5. SEE ALSO 11

• A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

• An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family offinite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

• An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specified bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

• A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

5.5 See also• Indexed family

• Class (set theory)

• Combinatorial design

• Russell’s paradox (or Set of sets that do not contain themselves)

5.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

5.7 References• Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0

• Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

• Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 6

Field of sets

“Set algebra” redirects here. For the basic properties and laws of sets, see Algebra of sets.

In mathematics a field of sets is a pair ⟨X,F⟩ whereX is a set andF is an algebra over X i.e., a non-empty subsetof the power set of X closed under the intersection and union of pairs of sets and under complements of individualsets. In other words F forms a subalgebra of the power set Boolean algebra of X . (Many authors refer to F itselfas a field of sets. The word “field” in “field of sets” is not used with the meaning of field from field theory.) Elementsof X are called points and those of F are called complexes and are said to be the admissible sets of X .Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can berepresented as a field of sets.

6.1 Fields of sets in the representation theory of Boolean algebras

6.1.1 Stone representation

Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each elementof the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power setrepresentation can be constructed more generally for any complete atomic Boolean algebra.In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representationby considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Booleanalgebra correspond to its ultrafilters and that an atom is below an element of a finite Boolean algebra if and only ifthat element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of aBoolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Booleanalgebra the set of ultrafilters containing that element. This construction does indeed produce a representation of theBoolean algebra as a field of sets and is known as the Stone representation. It is the basis of Stone’s representationtheorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or filters,similar to Dedekind cuts.Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexesby associating each element of the Boolean algebra with the set of such homomorphisms that map it to the topelement. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the topelements under these homomorphisms.) With this approach one sees that Stone representation can also be regardedas a generalization of the representation of finite Boolean algebras by truth tables.

6.1.2 Separative and compact fields of sets: towards Stone duality

• A field of sets is called separative (or differentiated) if and only if for every pair of distinct points there is acomplex containing one and not the other.

• Afield of sets is called compact if and only if for every proper filter overX the intersection of all the complexes

12

6.2. FIELDS OF SETS WITH ADDITIONAL STRUCTURE 13

contained in the filter is non-empty.

These definitions arise from considering the topology generated by the complexes of a field of sets. Given a field ofsets X = ⟨X,F⟩ the complexes form a base for a topology, we denote the corresponding topological space by T (X). Then

• T (X) is always a zero-dimensional space.

• T (X) is a Hausdorff space if and only if X is separative.

• T (X) is a compact space with compact open sets F if and only if X is compact.

• T (X) is a Boolean space with clopen sets F if and only if X is both separative and compact (in which case itis described as being descriptive)

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space isknown as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexesof the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stonerepresentation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a dualityexists between Boolean algebras and Boolean spaces.

6.2 Fields of sets with additional structure

6.2.1 Sigma algebras and measure spaces

If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebraand the corresponding field of sets is called a measurable space. The complexes of a measurable space are calledmeasurable sets.A measure space is a triple ⟨X,F , µ⟩ where ⟨X,F⟩ is a measurable space and µ is a measure defined on it. If µis in fact a probability measure we speak of a probability space and call its underlying measurable space a samplespace. The points of a sample space are called samples and represent potential outcomes while the measurable sets(complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Manyuse the term sample space simply for the underlying set of a probability space, particularly in the case where everysubset is an event.) Measure spaces and probability spaces play a foundational role in measure theory and probabilitytheory respectively.The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

6.2.2 Topological fields of sets

A topological field of sets is a triple ⟨X, T ,F⟩ where ⟨X, T ⟩ is a topological space and ⟨X,F⟩ is a field of setswhich is closed under the closure operator of T or equivalently under the interior operator i.e. the closure and interiorof every complex is also a complex. In other words F forms a subalgebra of the power set interior algebra on ⟨X, T ⟩.Every interior algebra can be represented as a topological field of sets with its interior and closure operators corre-sponding to those of the topological space.Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interiorand closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets.

Algebraic fields of sets and Stone fields

A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes.If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets areprecisely the open complexes. Moreover the open complexes form a base for the topology.

14 CHAPTER 6. FIELD OF SETS

Topological fields of sets that are separative, compact and algebraic are calledStone fields and provide a generalizationof the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation ofits underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated bythe complexes corresponding to the open elements of the interior algebra (which form a base for a topology). Thesecomplexes are then precisely the open complexes and the construction produces a Stone field representing the interioralgebra - the Stone representation.

6.2.3 Preorder fields

A preorder field is a triple ⟨X,≤,F⟩ where ⟨X,≤⟩ is a preordered set and ⟨X,F⟩ is a field of sets.Like the topological fields of sets, preorder fields play an important role in the representation theory of interior alge-bras. Every interior algebra can be represented as a preorder field with its interior and closure operators correspondingto those of the Alexandrov topology induced by the preorder. In other words

Int(S) = x ∈ X : there exists a y ∈ S with y ≤ x andCl(S) = x ∈ X : there exists a y ∈ S with x ≤ y for all S ∈ F

Preorder fields arise naturally in modal logic where the points represent the possible worlds in the Kripke semanticsof a theory in the modal logic S4 (a formal mathematical abstraction of epistemic logic), the preorder represents theaccessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worldsin which individual sentences in the theory hold, providing a representation of the Lindenbaum-Tarski algebra of thetheory.

Algebraic and canonical preorder fields

A preorder field is called algebraic if and only if it has a set of complexes A which determines the preorder in thefollowing manner: x ≤ y if and only if for every complex S ∈ A , x ∈ S implies y ∈ S . The preorder fieldsobtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possibleworlds in which the sentences of the theory closed under necessity hold.A separative compact algebraic preorder field is said to be canonical. Given an interior algebra, by replacing thetopology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain arepresentation of the interior algebra as a canonical preorder field. By replacing the preorder by its correspondingAlexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. (Thetopology of this "Alexandrov representation" is just the Alexandrov bi-coreflection of the topology of the Stonerepresentation.)

6.2.4 Complex algebras and fields of sets on relational structures

The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbi-trary (normal) Boolean algebras with operators. For this we consider structures ⟨X, (Ri)I ,F⟩ where ⟨X, (Ri)I⟩ is arelational structure i.e. a set with an indexed family of relations defined on it, and ⟨X,F⟩ is a field of sets. The com-plex algebra (or algebra of complexes) determined by a field of sets X = ⟨X, (Ri)I ,F⟩ on a relational structure,is the Boolean algebra with operators

C(X) = ⟨F ,∩,∪, ′, ∅, X, (fi)I⟩

where for all i ∈ I , if Ri is a relation of arity n+ 1 , then fi is an operator of arity n and for all S1, ..., Sn ∈ F

fi(S1, ..., Sn) = x ∈ X : there exist x1 ∈ S1, ..., xn ∈ Sn such that Ri(x1, ..., xn, x)

This construction can be generalized to fields of sets on arbitrary algebraic structures having both operators andrelations as operators can be viewed as a special case of relations. If F is the whole power set of X then C(X) iscalled a full complex algebra or power algebra.

6.3. SEE ALSO 15

Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in thesense that it is isomorphic to the complex algebra corresponding to the field.(Historically the term complex was first used in the case where the algebraic structure was a group and has its originsin 19th century group theory where a subset of a group was called a complex.)

6.3 See also• List of Boolean algebra topics

• Algebra of sets

• Sigma algebra

• Measure theory

• Probability theory

• Interior algebra

• Alexandrov topology

• Stone’s representation theorem for Boolean algebras

• Stone duality

• Boolean ring

• Preordered field

6.4 References• Goldblatt, R., Algebraic Polymodal Logic: A Survey, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450,July 2000

• Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, 44, p. 173-242, 1989

• Johnstone, Peter T. (1982). Stone spaces (3rd edition ed.). Cambridge: Cambridge University Press. ISBN0-521-33779-8.

• Naturman, C.A., Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Math-ematics, 1991

• Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., Handbook of Modal Logic, Volume 3 ofStudies in Logic and Practical Reasoning, Elsevier, 2006

6.5 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 7

Finite character

In mathematics, a family F of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

7.1 Properties

A family F of sets of finite character enjoys the following properties:

1. For each A ∈ F , every (finite or infinite) subset of A belongs to F .

2. Tukey’s lemma: In F , partially ordered by inclusion, the union of every chain of elements of F also belong toF , therefore, by Zorn’s lemma, F contains at least one maximal element.

7.2 Example

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finitecharacter (because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent). There-fore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family isa vector basis, every vector space has a (possibly infinite) vector basis.This article incorporates material from finite character on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

16

Chapter 8

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the finite intersectionproperty (FIP) if the intersection over any finite subcollection of A is nonempty. It has the strong finite intersectionproperty (SFIP) if the intersection over any finite subcollection of A is infinite.A centered system of sets is a collection of sets with the finite intersection property.

8.1 Definition

Let X be a set with A = Aii∈I a family of subsets of X. Then the collection A has the finite intersection property(FIP), if any finite subcollection J ⊆ I has non-empty intersection

∩i∈J Ai.

8.2 Discussion

Clearly the empty set cannot belong to any collection with the finite intersection property. The condition is triviallysatisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits.The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact ifand only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.[1]This formulation of compactness is used in some proofs of Tychonoff’s theorem and the uncountability of the realnumbers (see next section)

8.3 Applications

Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U ⊆ X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV ⊂ U whose closure doesn’t contain x (x may or may not be in U). Choose y in U different from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isn’t in U, this is possible sinceU is nonempty). Then by the Hausdorff condition, choose disjoint neighbourhoodsW and K of x and y respectively.Then K ∩ U will be a neighbourhood of y contained in U whose closure doesn’t contain x as desired.

Now suppose f : N → X is a bijection, and let xi : i ∈ N denote the image of f. Let X be the first open set

17

18 CHAPTER 8. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 ⊂ X whose closure doesn’t contain x1. Secondly, choose a neighbourhood U2 ⊂U1 whose closure doesn’t contain x2. Continue this process whereby choosing a neighbourhood Un₊₁ ⊂ Un whoseclosure doesn’t contain xn₊₁. Then the collection Ui : i ∈ N satisfies the finite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesn’t belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdorff condition; a countable set with the indiscrete topology is compact, has morethan one point, and satisfies the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a finite space given the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdorff space is uncountable.Proof. Let X be a perfect, compact, Hausdorff space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdorff space which is not compact, then the one-point compactification of X isa perfect, compact Hausdorff space. Therefore the one point compactification of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

8.4 Examples

A filter has the finite intersection property by definition.

8.5 Theorems

Let X be nonempty, F ⊆ 2X, F having the finite intersection property. Then there exists an F′ ultrafilter (in 2X) suchthat F ⊆ F′.See details and proof in Csirmaz & Hajnal (1994).[2] This result is known as ultrafilter lemma.

8.6 Variants

A family of sets A has the strong finite intersection property (sfip), if every finite subfamily of A has infiniteintersection.

8.7 References[1] A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.org.

[2] Csirmaz, László; Hajnal, András (1994), Matematikai logika (IN HUNGARIAN), Budapest: Eötvös Loránd University.

• Finite intersection property at PlanetMath.org.

Chapter 9

Greedoid

In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originallyintroduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of opti-mization problems that can be solved by greedy algorithms. Around 1980, Korte and Lovász introduced the greedoidto further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematicaloptimization, greedoids have also been connected to graph theory, language theory, poset theory, and other areas ofmathematics.

9.1 Definitions

A set system (F, E) is a collection F of subsets of a ground set E (i.e. F is a subset of the power set of E). Whenconsidering a greedoid, a member of F is called a feasible set. When considering a matroid, a feasible set is alsoknown as an independent set.An accessible set system (F, E) is a set system in which every nonempty feasible set X contains an element x suchthat X\x is feasible. This implies that any nonempty, finite accessible set system necessarily contains the empty set∅.A greedoid (F, E) is an accessible set system that satisfies the exchange property:

• for all X,Y ∈ F with |X| > |Y|, there is some x ∈ X\Y such that Y∪x ∈ F

(Note: Some people reserve the term exchange property for a condition on the bases of a greedoid, and prefer to callthe above condition the “Augmentation Property”.)A basis of a greedoid is a maximal feasible set, meaning it is a feasible set but not contained in any other one. Abasis of a subset X of E is a maximal feasible set contained in X.The rank of a greedoid is the size of a basis. By the exchange property, all bases have the same size. Thus, the rankfunction is well defined. The rank of a subset X of E is the size of a basis of X.

9.2 Classes of greedoids

Most classes of greedoids havemany equivalent definitions in terms of set system, language, poset, simplicial complex,and so on. The following description takes the traditional route of listing only a couple of the more well-knowncharacterizations.An interval greedoid (F, E) is a greedoid that satisfies the Interval Property:

• if A, B, C ∈ F with A ⊆ B ⊆ C, then, for all x ∈ E\C, (A∪x ∈ F and C∪x ∈ F) implies B∪x ∈ F

Equivalently, an interval greedoid is a greedoid such that the union of any two feasible sets is feasible if it is containedin another feasible set.

19

20 CHAPTER 9. GREEDOID

An antimatroid (F, E) is a greedoid that satisfies the Interval Property without Upper Bounds:

• if A, B ∈ F with A ⊆ B, then, for all x ∈ E\B, A∪x ∈ F implies B∪x ∈ F

Equivalently, an antimatroid is (i) a greedoid with a unique basis; or (ii) an accessible set system closed under union.It is easy to see that an antimatroid is also an interval greedoid.Amatroid (F, E) is a greedoid that satisfies the Interval Property without Lower Bounds:

• if B, C ∈ F with B ⊆ C, then, for all x ∈ E\C, C∪x ∈ F implies B∪x ∈ F

It is easy to see that a matroid is also an interval greedoid.

9.3 Examples

• Consider an undirected graph G. Let the ground set be the edges of G and the feasible sets be the edge set ofeach forest (i.e. subgraph containing no cycle) of G. This set system is called the cycle matroid. A set systemis said to be a graphic matroid if it is the cycle matroid of some graph. (Originally cycle matroid was definedon circuits, or minimal dependent sets. Hence the name cycle.)

• Consider a finite, undirected graph G rooted at the vertex r. Let the ground set be the vertices of G and thefeasible sets be the vertex subsets containing r that induce connected subgraphs of G. This is called the vertexsearch greedoid and is a kind of antimatroid.

• Consider a finite, directed graph D rooted at r. Let the ground set be the (directed) edges of D and the feasiblesets be the edge sets of each directed subtree rooted at r with all edges pointing away from r. This is called theline search greedoid, or directed branching greedoid. It is an interval greedoid, but neither an antimatroidnor a matroid.

• Consider an m-by-n matrix M. Let the ground set E be the indices of the columns from 1 to n and the feasiblesets be F = X ⊆ E: submatrix M₁,...,|X|,X is an invertible matrix. This is called theGaussian eliminationgreedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not aninterval greedoid.

9.4 Greedy algorithm

In general, a greedy algorithm is just an iterative process in which a locally best choice, usually an input of minimumweight, is chosen each round until all available choices have been exhausted. In order to describe a greedoid-basedcondition in which a greedy algorithm is optimal, we need some more common terminologies in greedoid theory.Without loss of generality, we consider a greedoid G = (F, E) with E finite.A subset X of E is rank feasible if the largest intersection of X with any feasible set has size equal to the rank of X.In a matroid, every subset of E is rank feasible. But the equality does not hold for greedoids in general.A function w: E → ℝ is R-compatible if x ∈ E: w(x) ≥ c is rank feasible for all real numbers c.An objective function f: 2S →ℝ is linear over a set S if, for all X ⊆ S, we have f(X) = Σₓ ∈ X w(x) for some weightfunction w: S → ℜ.Proposition. A greedy algorithm is optimal for every R-compatible linear objective function over a greedoid.The intuition behind this proposition is that, during the iterative process, each optimal exchange of minimum weightis made possible by the exchange property, and optimal results are obtainable from the feasible sets in the underlyinggreedoid. This result guarantees the optimality of many well-known algorithms. For example, a minimum spanningtree of a weighted graphmay be obtained using Kruskal’s algorithm, which is a greedy algorithm for the cycle matroid.

9.5. SEE ALSO 21

9.5 See also• Matroid

• Polymatroid

9.6 References• Björner, Anders; Ziegler, Günter M. (1992), White, Neil, ed., “Matroid Applications”, Matroid applications,Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press) 40: 284–357,doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537, Zbl 0772.05026 |chapter= ig-nored (help)

• Edmonds, Jack (1971), “Matroids and the greedy algorithm”,Mathematical Programming 1: 127–113, doi:10.1007/BF01584082,Zbl 0253.90027.

• Helman, Paul; Moret, Bernard M. E.; Shapiro, Henry D. (1993), “An exact characterization of greedy struc-tures”, SIAM Journal on Discrete Mathematics 6 (2): 274–283, doi:10.1137/0406021, Zbl 0798.68061.

• Korte, Bernhard; Lovász, László (1981), “Mathematical structures underlying greedy algorithms”, in Gecseg,Ferenc, Fundamentals of Computation Theory: Proceedings of the 1981 International FCT-Conference, Szeged,Hungaria, August 24–28, 1981, Lecture Notes in Computer Science 117, Berlin: Springer-Verlag, pp. 205–209, doi:10.1007/3-540-10854-8_22, Zbl 0473.68019.

• Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics 4, NewYork, Berlin: Springer-Verlag, ISBN 3-540-18190-3, Zbl 0733.05023.

• Oxley, James G. (1992),Matroid theory, Oxford Science Publications, Oxford: Oxford University Press, ISBN0-19-853563-5, Zbl 0784.05002.

• Whitney, Hassler (1935), “On the abstract properties of linear independence”, American Journal of Mathe-matics 57 (3): 509–533, doi:10.2307/2371182, JSTOR 2371182, Zbl 0012.00404.

9.7 External links• Introduction to Greedoids

• Theory of Greedy Algorithms

• Submodular Functions and Optimization

• Matchings, Matroids and Submodular Functions

Chapter 10

Partition of a set

For the partition calculus of sets, see infinitary combinatorics.In mathematics, a partition of a set is a grouping of the set’s elements into non-empty subsets, in such a way that

A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.

every element is included in one and only one of the subsets.

22

10.1. DEFINITION 23

10.1 Definition

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[1] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[2]

1. P does not contain the empty set.

2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)

3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

1. ∅ /∈ P

2.∪

A∈P A = X

3. if A,B ∈ P and A = B then A ∩B = ∅ ,

where ∅ is the empty set.The sets in P are called the blocks, parts or cells of the partition.[3]

The rank of P is |X| − |P|, if X is finite.

10.2 Examples• Every singleton set x has exactly one partition, namely x .

• For any nonempty set X, P = X is a partition of X, called the trivial partition.

• For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, A, U−A.

• The set 1, 2, 3 has these five partitions:

• 1, 2, 3 , sometimes written 1|2|3.• 1, 2, 3 , or 12|3.• 1, 3, 2 , or 13|2.• 1, 2, 3 , or 1|23.• 1, 2, 3 , or 123 (in contexts where there will be no confusion with the number).

• The following are not partitions of 1, 2, 3 :

• , 1, 3, 2 is not a partition (of any set) because one of its elements is the empty set.• 1, 2, 2, 3 is not a partition (of any set) because the element 2 is contained in more than one block.• 1, 2 is not a partition of 1, 2, 3 because none of its blocks contains 3; however, it is a partitionof 1, 2.

10.3 Partitions and equivalence relations

For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[4]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.

24 CHAPTER 10. PARTITION OF A SET

10.4 Refinement of partitions

A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarserthan α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentationof ρ. In that case, it is written that α ≤ ρ.This finer-than relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (forpartitions of a finite set) it is a geometric lattice.[5] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn − 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalenceclasses: the sets red cards and black cards. The 2-part partition corresponding to ~C has a refinement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes spades, diamonds, hearts, and clubs.

10.5 Noncrossing partitions

A partition of the set N = 1, 2, ..., n with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells’ C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.

10.6 Counting partitions

The total number of partitions of an n-element set is the Bell number Bn. The first several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion Bn+1 =

∑nk=0

(nk

)Bk

and have the exponential generating function

∞∑n=0

Bn

n!zn = ee

z−1.

The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

Cn =1

n+ 1

(2n

n

).

10.7. SEE ALSO 25

10.7 See also• Exact cover

• Cluster analysis

• Weak ordering (ordered set partition)

• Equivalence relation

• Partial equivalence relation

• Partition refinement

• List of partition topics

• Lamination (topology)

• Rhyme schemes by set partition

10.8 Notes[1] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.

[2] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732.

[3] Brualdi, pp. 44–45

[4] Schechter, p. 54

[5] Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.

10.9 References• Brualdi, Richard A. (2004). Introductory Combinatorics (4th edition ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.

• Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.

26 CHAPTER 10. PARTITION OF A SET

The 52 partitions of a set with 5 elements

10.9. REFERENCES 27

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning five elements.

28 CHAPTER 10. PARTITION OF A SET

Partitions of a 4-set ordered by refinement

10.9. REFERENCES 29

Construction of the Bell triangle

Chapter 11

Partition regularity

In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.Given a set X , a collection of subsets S ⊂ P(X) is called partition regular if every set A in the collection has theproperty that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong tothe collection. That is, for any A ∈ S , and any finite partition A = C1 ∪ C2 ∪ · · · ∪ Cn , there exists an i ≤ n, suchthat Ci belongs to S . Ramsey theory is sometimes characterized as the study of which collections S are partitionregular.

11.1 Examples• the collection of all infinite subsets of an infinite setX is a prototypical example. In this case partition regularityasserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.)

• sets with positive upper density inN : the upper density d(A) ofA ⊂ N is defined as d(A) = lim supn→∞|1,2,...,n∩A|

n .

• For any ultrafilter U on a setX , U is partition regular. If U ∋ A =∪n

1 Ci , then for exactly one i is Ci ∈ U .

• sets of recurrence: a set R of integers is called a set of recurrence if for any measure preserving transformationT of the probability space (Ω, β, μ) and A ∈ β of positive measure there is a nonzero n ∈ R so thatµ(A ∩ TnA) > 0 .

• Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then thecollection of a.p.-rich subsets is partition regular (Van der Waerden, 1927).

• Let [A]n be the set of all n-subsets of A ⊂ N . Let Sn =∪

A⊂N[A]n . For each n, Sn is partition regular.

(Ramsey, 1930).

• For each infinite cardinal κ , the collection of stationary sets of κ is partition regular. More is true: if S isstationary and S =

∪α<λ Sα for some λ < κ , then some Sα is stationary.

• the collection of∆ -sets: A ⊂ N is a∆ -set ifA contains the set of differences sm−sn : m,n ∈ N, n < mfor some sequence ⟨sn⟩ωn=1 .

• the set of barriers on N : call a collection B of finite subsets of N a barrier if:

• ∀X,Y ∈ B, X ⊂ Y and• for all infinite I ⊂ ∪B , there is someX ∈ B such that the elements of X are the smallest elements of I;i.e. X ⊂ I and ∀i ∈ I \X,∀x ∈ X,x < i .

This generalizes Ramsey’s theorem, as each [A]n is a barrier. (Nash-Williams, 1965)

30

11.2. REFERENCES 31

• finite products of infinite trees (Halpern–Läuchli, 1966)

• piecewise syndetic sets (Brown, 1968)

• Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finitesums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968).

• (m, p, c)-sets (Deuber, 1973)

• IP sets (Hindman, 1974, see also Hindman, Strauss, 1998)

• MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)

• central sets; i.e. the members of any minimal idempotent in βN , the Stone–Čech compactification of theintegers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)

11.2 References1. Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory

(Series A) 93 (2001), 18–36.

2. T. Brown, An interesting combinatorial method in the theory of locally finite semigroups, Pacific J. Math. 36,no. 2 (1971), 285–289.

3. W. Deuber, Mathematische Zeitschrift 133, (1973) 109–123

4. N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combinatorial Theory (Series A)17 (1974) 1–11.

5. C.St.J.A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965),33–39.

6. N. Hindman, D. Strauss, Algebra in the Stone–Čech compactification, De Gruyter, 1998

7. J.Sanders, A Generalization of Schur’s Theorem, Doctoral Dissertation, Yale University, 1968.

Chapter 12

Pi system

In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that

• P is non-empty.

• A ∩ B ∈ P whenever A and B are in P.

That is, P is a non-empty family of subsets of Ω that is closed under finite intersections. The importance of π-systemsarise from the fact that if two probability measures agree on a π-system, then they agree on the σ-algebra generatedby that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they holdfor the generated σ-algebra as well. This is the case whenever the collection of subsets for which the property holdsis a λ-system. π-systems are also useful for checking independence of random variables.This is desirable because in practice, π-systems are often simpler to work with than σ-algebras. For example, it maybe awkward to work with σ-algebras generated by infinitely many sets σ(E1, E2, . . .) . So instead we may examinethe union of all σ-algebras generated by finitely many sets

∪n σ(E1, . . . , En) . This forms a π-system that generates

the desired σ-algebra. Another example is the collection of all interval subsets of the real line, along with the emptyset, which is a π-system that generates the very important Borel σ-algebra of subsets of the real line.

12.1 Examples

• On the real line R , the intervals (−∞, a] form a π-system. Similarly, the intervals (a, b] form a π-system, ifthe empty set is also included.

• The topology (collection of open subsets) of any topological space is a π-system.

• For any collection Σ of subsets of Ω, there exists a π-system IΣ which is the unique smallest π-system of Ω tocontain every element of Σ, and is called the π-system generated by Σ.

• For any measurable function f : Ω → R , the set If =f−1 ((−∞, x]) : x ∈ R

defines a π-system, and is

called the π-system generated by f. (Alternatively,f−1 ((a, b]) : a, b ∈ R, a < b

∪∅ defines a π-system

generated by f .)

• If P1 and P2 are π-systems for Ω1 and Ω2, respectively, then A1 × A2 : A1 ∈ P1, A2 ∈ P2 is a π-systemfor the product space Ω1×Ω2.

• Any σ-algebra is a π-system.

12.2 Relationship to λ-Systems

A λ-system on Ω is a set D of subsets of Ω, satisfying

32

12.3. Π-SYSTEMS IN PROBABILITY 33

• Ω ∈ D ,

• if A ∈ D then Ac ∈ D ,

• if A1, A2, A3, . . . is a sequence of disjoint subsets in D then ∪∞n=1An ∈ D .

Whilst it is true that any σ-algebra satisfies the properties of being both a π-system and a λ-system, it is not truethat any π-system is a λ-system, and moreover it is not true that any π-system is a σ-algebra. However, a usefulclassification is that any set system which is both a λ-system and a π-system is a σ-algebra. This is used as a step inproving the π-λ theorem.

12.2.1 The π-λ Theorem

Let D be a λ-system, and let I ⊆ D be a π-system contained in D . The π-λ Theorem[1] states that the σ-algebraσ(I) generated by I is contained in D : σ(I) ⊂ D .The π-λ theorem can be used to prove many elementary measure theoretic results. For instance, it is used in provingthe uniqueness claim of the Carathéodory extension theorem for σ-finite measures.[2]

The π-λ theorem is closely related to the monotone class theorem, which provides a similar relationship betweenmonotone classes and algebras, and can be used to derive many of the same results. Since π-systems are simplerclasses than algebras, it can be easier to identify the sets that are in them while, on the other hand, checking whetherthe property under consideration determines a λ-system is often relatively easy. Despite the difference between thetwo theorems, the π-λ theorem is sometimes referred to as the monotone class theorem.[1]

Example

Let μ₁ , μ2 : F → R be two measures on the σ-algebra F, and suppose that F = σ(I) is generated by a π-system I. If

1. μ1(A) = μ2(A), ∀ A ∈ I, and

2. μ1(Ω) = μ2(Ω) < ∞,

then μ₁ = μ2. This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If thisresult does not seem very remarkable, consider the fact that it usually is very difficult or even impossible to fullydescribe every set in the σ-algebra, and so the problem of equating measures would be completely hopeless withoutsuch a tool.Idea of Proof[2] Define the collection of sets

D = A ∈ σ(I) : µ1(A) = µ2(A) .

By the first assumption, μ1 and μ2 agree on I and thus I D. By the second assumption, Ω ∈ D, and it can furtherbe shown that D is a λ-system. It follows from the π-λ theorem that σ(I) D σ(I), and so D = σ(I). That is to say,the measures agree on σ(I).

12.3 π-Systems in Probability

π-systems are more commonly used in the study of probability theory than in the general field of measure theory.This is primarily due to probabilistic notions such as independence, though it may also be a consequence of the factthat the π-λ theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically provethe same results via monotone classes, rather than π-systems.

34 CHAPTER 12. PI SYSTEM

12.3.1 Equality in Distribution

The π-λ theoremmotivates the common definition of the probability distribution of a random variableX : (Ω,F ,P) →R in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable isdefined as

FX(a) = P [X ≤ a] , a ∈ R

whereas the seemingly more general law of the variable is the probability measure

LX(B) = P[X−1(B)

], B ∈ B(R)

where B(R) is the Borel σ-algebra. We say that the random variables X : (Ω,F ,P) , and Y : (Ω, F , P) → R (ontwo possibly different probability spaces) are equal in distribution (or law),XD

=Y , if they have the same cumulativedistribution functions, FX = FY. The motivation for the definition stems from the observation that if FX = FY, thenthat is exactly to say that LX and LY agree on the π-system (−∞, a] : a ∈ R which generates B(R) , and so bythe example above: LX = LY .A similar result holds for the joint distribution of a random vector. For example, suppose X and Y are two randomvariables defined on the same probability space (Ω,F ,P) , with respectively generated π-systems IX and IY . Thejoint cumulative distribution function of (X,Y) is

FX,Y (a, b) = P [X ≤ a, Y ≤ b] = P[X−1((−∞, a]) ∩ Y −1((−∞, b])

], a, b ∈ R

However, A = X−1((−∞, a]) ∈ IX and B = Y −1((−∞, b]) ∈ IY . Since

IX,Y = A ∩B : A ∈ IX , B ∈ IY

is a π-system generated by the random pair (X,Y), the π-λ theorem is used to show that the joint cumulative distribu-tion function suffices to determine the joint law of (X,Y). In other words, (X,Y) and (W,Z) have the same distributionif and only if they have the same joint cumulative distribution function.In the theory of stochastic processes, two processes (Xt)t∈T , (Yt)t∈T are known to be equal in distribution if andonly if they agree on all finite-dimensional distributions. i.e. for all t1, . . . , tn ∈ T, n ∈ N .

(Xt1 , . . . , Xtn)D=(Yt1 , . . . , Ytn)

The proof of this is another application of the π-λ theorem.[3]

12.3.2 Independent Random Variables

The theory of π-system plays an important role in the probabilistic notion of independence. If X and Y are tworandom variables defined on the same probability space (Ω,F ,P) then the random variables are independent if andonly if their π-systems IX , IY satisfy

P [A ∩B] = P [A]P [B] , ∀A ∈ IX , B ∈ IY ,

which is to say that IX , IY are independent. This actually is a special case of the use of π-systems for determiningthe distribution of (X,Y).

12.4. SEE ALSO 35

Example

Let Z = (Z1, Z2) , where Z1, Z2 ∼ N (0, 1) are iid standard normal random variables. Define the radius andargument (arctan) variables

R =√

Z21 + Z2

2 , Θ = tan−1(Z2/Z1)

Then R and Θ are independent random variables.To prove this, it is sufficient to show that the π-systems IR, IΘ are independent: i.e.

P[R ≤ ρ,Θ ≤ θ] = P[R ≤ ρ]P[Θ ≤ θ] ∀ρ ∈ [0,∞), θ ∈ [0, 2π].

Confirming that this is the case is an exercise in changing variables. Fix ρ ∈ [0,∞), θ ∈ [0, 2π] , then the probabilitycan be expressed as an integral of the probability density function of Z .

P[R ≤ ρ,Θ ≤ θ] =

∫R≤ρ,Θ≤θ

1

2πexp

(−1

2(z21 + z22)

)dz1dz2

=

∫ θ

0

∫ ρ

0

1

2πe−

r2

2 rdrdθ

=

(∫ θ

0

1

2πdθ

)(∫ ρ

0

e−r2

2 rdr

)= P[Θ ≤ θ]P[R ≤ ρ].

12.4 See also• λ-systems

• σ-algebra

• Monotone class theorem

• Probability distribution

• Independence

12.5 Notes[1] Kallenberg, Foundations Of Modern Probability, p.2

[2] Durrett, Probability Theory and Examples, p.404

[3] Kallenberg, Foundations Of Modern probability, p. 48

12.6 References• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN0-387-22833-0.

• David Williams (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6.

Chapter 13

Ring of sets

Not to be confused with Ring (mathematics).

In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In ordertheory, a nonempty family of setsR is called a ring (of sets) if it is closed under intersection and union. That is, thefollowing two statements are true for all sets A and B ,

1. A,B ∈ R implies A ∩B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R. [1]

In measure theory, a ring of setsR is instead a nonempty family closed under unions and set-theoretic differences.[2]That is, the following two statements are true for all sets A and B (including when they are the same set),

1. A,B ∈ R implies A \B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R.

This implies the empty set is in R . It also implies that R is closed under symmetric difference and intersection,because of the identities

1. AB = (A \B) ∪ (B \A) and

2. A ∩B = A \ (A \B).

(So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, theseoperations give R the structure of a boolean ring. Conversely, every family of sets closed under both symmetricdifference and intersection is also closed under union and differences. This is due to the identities

1. A ∪B = (AB) (A ∩B) and

2. A \B = A (A ∩B).

13.1 Examples

If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.If (X,≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongsto an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions. However,in general it will not be closed under differences of sets.The open sets and closed sets of any topological space are closed under both unions and intersections.[1]

36

13.2. RELATED STRUCTURES 37

On the real line R, the family of sets consisting of the empty set and all finite unions of intervals of the form (a, b],a,b in R is a ring in the measure theory sense.If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed underboth unions and intersections.[1]

If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form aring of sets.[1]

13.2 Related structures

A ring of sets (in the order-theoretic sense) forms a distributive lattice in which the intersection and union operationscorrespond to the lattice’s meet and join operations, respectively. Conversely, every distributive lattice is isomorphicto a ring of sets; in the case of finite distributive lattices, this is Birkhoff’s representation theorem and the sets maybe taken as the lower sets of a partially ordered set.[1]

A field of subsets of X is a ring that contains X and is closed under relative complement. Every field, and so alsoevery σ-algebra, is a ring of sets in the measure theory sense.A semi-ring (of sets) is a family of sets S with the properties

1. ∅ ∈ S,

2. A,B ∈ S implies A ∩B ∈ S, and

3. A,B ∈ S implies A \B =∪n

i=1 Ci for some disjoint C1, . . . , Cn ∈ S.

Clearly, every ring (in the measure theory sense) is a semi-ring.A semi-field of subsets of X is a semi-ring that contains X.

13.3 References[1] Birkhoff, Garrett (1937), “Rings of sets”,DukeMathematical Journal 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-

X, MR 1546000.

[2] De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.

13.4 External links• Ring of sets at Encyclopedia of Mathematics

Chapter 14

Sigma-algebra

"Σ-algebra” redirects here. For an algebraic structure admitting a given signature Σ of operations, see Universal al-gebra.

In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a setX is a collection of subsets of X that is closed under countable-fold set operations (complement, union of countablymany sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitelymany set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations.The pair (X, Σ) is also a field of sets, called a measurable space.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for whicha given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as thefoundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events whichcan be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.In statistics, (sub) σ-algebras are needed for a formal mathematical definition of sufficient statistic,[1] particularlywhen the statistic is a function or a random process and the notion of conditional density is not applicable.If X = a, b, c, d, one possible σ-algebra on X is Σ = ∅, a, b, c, d, a, b, c, d , where ∅ is the empty set.However, a finite algebra is always a σ-algebra.If A1, A2, A3, … is a countable partition of X then the collection of all unions of sets in the partition (includingthe empty set) is a σ-algebra.A more useful example is the set of subsets of the real line formed by starting with all open intervals and addingin all countable unions, countable intersections, and relative complements and continuing this process (by transfiniteiteration through all countable ordinals) until the relevant closure properties are achieved (a construction known asthe Borel hierarchy).

14.1 Motivation

There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managingpartial information characterized by sets.

14.1.1 Measure

Ameasure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as makingprecise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of theirindividual sizes, even for an infinite sequence of disjoint sets.One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example theaxiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the realline, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead asmaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

38

14.1. MOTIVATION 39

operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable setand the countable union of measurable sets is a measurable set. Non-empty collections of sets with these propertiesare called σ-algebras.

14.1.2 Limits of Sets

Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences ofsets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows onσ-algebras.

• The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim supn→∞

An =∞∩

n=1

∞∪m=n

Am.

• The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim infn→∞

An =∞∪

n=1

∞∩m=n

Am.

• If, in fact,

lim infn→∞

An = lim supn→∞

An

then the limn→∞ An exists as that common set.

14.1.3 Sub σ-algebras

In much of probability, especially when conditional expectation is involved, one is concerned with sets that representonly part of all the possible information that can be observed. This partial information can be characterized with asmaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only toand determined only by the partial information. A simple example suffices to illustrate this idea.Imagine you are playing a game that involves flipping a coin repeatedly and observing whether it comes up Heads (H)or Tails (T). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last.This means the sample space Ω must consist of all possible infinite sequences of H or T :

Ω = H,T∞ = (x1, x2, x3, . . . ) : xi ∈ H,T, i ≥ 1

However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the nextflip. The observed information at that point can be described in terms of the 2n possibilities for the first n flips.Formally, since you need to use subsets of Ω, this is codified as the σ-algebra

Gn = A× H,T∞ : A ⊂ H,Tn

Observe that then

G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G∞

where G∞ is the smallest σ-algebra containing all the others.

40 CHAPTER 14. SIGMA-ALGEBRA

14.2 Definition and properties

14.2.1 Definition

Let X be some set, and let 2X represent its power set. Then a subset Σ ⊂ 2X is called a σ-algebra if it satisfies thefollowing three properties:[2]

1. X is in Σ.

2. Σ is closed under complementation: If A is in Σ, then so is its complement, X\A.

3. Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1 ∪ A2 ∪ A3 ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying DeMorgan’s laws).It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set,is also in Σ. Moreover, by (3) it follows as well that X, ∅ is the smallest possible σ-algebra.Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebraover X, is called a measurable space. A function between two measurable spaces is called a measurable function ifthe preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with themeasurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin’s theorem(below).

14.2.2 Dynkin’s π-λ theorem

This theorem (or the related monotone class theorem) is an essential tool for proving many results about propertiesof specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

A π-system P is a collection of subsets of Σ that is closed under finitely many intersections, anda Dynkin system (or λ-system) D is a collection of subsets of Σ that contains Σ and is closed undercomplement and under countable unions of disjoint subsets.

Dynkin’s π-λ theorem says, if P is a π-system and D is a Dynkin system that contains P then the σ-algebra σ(P)generated by P is contained in D. Since certain π-systems are relatively simple classes, it may not be hard to verifythat all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of allsubsets with the property is a Dynkin system can also be straightforward. Dynkin’s π-λ Theorem then implies thatall sets in σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P).One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures orintegrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integraltypically associated with computing the probability:

P(X ∈ A) =∫AF (dx) for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while P is a probability measure, defined ona σ-algebra Σ of subsets of some sample space Ω.

14.2.3 Combining σ-algebras

Suppose Σα : α ∈ A is a collection of σ-algebras on a space X.

• The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it oftenis denoted by:

14.2. DEFINITION AND PROPERTIES 41

∧α∈A

Σα.

Sketch of Proof: Let Σ∗ denote the intersection. Since X is in every Σα, Σ∗ is not empty. Closure undercomplement and countable unions for every Σα implies the same must be true for Σ∗. Therefore, Σ∗ is aσ-algebra.

• The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates aσ-algebra known as the join which typically is denoted

∨α∈A

Σα = σ

( ∪α∈A

Σα

).

A π-system that generates the join is

P =

n∩

i=1

Ai : Ai ∈ Σαi , αi ∈ A, n ≥ 1

.

Sketch of Proof: By the case n = 1, it is seen that each Σα ⊂ P , so∪α∈A

Σα ⊂ P.

This implies

σ

( ∪α∈A

Σα

)⊂ σ(P)

by the definition of a σ-algebra generated by a collection of subsets. On the other hand,

P ⊂ σ

( ∪α∈A

Σα

)

which, by Dynkin’s π-λ theorem, implies

σ(P) ⊂ σ

( ∪α∈A

Σα

).

14.2.4 σ-algebras for subspaces

Suppose Y is a subset of X and let (X, Σ) be a measurable space.

• The collection Y ∩ B: B ∈ Σ is a σ-algebra of subsets of Y.

• Suppose (Y, Λ) is a measurable space. The collection A ⊂ X : A ∩ Y ∈ Λ is a σ-algebra of subsets of X.

14.2.5 Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[3] A σ-ring need not be a σ-algebra, as for examplemeasurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real linehas infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takesmeasurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtainedby their countable union yet its measure is not finite.

42 CHAPTER 14. SIGMA-ALGEBRA

14.2.6 Typographic note

σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, Σ) may bedenoted as (X,F) or (X,F) . This is handy to avoid situations where the letter Σ may be confused for the summationoperator.

14.3 Examples

14.3.1 Simple set-based examples

Let X be any set.

• The family consisting only of the empty set and the set X, called the minimal or trivial σ-algebra over X.

• The power set of X, called the discrete σ-algebra.

• The collection ∅, A, Ac, X is a simple σ-algebra generated by the subset A.

• The collection of subsets of X which are countable or whose complements are countable is a σ-algebra (whichis distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by thesingletons of X. Note: “countable” includes finite or empty.

• The collection of all unions of sets in a countable partition of X is a σ-algebra.

14.3.2 Stopping time sigma-algebras

A stopping time τ can define a σ -algebraFτ , the so-called stopping time sigma-algebra, which in a filtered probabilityspace describes the information up to the random time τ in the sense that, if the filtered probability space is interpretedas a random experiment, the maximum information that can be found out about the experiment from arbitrarily oftenrepeating it until the time τ is Fτ .[4]

14.4 σ-algebras generated by families of sets

14.4.1 σ-algebra generated by an arbitrary family

Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every setin F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated byF.

For a simple example, consider the set X = 1, 2, 3. Then the σ-algebra generated by the single subset 1 isσ(1) = ∅, 1, 2, 3, 1, 2, 3. By an abuse of notation, when a collection of subsets contains only oneelement, A, one may write σ(A) instead of σ(A); in the prior example σ(1) instead of σ(1). Indeed, usingσ(A1, A2, ...) to mean σ(A1, A2, ...) is also quite common.There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

14.4.2 σ-algebra generated by a function

If f is a function from a set X to a set Y and B is a σ-algebra of subsets of Y, then the σ-algebra generated by thefunction f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B. i.e.

σ(f) = f−1(S) |S ∈ B.

14.4. Σ-ALGEBRAS GENERATED BY FAMILIES OF SETS 43

A function f from a set X to a set Y is measurable with respect to a σ-algebra Σ of subsets of X if and only if σ(f) isa subset of Σ.One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topologicalspace and B is the collection of Borel sets on Y.If f is a function from X to Rn then σ(f) is generated by the family of subsets which are inverse images of inter-vals/rectangles in Rn:

σ(f) = σ(f−1((a1, b1]× · · · × (an, bn]) : ai, bi ∈ R

).

A useful property is the following. Assume f is a measurable map from (X, ΣX) to (S, ΣS) and g is a measurable mapfrom (X, ΣX) to (T, ΣT). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂σ(g). If S is finite or countably infinite or if (S, ΣS) is a standard Borel space (e.g., a separable complete metric spacewith its associated Borel sets) then the converse is also true.[5] Examples of standard Borel spaces include Rn with itsBorel sets and R∞ with the cylinder σ-algebra described below.

14.4.3 Borel and Lebesgue σ-algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or,equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivialexample that is not a Borel set, see the Vitali set or Non-Borel sets.On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebracontains more sets than the Borel σ-algebra onRn and is preferred in integration theory, as it gives a complete measurespace.

14.4.4 Product σ-algebra

Let (X1,Σ1) and (X2,Σ2) be two measurable spaces. The σ-algebra for the corresponding product spaceX1 ×X2

is called the product σ-algebra and is defined by

Σ1 × Σ2 = σ(B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2).

Observe that B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2 is a π-system.The Borel σ-algebra for Rn is generated by half-infinite rectangles and by finite rectangles. For example,

B(Rn) = σ ((−∞, b1]× · · · × (−∞, bn] : bi ∈ R) = σ ((a1, b1]× · · · × (an, bn] : ai, bi ∈ R) .

For each of these two examples, the generating family is a π-system.

14.4.5 σ-algebra generated by cylinder sets

Suppose

X ⊂ RT = f : f(t) ∈ R, t ∈ T

is a set of real-valued functions. Let B(R) denote the Borel subsets ofR. A cylinder subset of X is a finitely restrictedset defined as

Ct1,...,tn(B1, . . . , Bn) = f ∈ X : f(ti) ∈ Bi, 1 ≤ i ≤ n.

Each

44 CHAPTER 14. SIGMA-ALGEBRA

Ct1,...,tn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n

is a π-system that generates a σ-algebra Σt1,...,tn . Then the family of subsets

FX =∞∪

n=1

∪ti∈T,i≤n

Σt1,...,tn

is an algebra that generates the cylinder σ-algebra for X. This σ-algebra is a subalgebra of the Borel σ-algebradetermined by the product topology of RT restricted to X.An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In thiscase, it suffices to consider the cylinder sets

Cn(B1, . . . , Bn) = (B1 × · · · ×Bn × R∞) ∩X = (x1, x2, . . . , xn, xn+1, . . . ) ∈ X : xi ∈ Bi, 1 ≤ i ≤ n,

for which

Σn = σ(Cn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n)

is a non-decreasing sequence of σ-algebras.

14.4.6 σ-algebra generated by random variable or vector

Suppose (Ω,Σ,P) is a probability space. If Y : Ω → Rn is measurable with respect to the Borel σ-algebra on Rn

then Y is called a random variable (n = 1) or random vector (n ≥ 1). The σ-algebra generated by Y is

σ(Y ) = Y −1(A) : A ∈ B(Rn).

14.4.7 σ-algebra generated by a stochastic process

Suppose (Ω,Σ,P) is a probability space and RT is the set of real-valued functions on T . If Y : Ω → X ⊂ RT ismeasurable with respect to the cylinder σ-algebra σ(FX) (see above) for X then Y is called a stochastic process orrandom process. The σ-algebra generated by Y is

σ(Y ) =Y −1(A) : A ∈ σ(FX)

= σ(Y −1(A) : A ∈ FX),

the σ-algebra generated by the inverse images of cylinder sets.

14.5 See also• Join (sigma algebra)

• Measurable function

• Sample space

• Separable sigma algebra

• Sigma ring

• Sigma additivity

14.6. REFERENCES 45

14.6 References[1] Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1118122372.

[2] Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

[3] Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 9780470317952.

[4] Fischer, Tom (2013). “On simple representations of stopping times and stopping time sigma-algebras”. Statistics andProbability Letters 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.

[5] Kallenberg, Olav (2001). Foundations of Modern Probability, 2nd ed. Springer. p. 7. ISBN 0-387-95313-2.

14.7 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Sigma Algebra from PlanetMath.

Chapter 15

Sigma-ideal

In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read “sigma,” means countable in thiscontext) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent applicationis perhaps in probability theory.Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if thefollowing properties are satisfied:(i) Ø ∈ N;(ii) When A ∈ N and B ∈ Σ , B ⊆ A⇒ B ∈ N;(iii) Ann∈N ⊆ N ⇒

∪n∈N An ∈ N.

Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. Theconcept of σ-ideal is dual to that of a countably complete (σ-) filter.If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.The notion can be generalized to preorders (P,≤,0) with a bottom element 0 as follows: I is a σ-ideal of P just when(i') 0 ∈ I,(ii') x ≤ y & y ∈ I ⇒ x ∈ I, and(iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ≤ y for each nThus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist).It is natural in this context to ask that P itself have countable suprema.

15.1 References• Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin,Germany.

46

Chapter 16

Sigma-ring

Inmathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countableunion and relative complementation.

16.1 Formal definition

LetR be a nonempty collection of sets. ThenR is a σ-ring if:

1.∪∞

n=1 An ∈ R if An ∈ R for all n ∈ N

2. A∖B ∈ R if A,B ∈ R

16.2 Properties

From these two properties we immediately see that

∩∞n=1 An ∈ R if An ∈ R for all n ∈ N

This is simply because ∩∞n=1An = A1 ∖ ∪∞

n=1(A1 ∖An) .

16.3 Similar concepts

If the first property is weakened to closure under finite union (i.e.,A∪B ∈ RwheneverA,B ∈ R ) but not countableunion, thenR is a ring but not a σ-ring.

16.4 Uses

σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one doesnot wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be aσ-field.A σ-ring R that is a collection of subsets of X induces a σ-field for X . Define A to be the collection of all subsetsof X that are elements ofR or whose complements are elements ofR . Then A is a σ-field over the set X . In factA is the minimal σ-field containingR since it must be contained in every σ-field containingR .

47

48 CHAPTER 16. SIGMA-RING

16.5 See also• Delta ring

• Ring of sets

• Sigma field

16.6 References• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-ringsin development of Lebesgue theory.

16.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 49

16.7 Text and image sources, contributors, and licenses

16.7.1 Text• Almost disjoint sets Source: http://en.wikipedia.org/wiki/Almost_disjoint_sets?oldid=607152448Contributors: Revolver, CharlesMatthews,

Kaol, Nickj, Oleg Alexandrov, Pol098, Salix alba, SmackBot, Maksim-e~enwiki, Dreadstar, CBM, Addbot, Yobot, SassoBot, Erik9bot,Kasterma and Anonymous: 3

• Delta-ring Source: http://en.wikipedia.org/wiki/Delta-ring?oldid=544536467Contributors: CharlesMatthews, Touriste, Salix alba, Keith111,Addbot and ZéroBot

• Disjoint sets Source: http://en.wikipedia.org/wiki/Disjoint_sets?oldid=655510316 Contributors: AxelBoldt, Mav, Tarquin, Jeronimo,Shd~enwiki, Arvindn, Toby Bartels, Michael Hardy, Wshun, Revolver, Charles Matthews, Fibonacci, Robbot, Sheskar~enwiki, TobiasBergemann, Giftlite, Fropuff, Achituv~enwiki, Almit39, Nickj, Jumbuck, Dirac1933, Salix alba, FlaBot, Jameshfisher, Chobot, Alge-braist, YurikBot, RobotE, Bota47, Light current, Raijinili, SmackBot, RDBury,Maksim-e~enwiki, DHN-bot~enwiki, Vina-iwbot~enwiki,Typinaway, Mets501, Iridescent, Devourer09, Ezrakilty, Christian75, Kilva, RobHar, Salgueiro~enwiki, JAnDbot, David Eppstein,Miker70741, J.delanoy, STBotD, VolkovBot, Jamelan, PanagosTheOther, Alexbot, DumZiBoT, WikHead, Addbot, LaaknorBot, Sp-Bot, Legobot, Luckas-bot, Yobot, GrouchoBot, Erik9bot, MastiBot, Peacedance, Tbhotch, Ripchip Bot, ZéroBot, 28bot, ClueBot NG,Prakhar.agrwl, Helpful Pixie Bot, BG19bot, Amyxz, JYBot, Stephan Kulla, Domez99, Flat Out and Anonymous: 34

• Dynkin system Source: http://en.wikipedia.org/wiki/Dynkin_system?oldid=598999646 Contributors: Charles Matthews, Giftlite, Lupin,Skylarth, Bob.v.R, Rich Farmbrough, Paul August, Tsirel, Btyner, Sin-man, Rjwilmsi, Salix alba, RussBot, Natalie Packham, Thijs!bot,Vanish2, Nm420, Jmath666, SieBot, Melcombe, Addbot, Tcncv, Angry bee, DSisyphBot, 777sms, Cncmaster, ChrisGualtieri, Rank-onemap, DarenCline and Anonymous: 7

• Family of sets Source: http://en.wikipedia.org/wiki/Family_of_sets?oldid=659156521 Contributors: Toby Bartels, Charles Matthews,Chris Howard, Oleg Alexandrov, Salix alba, Chobot, Wavelength, Arthur Rubin, Reedy, Mhss, CBM, RomanXNS, David Eppstein,JoergenB, Pomte, PixelBot, Avoided, Addbot, Matěj Grabovský, Calle, Erik9bot, DivineAlpha, NearSetAccount, Xnn, Sheerun, ClueBotNG, Wcherowi and Anonymous: 8

• Field of sets Source: http://en.wikipedia.org/wiki/Field_of_sets?oldid=655869630 Contributors: Charles Matthews, David Shay, To-bias Bergemann, Giftlite, William Elliot, Rich Farmbrough, Paul August, Touriste, DaveGorman, Kuratowski’s Ghost, Bart133, OlegAlexandrov, Salix alba, YurikBot, Trovatore, Mike Dillon, Arthur Rubin, That Guy, From That Show!, SmackBot, Mhss, Gala.martin,Stotr~enwiki, Mathematrucker, R'n'B, BotMultichill, VVVBot, Hans Adler, Addbot, DaughterofSun, Jarble, AnomieBOT, Citation bot,Kiefer.Wolfowitz, Yahia.barie, EmausBot, Tijfo098, ClueBot NG, MerlIwBot, Deltahedron, Mohammad Abubakar and Anonymous: 14

• Finite character Source: http://en.wikipedia.org/wiki/Finite_character?oldid=544108057Contributors: Michael Hardy, CharlesMatthews,Aleph4, Giftlite, Paul August, Salix alba, YurikBot, Arthur Rubin, Judicael, Dreadstar, David Eppstein, KittyHawker, Addbot, Ptbot-gourou, 777sms and Anonymous: 1

• Finite intersection property Source: http://en.wikipedia.org/wiki/Finite_intersection_property?oldid=658952118Contributors: MichaelHardy, Revolver, Charles Matthews, Dcoetzee, Ruakh, EmilJ, ABCD, Linas, Yuval Madar~enwiki, R.e.b., YurikBot, Ondenc, Kompik,SmackBot, Mhss, Nbarth, Physis, Zero sharp, Vaughan Pratt, Carl Turner, David Eppstein, Alighat~enwiki, JackSchmidt, J.Gowers,Addbot, Topology Expert, Ozob, Yobot, FrescoBot, Xnn, WikitanvirBot, Mark viking and Anonymous: 8

• Greedoid Source: http://en.wikipedia.org/wiki/Greedoid?oldid=643814038 Contributors: Edward, Michael Hardy, Charles Matthews,Dcoetzee, Zoicon5, Altenmann, Peterkwok, Peter Kwok, Zaslav, Nickj, Oleg Alexandrov, Mathbot, Gaius Cornelius, Claygate, Grin-Bot~enwiki, Bluebot, Mhym, Dreadstar, E-Kartoffel, Sytelus, Headbomb, LachlanA, David Eppstein, LokiClock, Justin W Smith, HansAdler, Addbot, Citation bot, Citation bot 1, Kiefer.Wolfowitz, SporkBot, Desikblack, Nosuchforever, Deltahedron, Dillon128, JMP EAXand Anonymous: 11

• Partition of a set Source: http://en.wikipedia.org/wiki/Partition_of_a_set?oldid=663746516 Contributors: AxelBoldt, Tomo, Patrick,Michael Hardy, Wshun, Kku, Revolver, Charles Matthews, Zero0000, Robbot, MathMartin, Ruakh, Tobias Bergemann, Giftlite, Smjg,Arved, Fropuff, Gubbubu, Mennucc, Sam Hocevar, Tsemii, TedPavlic, Paul August, Zaslav, Elwikipedista~enwiki, El C, PhilHibbs,Corvi42, Oleg Alexandrov, Stemonitis, Bobrayner, Kelly Martin, Linas, MFH, Mayumashu, Salix alba, R.e.b., FlaBot, Mathbot, Yurik-Bot, Laurentius, Gaius Cornelius, StevenL, Pred, Finell, Capitalist, That Guy, From That Show!, Adam majewski, Mcld, Mhss, Tsca.bot,Mhym, Armend, Jon Awbrey, Rob Zako, CRGreathouse, Sopoforic, Escarbot, Magioladitis, Jiejunkong, David Eppstein, Lantonov,Elenseel, TXiKiBoT, Anonymous Dissident, PaulTanenbaum, Jamelan, Skippydo, PipepBot, DragonBot, Watchduck, Addbot, AkhtaBot,Legobot, Luckas-bot, Yobot, Bunnyhop11, Calle, ArthurBot, Stereospan, MastiBot, EmausBot, MartinThoma, ZéroBot, D.Lazard, Or-ange Suede Sofa, ClueBot NG,Mesoderm, Helpful Pixie Bot, BG19bot, MinatureCookie, Mark viking, Cepphus, BethNaught and Anony-mous: 45

• Partition regularity Source: http://en.wikipedia.org/wiki/Partition_regularity?oldid=607156121 Contributors: Michael Hardy, CharlesMatthews, Altenmann, Tobias Bergemann, Giftlite, Redquark, Greg321, JdH, Landonproctor, Ksoileau, FizzyP, Sullivan.t.j, David Epp-stein, Darnedfrenchman, Addbot, Yobot, Kiefer.Wolfowitz, Xnn, WikitanvirBot and Anonymous: 6

• Pi system Source: http://en.wikipedia.org/wiki/Pi_system?oldid=638261010 Contributors: Charles Matthews, Salix alba, Wavelength,Spacepotato, Dmharvey, Stifle, Mattroberts, Ru elio, Vanish2, Sullivan.t.j, David Eppstein, TXiKiBoT, Kinrayfchen, Addbot, Aroch,Yobot, Hairer, DrilBot, Helpful Pixie Bot, BG19bot, Odaniel1, Savick01, Bwangaa, Egreif1, DarenCline and Anonymous: 9

• Ring of sets Source: http://en.wikipedia.org/wiki/Ring_of_sets?oldid=654807888 Contributors: Michael Hardy, Charles Matthews,Chentianran~enwiki, Lethe, DemonThing, EmilJ, Touriste, Keenan Pepper, Salix alba, Trovatore, SmackBot, Keith111, David Epp-stein, VolkovBot, Joeldl, DragonBot, PixelBot, Addbot, Jarble, Luckas-bot, Erik9bot, ZéroBot, Zephyrus Tavvier, Wlelsing, DarenClineand Anonymous: 8

• Sigma-algebra Source: http://en.wikipedia.org/wiki/Sigma-algebra?oldid=663323104 Contributors: AxelBoldt, Zundark, Tarquin, Iwn-bap, Miguel~enwiki, Michael Hardy, Chinju, Karada, Stevan White, Charles Matthews, Dysprosia, Vrable, AndrewKepert, Fibonacci,Robbot, Romanm, Ruakh, Giftlite, Lethe, MathKnight, Mboverload, Gubbubu, Gauss, Barnaby dawson, Vivacissamamente, WilliamElliot, ArnoldReinhold, Paul August, Zaslav, Elwikipedista~enwiki, MisterSheik, EmilJ, SgtThroat, Jung dalglish, Tsirel, Passw0rd,Msh210, Jheald, Cmapm, Ultramarine, Oleg Alexandrov, Linas, Graham87, Salix alba, FlaBot, Mathbot, Jrtayloriv, Chobot, Jayme,YurikBot, Lucinos~enwiki, Archelon, Trovatore,Mindthief, Solstag, Crasshopper, Dinno~enwiki, Nielses, SmackBot, Melchoir, JanusDC,

50 CHAPTER 16. SIGMA-RING

Object01, Dan Hoey, MalafayaBot, RayAYang, Nbarth, DHN-bot~enwiki, Javalenok, Gala.martin, Henning Makholm, Lambiam, Dbtfz,Jim.belk, Mets501, Stotr~enwiki, Madmath789, CRGreathouse, CBM, David Cooke, Mct mht, Xantharius, , Thijs!bot, Escarbot,Keith111, Forgetfulfunctor, Quentar~enwiki, MSBOT, Magioladitis, RogierBrussee, Paartha, Joeabauer, Hippasus, Policron, Cerberus0,Digby Tantrum, Jmath666, Alfredo J. Herrera Lago, StevenJohnston, Ocsenave, Tcamps42, SieBot, Melcombe, MicoFilós~enwiki, An-drewbt, The Thing That Should Not Be, BearMachine, 1ForTheMoney, DumZiBoT, Addbot, Luckas-bot, Yobot, Li3939108, Amirmath,Godvjrt, Xqbot, RibotBOT, Charvest, FrescoBot, BrideOfKripkenstein, AstaBOTh15, Stpasha, RedBot, Soumyakundu, Stj6, TjBot,Max139, KHamsun, Rafi5749, ClueBot NG, Thegamer 298, QuarkyPi, Brad7777, AntanO, Shalapaws, Crawfoal, Dexbot, Y256, JochenBurghardt, A koyee314, Limit-theorem, Mark viking, NumSPDE,Moyaccercchi, Killaman slaughtermaster, DarenCline and Anonymous:81

• Sigma-ideal Source: http://en.wikipedia.org/wiki/Sigma-ideal?oldid=607167568 Contributors: Charles Matthews, RickK, Vivacissama-mente, Linas, Magister Mathematicae, RussBot, SmackBot, Valfontis, Yobot and Anonymous: 3

• Sigma-ring Source: http://en.wikipedia.org/wiki/Sigma-ring?oldid=635505260 Contributors: Charles Matthews, Touriste, Salix alba,Maxbaroi, Konradek, Keith111, Addbot, Luckas-bot, Xqbot, ZéroBot, Zephyrus Tavvier, Jim Sukwutput, Brirush, DarenCline andAnonymous: 3

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