Ring of sets 1_3

Ring of sets 1From Wikipedia, the free encyclopedia

Contents

1 Birkhos representation theorem 11.1 Understanding the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The partial order of join-irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Birkhos theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Rings of sets and preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Borel set 72.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Alternative non-equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Distributive lattice 113.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Characteristic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Free distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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4 Dynkin system 174.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Empty set 195.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Family of sets 256.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Field of sets 277.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 27

7.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.1.2 Separative and compact elds of sets: towards Stone duality . . . . . . . . . . . . . . . . . 27

7.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2.2 Topological elds of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2.3 Preorder elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2.4 Complex algebras and elds of sets on relational structures . . . . . . . . . . . . . . . . . . 29

7.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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7.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Intersection (set theory) 318.1 Basic denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Lebesgue measure 379.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

10 Mathematics 4210.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 4910.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11 Measure (mathematics) 60

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11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.3.2 Measures of innite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 6211.3.3 Measures of innite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 62

11.4 Sigma-nite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12 Pi system 6812.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

12.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13 Ring of sets 7213.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

14 Semiring 7414.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7414.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.2.1 In general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514.2.2 Specic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14.3 Semiring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.5 Complete and continuous semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7614.6 Star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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14.6.1 Complete star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.7 Further generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.8 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.9 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

15 Set (mathematics) 8115.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

15.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

15.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

15.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

15.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16 Sigma-algebra 9316.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

16.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

vi CONTENTS

16.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

16.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

16.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 9916.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 99

16.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9916.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10016.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

17 Subset 10117.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10317.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10317.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10317.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

18 Unit interval 10518.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

18.1.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10618.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10618.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10618.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

19 Upper set 10719.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 109

19.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10919.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11319.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Chapter 1

Birkhos representation theorem

This is about lattice theory. For other similarly named results, see Birkhos theorem (disambiguation).

In mathematics, Birkhos representation theorem for distributive lattices states that the elements of any nitedistributive lattice can be represented as nite sets, in such a way that the lattice operations correspond to unions andintersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributivelattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between nite topological spacesand preorders. It is named after Garrett Birkho, who published a proof of it in 1937.[1]

The name Birkhos representation theorem has also been applied to two other results of Birkho, one from 1935on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called elds of sets, closely related to the rings of sets used by Birkho to represent distributive lattices), and BirkhosHSP theorem representing algebras as products of irreducible algebras. Birkhos representation theorem has alsobeen called the fundamental theorem for nite distributive lattices.[2]

1.1 Understanding the theoremMany lattices can be dened in such a way that the elements of the lattice are represented by sets, the join operationof the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. Forinstance, the Boolean lattice dened from the family of all subsets of a nite set has this property. More generally anynite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey thedistributive law, any lattice dened in this way is a distributive lattice. Birkhos theorem states that in fact all nitedistributive lattices can be obtained this way, and later generalizations of Birkhos theorem state a similar thing forinnite distributive lattices.

1.2 ExamplesConsider the divisors of some composite number, such as (in the gure) 120, partially ordered by divisibility. Anytwo divisors of 120, such as 12 and 20, have a unique greatest common factor 12 20 = 4, the largest number thatdivides both of them, and a unique least common multiple 12 20 = 60; both of these numbers are also divisors of120. These two operations and satisfy the distributive law, in either of two equivalent forms: (x y) z = (x z) (y z) and (x y) z = (x z) (y z), for all x, y, and z. Therefore, the divisors form a nite distributive lattice.One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4},while 20 is associated with the set {2,4,5}. Then 12 20 = 4 is associated with the set {2,3,4} {2,4,5} = {2,4},while 12 20 = 60 is associated with the set {2,3,4} {2,4,5} = {2,3,4,5}, so the join and meet operations of thelattice correspond to union and intersection of sets.The prime powers 2, 3, 4, 5, and 8 appearing as elements in these sets may themselves be partially ordered bydivisibility; in this smaller partial order, 2 4 8 and there are no order relations between other pairs. The 16 setsthat are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such thatif x y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover the

1

2 CHAPTER 1. BIRKHOFFS REPRESENTATION THEOREM

{2} {3} {5}

{2,4} {2,3} {2,5} {3,5}

{2,4,8} {2,3,4} {2,4,5} {2,3,5}

{2,3,4,8} {2,4,5,8} {2,3,4,5}

{2,3,4,5,8}

1

2 3 5

4 6 10 15

8 12 20 30

24 40 60

120

The distributive lattice of divisors of 120, and its representation as sets of prime powers.

associated divisor by computing the least common multiple of the prime powers in L. Thus, the partial order on theve prime powers 2, 3, 4, 5, and 8 carries enough information to recover the entire original 16-element divisibilitylattice.Birkhos theorem states that this relation between the operations and of the lattice of divisors and the operations and of the associated sets of prime powers is not coincidental, and not dependent on the specic properties ofprime numbers and divisibility: the elements of any nite distributive lattice may be associated with lower sets of apartial order in the same way.As another example, the application of Birkhos theorem to the family of subsets of an n-element set, partiallyordered by inclusion, produces the free distributive lattice with n generators. The number of elements in this latticeis given by the Dedekind numbers.

1.3 The partial order of join-irreduciblesIn a lattice, an element x is join-irreducible if x is not the join of a nite set of other elements. Equivalently, x isjoin-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any twosmaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 isjoin-irreducible. An element x is join-prime if, whenever x y z, either x y or x z. In the same lattice, 4 isjoin-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4.In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible isnot join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y z. But then x y z, and x is not less than or equal to either y or z, showing that it is not join-prime.There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but ina distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x y z.This inequality is equivalent to the statement that x = x (y z), and by the distributive law x = (x y) (x z).But since x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x y(equivalently x y) or x = x z (equivalently x z).The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhos theorem states thatthe lattice itself can be recovered from the lower sets of this partial order.

1.4 Birkhos theoremIn any partial order, the lower sets form a lattice in which the lattices partial ordering is given by set inclusion, thejoin operation corresponds to set union, and the meet operation corresponds to set intersection, because unions andintersections preserve the property of being a lower set. Because set unions and intersections obey the distributive

1.5. RINGS OF SETS AND PREORDERS 3

law, this is a distributive lattice. Birkhos theorem states that any nite distributive lattice can be constructed in thisway.

Theorem. Any nite distributive lattice L is isomorphic to the lattice of lower sets of the partial orderof the join-irreducible elements of L.

That is, there is a one-to-one order-preserving correspondence between elements of L and lower sets of the partialorder. The lower set corresponding to an element x of L is simply the set of join-irreducible elements of L that areless than or equal to x, and the element of L corresponding to a lower set S of join-irreducible elements is the join ofS.If one starts with a lower set S of join-irreducible elements, lets x be the join of S, and constructs lower set T of thejoin-irreducible elements less than or equal to x, then S = T. For, every element of S clearly belongs to T, and anyjoin-irreducible element less than or equal to x must (by join-primality) be less than or equal to one of the membersof S, and therefore must (by the assumption that S is a lower set) belong to S itself. Conversely, if one starts with anelement x of L, lets S be the join-irreducible elements less than or equal to x, and constructs y as the join of S, thenx = y. For, as a join of elements less than or equal to x, y can be no greater than x itself, but if x is join-irreduciblethen x belongs to S while if x is the join of two or more join-irreducible items then they must again belong to S, so y x. Therefore, the correspondence is one-to-one and the theorem is proved.

1.5 Rings of sets and preordersBirkho (1937) dened a ring of sets to be a family of sets that is closed under the operations of set unions and setintersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called thesame structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form adistributive lattice. The elements of the sets may be given a preorder in which x y whenever some set in the ringcontains x but not y. The ring of sets itself is then the family of lower sets of this preorder, and any preorder givesrise to a ring of sets in this way.

1.6 FunctorialityBirkhos theorem, as stated above, is a correspondence between individual partial orders and distributive lattices.However, it can also be extended to a correspondence between order-preserving functions of partial orders andbounded homomorphisms of the corresponding distributive lattices. The direction of these maps is reversed in thiscorrespondence.Let 2 denote the partial order on the two-element set {0, 1}, with the order relation 0 < 1, and (following Stanley)let J(P) denote the distributive lattice of lower sets of a nite partial order P. Then the elements of J(P) correspondone-for-one to the order-preserving functions from P to 2.[2] For, if is such a function, 1(0) forms a lower set,and conversely if L is a lower set one may dene an order-preserving function L that maps L to 0 and that mapsthe remaining elements of P to 1. If g is any order-preserving function from Q to P, one may dene a function g*from J(P) to J(Q) that uses the composition of functions to map any element L of J(P) to L g. This compositefunction maps Q to 2 and therefore corresponds to an element g*(L) = (L g)1(0) of J(Q). Further, for any x andy in J(P), g*(x y) = g*(x) g*(y) (an element of Q is mapped by g to the lower set x y if and only if belongsboth to the set of elements mapped to x and the set of elements mapped to y) and symmetrically g*(x y) = g*(x) g*(y). Additionally, the bottom element of J(P) (the function that maps all elements of P to 0) is mapped by g* tothe bottom element of J(Q), and the top element of J(P) is mapped by g* to the top element of J(Q). That is, g* is ahomomorphism of bounded lattices.However, the elements of P themselves correspond one-for-one with bounded lattice homomorphisms from J(P) to 2.For, if x is any element of P, one may dene a bounded lattice homomorphism jx that maps all lower sets containingx to 1 and all other lower sets to 0. And, for any lattice homomorphism from J(P) to 2, the elements of J(P) thatare mapped to 1 must have a unique minimal element x (the meet of all elements mapped to 1), which must bejoin-irreducible (it cannot be the join of any set of elements mapped to 0), so every lattice homomorphism has theform jx for some x. Again, from any bounded lattice homomorphism h from J(P) to J(Q) one may use composition offunctions to dene an order-preserving map h* from Q to P. It may be veried that g** = g for any order-preservingmap g from Q to P and that and h** = h for any bounded lattice homomorphism h from J(P) to J(Q).


In category theoretic terminology, J is a contravariant hom-functor J = Hom(,2) that denes a duality of categoriesbetween, on the one hand, the category of nite partial orders and order-preserving maps, and on the other hand thecategory of nite distributive lattices and bounded lattice homomorphisms.

1.7 GeneralizationsIn an innite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are inone-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, forinstance, in the lattice of all natural numbers, ordered with the reverse of the usual divisibility ordering (so x ywhen y divides x): any number x can be expressed as the join of numbers xp and xq where p and q are distinct primenumbers. However, elements in innite distributive lattices may still be represented as sets via Stones representationtheorem for distributive lattices, a form of Stone duality in which each lattice element corresponds to a compact openset in a certain topological space. This generalized representation theorem can be expressed as a category-theoreticduality between distributive lattices and coherent spaces (sometimes called spectral spaces), topological spaces inwhich the compact open sets are closed under intersection and form a base for the topology.[3] Hilary Priestley showedthat Stones representation theorem could be interpreted as an extension of the idea of representing lattice elementsby lower sets of a partial order, using Nachbins idea of ordered topological spaces. Stone spaces with an additionalpartial order linked with the topology via Priestley separation axiom can also be used to represent bounded distributivelattices. Such spaces are known as Priestley spaces. Further, certain bitopological spaces, namely pairwise Stonespaces, generalize Stones original approach by utilizing two topologies on a set to represent an abstract distributvelattice. Thus, Birkhos representation theorem extends to the case of innite (bounded) distributive lattices in atleast three dierent ways, summed up in duality theory for distributive lattices.Birkhos representation theorem may also be generalized to nite structures other than distributive lattices. In adistributive lattice, the self-dual median operation[4]

m(x; y; z) = (x _ y) ^ (x _ z) ^ (y _ z) = (x ^ y) _ (x ^ z) _ (y ^ z)

gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebrasand median graphs have a dual structure as the set of solutions of a 2-satisability instance; Barthlemy & Constantin(1993) formulate this structure equivalently as the family of initial stable sets in a mixed graph.[5] For a distributivelattice, the corresponding mixed graph has no undirected edges, and the initial stable sets are just the lower sets ofthe transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the 2-satisabilityinstance can be partitioned into two connected components, one on the positive variables of the instance and theother on the negative variables; the transitive closure of the positive component is the underlying partial order of thedistributive lattice.Another result analogous to Birkhos representation theorem, but applying to a broader class of lattices, is thetheorem of Edelman (1980) that any nite join-distributive lattice may be represented as an antimatroid, a familyof sets closed under unions but in which closure under intersections has been replaced by the property that eachnonempty set has a removable element.

1.8 Notes

[1] Birkho (1937).

[2] (Stanley 1997).

[3] Johnstone (1982).

[4] Birkho & Kiss (1947).

[5] A minor dierence between the 2-SAT and initial stable set formulations is that the latter presupposes the choice of a xedbase point from the median graph that corresponds to the empty initial stable set.

1.9. REFERENCES 5

1.9 References Barthlemy, J.-P.; Constantin, J. (1993), Median graphs, parallelism and posets, Discrete Mathematics 111(13): 4963, doi:10.1016/0012-365X(93)90140-O.

Birkho, Garrett (1937), Rings of sets, Duke Mathematical Journal 3 (3): 443454, doi:10.1215/S0012-7094-37-00334-X.

Birkho, Garrett; Kiss, S. A. (1947), A ternary operation in distributive lattices, Bulletin of the AmericanMathematical Society 53 (1): 749752, doi:10.1090/S0002-9904-1947-08864-9, MR 0021540.

Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2. Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(1): 290299, doi:10.1007/BF02482912.

Johnstone, Peter (1982), II.3 Coherent locales, Stone Spaces, Cambridge University Press, pp. 6269, ISBN978-0-521-33779-3.

Priestley, H. A. (1970), Representation of distributive lattices by means of ordered Stone spaces, Bulletin ofthe London Mathematical Society 2 (2): 186190, doi:10.1112/blms/2.2.186.

Priestley, H. A. (1972), Ordered topological spaces and the representation of distributive lattices, Proceedingsof the London Mathematical Society 24 (3): 507530, doi:10.1112/plms/s3-24.3.507.

Stanley, R. P. (1997), Enumerative Combinatorics, Volume I, Cambridge Studies in Advanced Mathematics 49,Cambridge University Press, pp. 104112.


Distributive example lattice, with join-irreducible elements a,...,g (shadowed nodes). The lower set a node corresponds to by Birkhosisomorphism is shown in light blue.

Chapter 2

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, fromclosed sets) through the operations of countable union, countable intersection, and relative complement. Borel setsare named after mile Borel.For a topological space X, the collection of all Borel sets on X forms a -algebra, known as the Borel algebra orBorel -algebra. The Borel algebra on X is the smallest -algebra containing all open sets (or, equivalently, all closedsets).Borel sets are important in measure theory, since any measure dened on the open sets of a space, or on the closedsets of a space, must also be dened on all Borel sets of that space. Any measure dened on the Borel sets is called aBorel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.In some contexts, Borel sets are dened to be generated by the compact sets of the topological space, rather thanthe open sets. The two denitions are equivalent for many well-behaved spaces, including all Hausdor -compactspaces, but can be dierent in more pathological spaces.

2.1 Generating the Borel algebraIn the case X is a metric space, the Borel algebra in the rst sense may be described generatively as follows.For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

T be all countable unions of elements of T T be all countable intersections of elements of T T = (T):

Now dene by transnite induction a sequence Gm, where m is an ordinal number, in the following manner:

For the base case of the denition, let G0 be the collection of open subsets of X. If i is not a limit ordinal, then i has an immediately preceding ordinal i 1. Let

Gi = [Gi1]:

If i is a limit ordinal, set

Gi =[j

8 CHAPTER 2. BOREL SET

G 7! G:to the rst uncountable ordinal.To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets.In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountablelimit ordinal, Gm is closed under countable unions.Note that for each Borel set B, there is some countable ordinal B such that B can be obtained by iterating theoperation over B. However, as B varies over all Borel sets, B will vary over all the countable ordinals, and thus therst ordinal at which all the Borel sets are obtained is 1, the rst uncountable ordinal.

2.1.1 ExampleAn important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It isthe algebra on which the Borel measure is dened. Given a real random variable dened on a probability space, itsprobability distribution is by denition also a measure on the Borel algebra.The Borel algebra on the reals is the smallest -algebra on R which contains all the intervals.In the construction by transnite induction, it can be shown that, in each step, the number of sets is, at most, thepower of the continuum. So, the total number of Borel sets is less than or equal to

@1 2@0 = 2@0 :

2.2 Standard Borel spaces and Kuratowski theoremsLet X be a topological space. The Borel space associated to X is the pair (X,B), where B is the -algebra of Borelsets of X.Mackey dened a Borel space somewhat dierently, writing that it is a set together with a distinguished -eld ofsubsets called its Borel sets. [1] However, modern usage is to call the distinguished sub-algebra measurable sets andsuch spaces measurable spaces. The reason for this distinction is that the Borel sets are the -algebra generated byopen sets (of a topological space), whereas Mackeys denition refers to a set equipped with an arbitrary -algebra.There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Afunction f : X ! Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f1(B) is ameasurable set in X.Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which denes thetopology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to oneof (1) R, (2) Z or (3) a nite space. (This result is reminiscent of Maharams theorem.)Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized upto isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injectivemaps dened on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.See analytic set.Every probability measure on a standard Borel space turns it into a standard probability space.

2.3 Non-Borel setsAn example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 7678), is describedbelow. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

2.4. ALTERNATIVE NON-EQUIVALENT DEFINITIONS 9

Every irrational number has a unique representation by a continued fraction

x = a0 +1

a1 +1

a2 +1

a3 +1

. . .where a0 is some integer and all the other numbers ak are positive integers. Let A be the set of all irrationalnumbers that correspond to sequences (a0; a1; : : : ) with the following property: there exists an innite subsequence(ak0 ; ak1 ; : : : ) such that each element is a divisor of the next element. This set A is not Borel. In fact, it is analytic,and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris,especially Exercise (27.2) on page 209, Denition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.Another non-Borel set is an inverse image f1[0] of an innite parity function f : f0; 1g! ! f0; 1g . However, thisis a proof of existence (via the axiom of choice), not an explicit example.

2.4 Alternative non-equivalent denitionsAccording to Halmos (Halmos 1950, page 219), a subset of a locally compact Hausdor topological space is calleda Borel set if it belongs to the smallest ring containing all compact sets.Norberg and Vervaat [5] redene the Borel algebra of a topological space X as the algebra generated by its opensubsets and its compact saturated subsets. This denition is well-suited for applications in the case where X is notHausdor. It coincides with the usual denition if X is second countable or if every compact saturated subset isclosed (which is the case in particular if X is Hausdor).

2.5 See also Baire set Cylindrical -algebra Polish space Descriptive set theory Borel hierarchy

2.6 References William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent expo-sition of Polish topology)

Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

Halmos, Paul R. (1950). Measure theory. D. van Nostrand Co. See especially Sect. 51 Borel sets and Bairesets.

Halsey Royden, Real Analysis, Prentice Hall, 1988

Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol.156)

10 CHAPTER 2. BOREL SET

2.7 Notes[1] Mackey, G.W. (1966), Ergodic Theory and Virtual Groups, Math. Annalen. (Springer-Verlag) 166 (3): 187207,

doi:10.1007/BF01361167, ISSN 0025-5831, (subscription required (help))

[2] Jochen Wengenroth (mathoverflow.net/users/21051), Is every sigma-algebra the Borel algebra of a topology?, http://mathoverflow.net/questions/87888 (version: 2012-02-09)

[3] Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7

[4] Lusin, Nicolas (1927), Sur les ensembles analytiques, Fundamenta Mathematicae (Institute of mathematics, Polishacademy of sciences) 10: 195.

[5] Tommy Norberg and Wim Vervaat, Capacities on non-Hausdor spaces, in: Probability and Lattices, in: CWI Tract, vol.110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

2.8 External links Hazewinkel, Michiel, ed. (2001), Borel set, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Formal denition of Borel Sets in the Mizar system, and the list of theorems that have been formally provedabout it.

Weisstein, Eric W., Borel Set, MathWorld.

Chapter 3

Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.The prototypical examples of such structures are collections of sets for which the lattice operations can be given byset union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is up to isomorphism given as such a lattice of sets.

3.1 DenitionAs in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of ordertheory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. Inthe present situation, the algebraic description appears to be more convenient:A lattice (L,,) is distributive if the following additional identity holds for all x, y, and z in L:

x (y z) = (x y) (x z).

Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty nite joins. It is abasic fact of lattice theory that the above condition is equivalent to its dual:[1]

x (y z) = (x y) (x z) for all x, y, and z in L.[2]

In every lattice, dening pq as usual to mean pq=p, the inequation x (y z) (x y) (x z) holds as well asits dual inequation x (y z) (x y) (x z). A lattice is distributive if one of the converse inequations holds,too. More information on the relationship of this condition to other distributivity conditions of order theory can befound in the article on distributivity (order theory).

3.2 MorphismsA morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i.e. a functionthat is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure,it will consequently also preserve the distributivity (and thus be a morphism of distributive lattices).

3.3 ExamplesDistributive lattices are ubiquitous but also rather specic structures. As already mentioned the main example fordistributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Furtherexamples include:

11

12 CHAPTER 3. DISTRIBUTIVE LATTICE

Youngs lattice

The Lindenbaum algebra of most logics that support conjunction and disjunction is a distributive lattice, i.e.and distributes over or and vice versa.

Every Boolean algebra is a distributive lattice.

Every Heyting algebra is a distributive lattice. Especially this includes all locales and hence all open set latticesof topological spaces. Also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionisticlogic, which makes them a special case of the above example.

Every totally ordered set is a distributive lattice with max as join and min as meet.

The natural numbers form a distributive lattice (complete as a meet-semilattice) with the greatest commondivisor as meet and the least common multiple as join.

Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatestcommon divisor as meet and the least common multiple as join. This is a Boolean algebra if and only if n issquare-free.

A lattice-ordered vector space is a distributive lattice.

Youngs lattice given by the inclusion ordering ofYoung diagrams representing integer partitions is a distributivelattice.

Early in the development of the lattice theory Charles S. Peirce believed that all lattices are distributive, that is,distributivity follows from the rest of the lattice axioms.[3][4] However, independence proofs were given by Schrder,Voigt,(de) Lroth, Korselt,[5] and Dedekind.[3]

3.4. CHARACTERISTIC PROPERTIES 13

3.4 Characteristic propertiesVarious equivalent formulations to the above denition exist. For example, L is distributive if and only if the followingholds for all elements x, y, z in L:

(x ^ y) _ (y ^ z) _ (z ^ x) = (x _ y) ^ (y _ z) ^ (z _ x).

Similarly, L is distributive if and only if

x ^ z = y ^ z and x _ z = y _ z always imply x=y.

Hasse diagrams of the two prototypical non-distributive lattices The diamond lattice M3 is non-distributive: x (y z) = x 1 = x 0 = 0 0 = (x y) (x z). The pentagon lattice N5 is non-distributive: x (y z) = x 1 = x z = 0 z = (x y) (x z).

Distributive lattice which contains N5 (solid lines, left) and M3 (right) as subset, but not as sublattice, respectively

The simplest non-distributive lattices are M3, the diamond lattice, and N5, the pentagon lattice. A lattice isdistributive if and only if none of its sublattices is isomorphic toM3 or N5; a sublattice is a subset that is closed underthe meet and join operations of the original lattice. Note that this is not the same as being a subset that is a latticeunder the original order (but possibly with dierent join and meet operations). Further characterizations derive fromthe representation theory in the next section.Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice ismeet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, thesame is true for join-prime and join-irreducible elements.[6] If a lattice is distributive, its covering relation forms amedian graph.[7]

Furthermore, every distributive lattice is also modular.

3.5 Representation theoryThe introduction already hinted at the most important characterization for distributive lattices: a lattice is distributiveif and only if it is isomorphic to a lattice of sets (closed under set union and intersection). That set union andintersection are indeed distributive in the above sense is an elementary fact. The other direction is less trivial, inthat it requires the representation theorems stated below. The important insight from this characterization is that theidentities (equations) that hold in all distributive lattices are exactly the ones that hold in all lattices of sets in the abovesense.


Birkhos representation theorem for distributive lattices states that every nite distributive lattice is isomorphic tothe lattice of lower sets of the poset of its join-prime (equivalently: join-irreducible) elements. This establishes abijection (up to isomorphism) between the class of all nite posets and the class of all nite distributive lattices.This bijection can be extended to a duality of categories between homomorphisms of nite distributive lattices andmonotone functions of nite posets. Generalizing this result to innite lattices, however, requires adding furtherstructure.Another early representation theorem is now known as Stones representation theorem for distributive lattices (thename honors Marshall Harvey Stone, who rst proved it). It characterizes distributive lattices as the lattices ofcompact open sets of certain topological spaces. This result can be viewed both as a generalization of Stones famousrepresentation theorem for Boolean algebras and as a specialization of the general setting of Stone duality.A further important representation was established by Hilary Priestley in her representation theorem for distributivelattices. In this formulation, a distributive lattice is used to construct a topological space with an additional partialorder on its points, yielding a (completely order-separated) ordered Stone space (or Priestley space). The originallattice is recovered as the collection of clopen lower sets of this space.As a consequence of Stones and Priestleys theorems, one easily sees that any distributive lattice is really isomorphicto a lattice of sets. However, the proofs of both statements require the Boolean prime ideal theorem, a weak form ofthe axiom of choice.

3.6 Free distributive lattices

(x y)

(x z)

(x y)

(x z)(x y)

(y z)

(x z)

(y z)

(x y)

(y z)(x z)

(y z)

x y

x y

x z y z

x y z

1

1

1

1

0

0

0

0

xxx yy zmajority

x y z

x zx y

x y

y z

Free distributive lattices on zero, one, two, and three generators. The elements labeled 0 and 1 are the empty join and meet, andthe element labeled majority is (x y) (x z) (y z) = (x y) (x z) (y z).

The free distributive lattice over a set of generators G can be constructed much more easily than a general free lattice.The rst observation is that, using the laws of distributivity, every term formed by the binary operations _ and ^ ona set of generators can be transformed into the following equivalent normal form:

3.7. SEE ALSO 15

M1 _ M2 _ ... _ Mn

where theMi are nite meets of elements ofG. Moreover, since both meet and join are commutative and idempotent,one can ignore duplicates and order, and represent a join of meets like the one above as a set of sets:

{N1, N2, ..., Nn},

where the Ni are nite subsets of G. However, it is still possible that two such terms denote the same element of thedistributive lattice. This occurs when there are indices j and k such that Nj is a subset of Nk. In this case the meetof Nk will be below the meet of Nj, and hence one can safely remove the redundant set Nk without changing theinterpretation of the whole term. Consequently, a set of nite subsets of G will be called irredundant whenever all ofits elements Ni are mutually incomparable (with respect to the subset ordering); that is, when it forms an antichainof nite sets.Now the free distributive lattice over a set of generators G is dened on the set of all nite irredundant sets of nitesubsets of G. The join of two nite irredundant sets is obtained from their union by removing all redundant sets.Likewise the meet of two sets S and T is the irredundant version of { N [M | N in S,M in T}. The verication thatthis structure is a distributive lattice with the required universal property is routine.The number of elements in free distributive lattices with n generators is given by the Dedekind numbers. Thesenumbers grow rapidly, and are known only for n 8; they are

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 inOEIS).

The numbers above count the number of free distributive lattices in which the lattice operations are joins and meetsof nite sets of elements, including the empty set. If empty joins and empty meets are disallowed, the resulting freedistributive lattices have two fewer elements; their numbers of elements form the sequence

1, 4, 18, 166, 7579, 7828352, 2414682040996, 56130437228687557907786 (sequence A007153 inOEIS).

3.7 See also Completely distributive lattice a lattice in which innite joins distribute over innite meets Duality theory for distributive lattices Spectral space

3.8 References[1] Garrett Birkho (1967). Lattice Theory. Colloquium Publications 25. Am. Math. Soc.; here: 5-6, p.8-12

[2] For individual elements x, y, z, e.g. the rst equation may be violated, but the second may hold; see the N5 picture for anexample.

[3] Peirce, Charles S.; Fisch, M. H.; Kloesel, C. J. W. (1989), Writings of Charles S. Peirce: 18791884, Indiana UniversityPress, p. xlvii.

[4] Charles S. Peirce (1880). On theAlgebra of Logic (PDF).American Journal ofMathematics 3: 1557. doi:10.2307/2369442.,p. 33 bottom

[5] A.Korselt (1894). Bemerkung zurAlgebra der Logik (PDF).Mathematische Annalen 44: 156157. doi:10.1007/bf01446978.Korselts non-distributive lattice example is a variant of M3, with 0, 1, and x, y, z corresponding to the empty set, a line,and three distinct points on it, respectively.

[6] See Birkhos representation theorem#The partial order of join-irreducibles.

[7] Birkho, Garrett; Kiss, S. A. (1947), A ternary operation in distributive lattices, Bulletin of the American MathematicalSociety 53 (1): 749752, doi:10.1090/S0002-9904-1947-08864-9, MR 0021540.


3.9 Further reading Burris, Stanley N.; Sankappanavar, H.P. (1981). A Course in Universal Algebra. Springer-Verlag. ISBN3-540-90578-2.

Chapter 4

Dynkin system

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

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

1. 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 for all n 1, then

S1n=1An 2 D .

Equivalently, D is a Dynkin system if

1. D ,2. if A D, then Ac D,3. if A1, A2, A3, ... is a sequence of subsets in D such that Ai Aj = for all i j, then

S1n=1An 2 D .

The second denition is generally preferred as it usually is easier to check.An important fact is that a Dynkin systemwhich is also a -system (i.e., closed under nite intersection) is a -algebra.This can be veried by noting that condition 3 and closure under nite intersection implies closure under countableunions.Given any collection J of subsets of , there exists a unique Dynkin system denotedDfJ g which is minimal withrespect to containing J . That is, if ~D is any Dynkin system containing J , then DfJ g ~D . DfJ g is called theDynkin system generated by J . Note Df;g = f;;g . For another example, let = f1; 2; 3; 4g and J = f1g ;then DfJ g = f;; f1g; f2; 3; 4g;g .

4.2 Dynkins - theoremIf P is a -system andD is a Dynkin system with P D , then fPg D . In other words, the -algebra generatedby P is contained in D .One application of Dynkins - theorem is the uniqueness of a measure that evaluates the length of an interval(known as the Lebesgue measure):

17

18 CHAPTER 4. DYNKIN SYSTEM

Let (, B, ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let be another measure on satisfying [(a,b)] = b a, and letD be the family of sets S such that [S] = [S]. Let I = { (a,b),[a,b),(a,b],[a,b] : 0

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