Pi system 1From Wikipedia, the free encyclopedia

Contents

1 Borel set 11.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Alternative non-equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Dynkin system 52.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Empty set 73.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

i

ii CONTENTS

4 Intersection (set theory) 134.1 Basic denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Lebesgue measure 195.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Mathematics 246.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 316.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Measure (mathematics) 427.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

CONTENTS iii

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.3.2 Measures of innite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 447.3.3 Measures of innite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 44

7.4 Sigma-nite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8 Pi system 508.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

9 Set (mathematics) 549.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

9.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

9.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

iv CONTENTS

9.10 De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

10 Sigma-algebra 6610.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

10.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

10.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6810.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6810.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6810.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6810.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7110.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7110.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7110.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 7210.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 72

10.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7210.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7310.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

11 Subset 7411.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

CONTENTS v

12 Unit interval 7812.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

12.1.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7812.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 80

12.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8012.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8412.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Chapter 1

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.

1.1 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

2 CHAPTER 1. 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.

1.1.1 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 :

1.2 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.

1.3 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.

1.4. ALTERNATIVE NON-EQUIVALENT DEFINITIONS 3

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.

1.4 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).

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

1.6 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)

4 CHAPTER 1. BOREL SET

1.7 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

1.8 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 2

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.

2.1 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 .

2.2 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):

5

6 CHAPTER 2. 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