Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction: Donsker's Theorem, Metric Entropy, and Inequalities
- 2 Gaussian Measures and Processes; Sample Continuity
- 3 Foundations of Uniform Central Limit Theorems: Donsker Classes
- 4 Vapnik-Červonenkis Combinatorics
- 5 Measurability
- 6 Limit Theorems for Vapnik-Červonenkis and Related Classes
- 7 Metric Entropy, with Inclusion and Bracketing
- 8 Approximation of Functions and Sets
- 9 Sums in General Banach Spaces and Invariance Principles
- 10 Universal and Uniform Central Limit Theorems
- 11 The Two-Sample Case, the Bootstrap, and Confidence Sets
- 12 Classes of Sets or Functions Too Large for Central Limit Theorems
- Appendix A Differentiating under an Integral Sign
- Appendix B Multinomial Distributions
- Appendix C Measures on Nonseparable Metric Spaces
- Appendix D An Extension of Lusin's Theorem
- Appendix E Bochner and Pettis Integrals
- Appendix F Nonexistence of Types of Linear Forms on Some Spaces
- Appendix G Separation of Analytic Sets; Borel Injections
- Appendix H Young-Orlicz Spaces
- Appendix I Modifications and Versions of Isonormal Processes
- Subject Index
- Author Index
- Index of Notation
5 - Measurability
Published online by Cambridge University Press: 20 May 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction: Donsker's Theorem, Metric Entropy, and Inequalities
- 2 Gaussian Measures and Processes; Sample Continuity
- 3 Foundations of Uniform Central Limit Theorems: Donsker Classes
- 4 Vapnik-Červonenkis Combinatorics
- 5 Measurability
- 6 Limit Theorems for Vapnik-Červonenkis and Related Classes
- 7 Metric Entropy, with Inclusion and Bracketing
- 8 Approximation of Functions and Sets
- 9 Sums in General Banach Spaces and Invariance Principles
- 10 Universal and Uniform Central Limit Theorems
- 11 The Two-Sample Case, the Bootstrap, and Confidence Sets
- 12 Classes of Sets or Functions Too Large for Central Limit Theorems
- Appendix A Differentiating under an Integral Sign
- Appendix B Multinomial Distributions
- Appendix C Measures on Nonseparable Metric Spaces
- Appendix D An Extension of Lusin's Theorem
- Appendix E Bochner and Pettis Integrals
- Appendix F Nonexistence of Types of Linear Forms on Some Spaces
- Appendix G Separation of Analytic Sets; Borel Injections
- Appendix H Young-Orlicz Spaces
- Appendix I Modifications and Versions of Isonormal Processes
- Subject Index
- Author Index
- Index of Notation
Summary
Let A be the set of all possible empirical distribution functions F1 for one observation x ∈ [0, 1], namely F1(t) = 0 for t < x and F1{t) = 1 for t ≥ x. We noted previously that A in the supremum norm is nonseparable: it is an uncountable set, in which any two points are at a distance 1 apart. Thus A and all its subsets are closed. If x ≔ X1 has a continuous distribution such as the uniform distribution U[0, 1] on [0, 1], then x → (t → 1t≥x) takes [0, 1] onto A, but it is not continuous for the supremum norm. Also, it is not measurable for the Borel σ-algebra on the range space. So, in Chapter 3, functions f* and upper expectations E* were used to get around measurability problems.
Here is a different kind of example. It is related to the basic “ordinal triangle” counterexample in integration theory, showing why measurability is needed in the Tonelli-Fubini theorem on Cartesian product integrals. Let (Ω, ≤) be an uncountable well-ordered set such that for each x ∈ Ω, the initial segment Ix ≔ {y : y ≤ x} is countable. (In terms of ordinals, Ω is, or is orderisomorphic to, the least uncountable ordinal.) Let S be the σ-algebra of subsets of Ω consisting of sets that are countable or have countable complement. Let P be the probability measure on S which is 0 on countable sets and 1 on sets with countable complement.
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- Uniform Central Limit Theorems , pp. 170 - 195Publisher: Cambridge University PressPrint publication year: 1999