Part 1. Finite additivity 7
Chapter 1. Measure & probability: finite additivity 9
1.1. Introduction: Measure and integration 9
1.2. Probability, events, and sample spaces 16
1.3. Semirings, rings and σ-algebras 26
1.4. The Borel sets and the Principle of Appropriate Sets 36
1.5. Additive set functions in classical probability 44
1.6. More examples of additive set functions 59
Notes and references on Chapter 1 67
Chapter 2. Finitely additive integration 69
2.1. Integration on semirings 69
2.2. Random variables and (mathematical) expectations 75
2.3. Properties of additive set functions on semirings 86
2.4. Bernoulli’s Theorem (The WLLNs) and expectations 97
2.5. De Moivre, Laplace and Stirling star in The Normal Curve 108
Notes and references on Chapter 2 127
Part 2. Countable additivity 131
Chapter 3. Measure and probability: countable additivity 133
3.1. Introduction: What is a measurable set? 133
3.2. Countably additive set functions on semirings 137
3.3. Infinite product spaces and properties of countably additivity 144
3.4. Outer measures, measures, and Carath′eodory’s idea 158
3.5. The Extension Theorem and regularity properties of measures 168
Notes and references on Chapter 3 178
Chapter 4. Reactions to the Extension & Regularity Theorems 181
4.1. Probability, Bernoulli sequences and Borel-Cantelli 181
4.2. Borel’s Strong Law of Large Numbers 190
4.3. Measurability and Littlewood’s First Principle(s) 198
4.4. Geometry, Vitali’s nonmeasurable set, and paradoxes 210
4.5. The Cantor set 227
Notes and references on Chapter 4 239
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iv CONTENTS
Part 3. Integration 245
Chapter 5. Basics of Integration Theory 247
5.1. Introduction: Interchanging limits and integrals 247
5.2. Measurable functions and Littlewood’s second principle 252
5.3. Sequences of functions and Littlewood’s third principle 264
5.4. Lebesgue’s definition of the integral and the MCT 275
5.5. Features of the integral and the Principle of Appropriate Functions 285
5.6. The DCT and Osgood’s Principle 297
Notes and references on Chapter 5 312
Chapter 6. Some applications of integration 315
6.1. Practice with the DCT and its corollaries 315
6.2. Lebesgue, Riemann and Stieltjes integration 332
6.3. Probability distributions, mass functions and pdfs 344
6.4. Independence and identically distributed random variables 354
6.5. Approximations and the Stone-Weierstrass theorem 365
6.6. The Law of Large Numbers and Normal Numbers 373
Notes and references on Chapter 6 386
Part 4. Further results in measure and integration 389
Chapter 7. Fubini’s theorem and Change of Variables 391
7.1. Introduction: Iterated integration 391
7.2. Product measures, volumes by slices, and volumes of balls 396
7.3. The Fubini-Tonelli Theorems on iterated integrals 409
7.4. Change of variables in multiple integrals 422
7.5. Some applications of change of variables 437
7.6. Polar coordinates and integration over spheres 448
Notes and references on Chapter 7