Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1. Constructions and extensions of measures . . . . . . . . . . . . 1
1.1. Measurement of length: introductory remarks . . . . . . . . . . . . . . . . . 1
1.2. Algebras and ヲ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
1.3. Additivity and countable additivity of measures . . . . . . . . . . . . . . . 9
1.4. Compact classes and countable additivity . . . . . . . . . . . . . . . . . . . . 13
1.5. Outer measure and the Lebesgue extension of measures. . . . . . .16
1.6. Infinite and ヲ-finite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7. Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
1.8. Lebesgue-Stieltjes measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.9. Monotone and ヲ-additive classes of sets . . . . . . . . . . . . . . . . . . . . . . 33
1.10. Souslin sets and the A-operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.11. Caratheodory outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.12. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Set operations (48). Compact classes (50). Metric Boolean algebra (53).
Measurable envelope, measurable kernel and inner measure (56).
Extensions of measures (58). Some interesting sets (61). Additive, but
not countably additive measures (67). Abstract inner measures (70).
Measures on lattices of sets (75). Set-theoretic problems in measure
theory (77). Invariant extensions of Lebesgue measure (80). Whitney’s
decomposition (82). Exercises (83).
Chapter 2. The Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.1. Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.2. Convergence in measure and almost everywhere . . . . . . . . . . . . . 110
2.3. The integral for simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.4. The general definition of the Lebesgue integral . . . . . . . . . . . . . . 118
2.5. Basic properties of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.6. Integration with respect to infinite measures . . . . . . . . . . . . . . . . 124
2.7. The completeness of the space L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.8. Convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.9. Criteria of integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.10. Connections with the Riemann integral . . . . . . . . . . . . . . . . . . . . . 138
2.11. The H¨older and Minkowski inequalities . . . . . . . . . . . . . . . . . . . . . .139
xii Contents
2.12. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
The ヲ-algebra generated by a class of functions (143). Borel mappings
on IRn (145). The functional monotone class theorem (146). Baire
classes of functions (148). Mean value theorems (150). The Lebesgue.
Stieltjes integral (152). Integral inequalities (153). Exercises (156).
Chapter 3. Operations on measures and functions . . . . . . . . . . . . . .175
3.1. Decomposition of signed measures. . . . . . . . . . . . . . . . . . . . . . . . . . .175
3.2. The Radon.Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3.3. Products of measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.4. Fubini’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.5. Infinite products of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
3.6. Images of measures under mappings. . . . . . . . . . . . . . . . . . . . . . . . .190
3.7. Change of variables in IRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.8. The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.9. Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
3.10. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
On Fubini’s theorem and products of ヲ-algebras (209). Steiner’s
symmetrization (212). Hausdorff measures (215). Decompositions of
set functions (218). Properties of positive definite functions (220).
The Brunn.Minkowski inequality and its generalizations (222).
Mixed volumes (226). Exercises (228).
Chapter 4. The spaces Lp and spaces of measures . . . . . . . . . . . . . . 249
4.1. The spaces Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
4.2. Approximations in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251
4.3. The Hilbert space L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4.4. Duality of the spaces Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.5. Uniform integrability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266
4.6. Convergence of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273
4.7. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
The spaces Lp and the space of measures as structures (277). The weak
topology in Lp (280). Uniform convexity (283). Uniform integrability
and weak compactness in L1 (285). The topology of setwise convergence
of measures (291). Norm compactness and approximations in Lp (294).
Certain conditions of convergence in Lp (298). Hellinger’s integral and
Hellinger’s distance (299). Additive set functions (302). Exercises (303).
Chapter 5. Connections between the integral and derivative . . 329
5.1. Differentiability of functions on the real line . . . . . . . . . . . . . . . . .329
5.2. Functions of bounded variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332
5.3. Absolutely continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337
5.4. The Newton.Leibniz formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341
5.5. Covering theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
5.6. The maximal function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .349
5.7. The Henstock.Kurzweil integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353
Contents xiii
5.8. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Covering theorems (361). Density points and Lebesgue points (366).
Differentiation of measures on IRn (367). The approximate
continuity (369). Derivates and the approximate differentiability (370).
The class BMO (373). Weighted inequalities (374). Measures with
the doubling property (375). Sobolev derivatives (376). The area and
coarea formulas and change of variables (379). Surface measures (383).
The Calderⅴon.Zygmund decomposition (385). Exercises (386).
Bibliographical and Historical Comments. . . . . . . . . . . . . . . . . . . . . . . . .409
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Contents of Volume 2
Preface to Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 6. Borel, Baire and Souslin sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
6.1. Metric and topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
6.2. Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6.3. Baire sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.4. Products of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
6.5. Countably generated ヲ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.6. Souslin sets and their separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.7. Sets in Souslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.8. Mappings of Souslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
6.9. Measurable choice theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.10. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Borel and Baire sets (43). Souslin sets as projections (46). K-analytic
and F-analytic sets (49). Blackwell spaces (50). Mappings of Souslin
spaces (51). Measurability in normed spaces (52). The Skorohod
space (53). Exercises (54).
Chapter 7. Measures on topological spaces . . . . . . . . . . . . . . . . . . . . . . . 67
7.1. Borel, Baire and Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.2. ン -additive measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3. Extensions of measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
7.4. Measures on Souslin spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
7.5. Perfect measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.6. Products of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.7. The Kolmogorov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.8. The Daniell integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
7.9. Measures as functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.10. The regularity of measures in terms of functionals . . . . . . . . . . .111
7.11. Measures on locally compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.12. Measures on linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.13. Characteristic functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.14. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Extensions of product measures (126). Measurability on products (129).
Maⅶrⅴ━k spaces (130). Separable measures (132). Diffused and atomless
xvi Contents
measures (133). Completion regular measures (133). Radon
spaces (135). Supports of measures (136). Generalizations of Lusin’s
theorem (137). Metric outer measures (140). Capacities (142).
Covariance operators and means of measures (142). The Choquet
representation (145). Convolutions (146). Measurable linear
functions (149). Convex measures (149). Pointwise convergence (151).
Infinite Radon measures (154). Exercises (155).
Chapter 8. Weak convergence of measure . . . . . . . . . . . . . . . . . . . . . . . 175
8.1. The definition of weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . .175
8.2. Weak convergence of nonnegative measures. . . . . . . . . . . . . . . . . .182
8.3. The case of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.4. Some properties of weak convergence . . . . . . . . . . . . . . . . . . . . . . . .194
8.5. The Skorohod representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199
8.6. Weak compactness and the Prohorov theorem . . . . . . . . . . . . . . . 202
8.7. Weak sequential completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209
8.8. Weak convergence and the Fourier transform . . . . . . . . . . . . . . . . 210
8.9. Spaces of measures with the weak topology. . . . . . . . . . . . . . . . . .211
8.10. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Weak compactness (217). Prohorov spaces (219). Weak sequential
completeness of spaces of measures (226). The A-topology (226).
Continuous mappings of spaces of measures (227). The separability
of spaces of measures (230). Young measures (231). Metrics on
spaces of measures (232). Uniformly distributed sequences (237).
Setwise convergence of measures (241). Stable convergence and
ws-topology (246). Exercises (249)
Chapter 9. Transformations of measures and isomorphisms . . . 267
9.1. Images and preimages of measures . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.2. Isomorphisms of measure spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
9.3. Isomorphisms of measure algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .277
9.4. Induced point isomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280
9.5. Lebesgue.Rohlin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284
9.6. Topologically equivalent measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9.7. Continuous images of Lebesgue measure. . . . . . . . . . . . . . . . . . . . .288
9.8. Connections with extensions of measures . . . . . . . . . . . . . . . . . . . . 291
9.9. Absolute continuity of the images of measures . . . . . . . . . . . . . . .292
9.10. Shifts of measures along integral curves . . . . . . . . . . . . . . . . . . . . . 297
9.11. Invariant measures and Haar measures . . . . . . . . . . . . . . . . . . . . . . 303
9.12. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Projective systems of measures (308). Extremal preimages of
measures and uniqueness (310). Existence of atomless measures (317).
Invariant and quasi-invariant measures of transformations (318). Point
and Boolean isomorphisms (320). Almost homeomorphisms (323).
Measures with given marginal projections (324). The Stone
representation (325). The Lyapunov theorem (326). Exercises (329)
Contents xvii
Chapter 10. Conditional measures and conditional
expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
10.1. Conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .339
10.2. Convergence of conditional expectations. . . . . . . . . . . . . . . . . . . .346
10.3. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .348
10.4. Regular conditional measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
10.5. Liftings and conditional measures . . . . . . . . . . . . . . . . . . . . . . . . . . 371
10.6. Disintegration of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
10.7. Transition measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384
10.8. Measurable partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389
10.9. Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
10.10. Supplements and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Independence (398). Disintegrations (403). Strong liftings (406).
Zero.one laws (407). Laws of large numbers (410). Gibbs
measures (416). Triangular mappings (417). Exercises (427).
Bibliographical and Historical Comments. . . . . . . . . . . . . . . . . . . . . . . . .439
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561