Measures, integrals and martingales

René L. Schilling9780521615259, 0521615259, 0521850150, 9780521850155

This is a concise and elementary introduction to contemporary measure and integration theory as it is needed in many parts of analysis and probability theory. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites, making the text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included to consolidate what has already been learned and to discover variants and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached at http://www.motapa.de/measures_integrals_and_martingales/index.html

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Prelude……Page 10
1 Prologue……Page 15
Problems……Page 18
2 The pleasures of counting……Page 19
Problems……Page 27
3 Sigma-algebras……Page 29
Problems……Page 34
4 Measures……Page 36
Problems……Page 42
5 Uniqueness of measures……Page 45
Problems……Page 50
6 Existence of measures……Page 51
Problems……Page 60
7 Measurable mappings……Page 63
Problems……Page 68
8 Measurable functions……Page 71
Problems……Page 79
9 Integration of positive functions……Page 81
Problems……Page 87
10 Integrals of measurable functions and null sets……Page 90
Problems……Page 98
11 Convergence theorems and their applications……Page 102
Parameter-dependent integrals……Page 105
Riemann vs. Lebesgue integration……Page 106
Examples……Page 112
Problems……Page 114
12 The function spaces………Page 119
Problems……Page 130
13 Product measures and Fubini’s theorem……Page 134
More on measurable functions……Page 141
Distribution functions……Page 142
Problems……Page 144
14 Integrals with respect to image measures……Page 148
Problems……Page 154
15 Integrals of images and Jacobi’s transformation rule……Page 156
Jacobi’s transformation formula……Page 161
Spherical coordinates and the volume of the unit ball……Page 166
Continuous functions are dense in………Page 170
Regular measures……Page 172
Problems……Page 173
16 Uniform integrability and Vitali’s convergence theorem……Page 177
Problems……Page 187
17 Martingales……Page 190
Problems……Page 202
18 Martingale convergence theorems……Page 204
Problems……Page 214
The Radon–Nikodým theorem……Page 216
Martingale inequalities……Page 225
The Hardy–Littlewood maximal theorem……Page 227
Lebesgue’s differentiation theorem……Page 232
The Calderón–Zygmund lemma……Page 235
Problems……Page 236
20 Inner product spaces……Page 240
Problems……Page 246
21 Hilbert space………Page 248
Gram–Schmidt orthonormalization procedure……Page 258
Problems……Page 260
22 Conditional expectations in L……Page 262
On the structure of subspaces of L……Page 267
Problems……Page 271
23 Conditional expectations in L……Page 272
Classical conditional expectations……Page 277
Separability criteria for the spaces Lp X……Page 283
Problems……Page 288
Orthogonal polynomials……Page 290
The trigonometric system and Fourier series……Page 297
The Haar system……Page 303
The Haar wavelet……Page 309
The Rademacher functions……Page 313
Well-behaved orthonormal systems……Page 316
Problems……Page 326
Appendix A lim inf and lim sup……Page 327
Appendix B Some facts from point-set topology……Page 332
Topological spaces……Page 333
Metric spaces……Page 336
Normed spaces……Page 339
Appendix C The volume of a parallelepiped……Page 342
Appendix D Non-measurable sets……Page 344
The (proper) Riemann integral……Page 351
Integrals and limits……Page 365
Improper Riemann integrals……Page 367
A. Improper Riemann integrals of the type………Page 368
B. Improper Riemann integrals with unbounded integrands……Page 372
C. Improper Riemann integrals where both limits are critical……Page 373
Further reading……Page 374
Fourier series, harmonic analysis, orthonormal systems, wavelets……Page 375
Probability theory (in particular probabilistic measure theory)……Page 376
Martingales and their applications……Page 377
References……Page 378
Notation index……Page 381
Name and subject index……Page 385