دانلود کتاب Measure, integration and real analysis

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About the Author......Page 7
Contents......Page 8
Preface for Students......Page 14
Preface for Instructors......Page 15
Acknowledgments......Page 19
Riemann Integration......Page 20
Review: Riemann Integral......Page 21
Exercises 1A......Page 26
Riemann Integral Is Not Good Enough......Page 28
Exercises 1B......Page 31
Measures......Page 32
Motivation and Definition of Outer Measure......Page 33
Good Properties of Outer Measure......Page 34
Outer Measure of Closed Bounded Interval......Page 37
Outer Measure is Not Additive......Page 40
Exercises 2A......Page 42
Measurable Spaces and Functions......Page 44
-Algebras......Page 45
Borel Subsets of R......Page 47
Inverse Images......Page 48
Measurable Functions......Page 50
Exercises 2B......Page 57
Definition and Examples of Measures......Page 60
Properties of Measures......Page 61
Exercises 2C......Page 64
Additivity of Outer Measure on Borel Sets......Page 66
Lebesgue Measurable Sets......Page 71
Cantor Set and Cantor Function......Page 74
Exercises 2D......Page 79
Pointwise and Uniform Convergence......Page 81
Egorov's Theorem......Page 82
Approximation by Simple Functions......Page 84
Luzin's Theorem......Page 85
Lebesgue Measurable Functions......Page 88
Exercises 2E......Page 90
Integration......Page 92
Integration of Nonnegative Functions......Page 93
Monotone Convergence Theorem......Page 96
Integration of Real-Valued Functions......Page 100
Exercises 3A......Page 103
Bounded Convergence Theorem......Page 107
Sets of Measure 0 in Integration Theorems......Page 108
Dominated Convergence Theorem......Page 109
Riemann Integrals and Lebesgue Integrals......Page 112
Approximation by Nice Functions......Page 114
Exercises 3B......Page 118
Differentiation......Page 120
Markov's Inequality......Page 121
Vitali Covering Lemma......Page 122
Hardy–Littlewood Maximal Inequality......Page 123
Exercises 4A......Page 125
Lebesgue Differentiation Theorem......Page 127
Derivatives......Page 129
Density......Page 131
Exercises 4B......Page 134
Product Measures......Page 135
Products of -Algebras......Page 136
Monotone Class Theorem......Page 139
Products of Measures......Page 142
Exercises 5A......Page 147
Tonelli's Theorem......Page 148
Fubini's Theorem......Page 150
Area Under Graph......Page 152
Exercises 5B......Page 154
Borel Subsets of Rn......Page 155
Lebesgue Measure on Rn......Page 158
Volume of Unit Ball in Rn......Page 159
Equality of Mixed Partial Derivatives Via Fubini's Theorem......Page 161
Exercises 5C......Page 163
Banach Spaces......Page 165
Open Sets, Closed Sets, and Continuity......Page 166
Cauchy Sequences and Completeness......Page 170
Exercises 6A......Page 172
Integration of Complex-Valued Functions......Page 174
Vector Spaces and Subspaces......Page 178
Exercises 6B......Page 181
Norms and Complete Norms......Page 182
Bounded Linear Maps......Page 186
Exercises 6C......Page 189
Bounded Linear Functionals......Page 191
Discontinuous Linear Functionals......Page 193
Hahn–Banach Theorem......Page 196
Exercises 6D......Page 200
Baire's Theorem......Page 203
Open Mapping Theorem and Inverse Mapping Theorem......Page 205
Closed Graph Theorem......Page 207
Principle of Uniform Boundedness......Page 208
Exercises 6E......Page 209
Lp Spaces......Page 212
Hölder's Inequality......Page 213
Minkowski's Inequality......Page 217
Exercises 7A......Page 218
Definition of Lp()......Page 221
Lp() Is a Banach Space......Page 223
Duality......Page 225
Exercises 7B......Page 227
Hilbert Spaces......Page 230
Inner Products......Page 231
Cauchy–Schwarz Inequality and Triangle Inequality......Page 233
Exercises 8A......Page 240
Orthogonal Projections......Page 243
Orthogonal Complements......Page 248
Riesz Representation Theorem......Page 252
Exercises 8B......Page 253
Bessel's Inequality......Page 256
Parseval's Identity......Page 262
Gram–Schmidt Process and Existence of Orthonormal Bases......Page 264
Riesz Representation Theorem, Revisited......Page 269
Exercises 8C......Page 270
Real and Complex Measures......Page 274
Properties of Real and Complex Measures......Page 275
Total Variation Measure......Page 278
The Banach Space of Measures......Page 281
Exercises 9A......Page 284
Hahn Decomposition Theorem......Page 286
Jordan Decomposition Theorem......Page 287
Lebesgue Decomposition Theorem......Page 289
Radon–Nikodym Theorem......Page 291
Dual Space of Lp()......Page 294
Exercises 9B......Page 297
Linear Maps on Hilbert Spaces......Page 299
Adjoints of Linear Maps on Hilbert Spaces......Page 300
Null Spaces and Ranges in Terms of Adjoints......Page 304
Invertibility of Operators......Page 305
Exercises 10A......Page 311
Spectrum of an Operator......Page 313
Self-adjoint Operators......Page 318
Normal Operators......Page 321
Isometries and Unitary Operators......Page 324
Exercises 10B......Page 328
The Ideal of Compact Operators......Page 331
Spectrum of Compact Operator and Fredholm Alternative......Page 335
Exercises 10C......Page 342
Orthonormal Bases Consisting of Eigenvectors......Page 345
Singular Value Decomposition......Page 351
Exercises 10D......Page 355
Fourier Analysis......Page 358
Fourier Coefficients and Riemann–Lebesgue Lemma......Page 359
Poisson Kernel......Page 363
Solution to Dirichlet Problem on Disk......Page 367
Fourier Series of Smooth Functions......Page 369
Exercises 11A......Page 371
Orthonormal Basis for L2 of Unit Circle......Page 374
Convolution on Unit Circle......Page 376
Exercises 11B......Page 380
Fourier Transform on L1(R)......Page 382
Convolution on R......Page 387
Poisson Kernel on Upper Half-Plane......Page 389
Fourier Inversion Formula......Page 393
Extending Fourier Transform to L2(R)......Page 394
Exercises 11C......Page 396
Probability Measures......Page 399
Probability Spaces......Page 400
Independent Events and Independent Random Variables......Page 402
Variance and Standard Deviation......Page 407
Conditional Probability and Bayes' Theorem......Page 409
Distribution and Density Functions of Random Variables......Page 411
Weak Law of Large Numbers......Page 415
Exercises 12......Page 417
Photo Credits......Page 419
Bibliography......Page 421
Notation Index......Page 422
Index......Page 425
Colophon: Notes on Typesetting......Page 430