This lively introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems, highlighting the difficulties mathematicians encountered as these ideas were refined. The story begins with Riemann's definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemann's definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis.
Author(s): David M. Bressoud
Series: Mathematical Association of America Textbooks
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 344
Tags: Differential Equations;Applied;Mathematics;Science & Math;Abstract;Algebra;Pure Mathematics;Mathematics;Science & Math;Calculus;Pure Mathematics;Mathematics;Science & Math;Algebra & Trigonometry;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique;Calculus;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique
Cover ... 1
About the Book and the Author ... 2
Editors and List of Publications ... 4
A RADICAL APPROACH TO LEBESGUE'S THEORY OF INTEGRATION ... 6
Copyright ... 7
© David M. Bressoud 2008 ... 7
ISBN-13: 978-0-521-88474-7 (hardback) ... 7
ISBN-13: 978-0-521-71183-8 (pbk.) ... 7
QA312.B67 2008 5 15'.42—dc22 ... 7
Dedication ... 8
Contents ... 10
Preface ... 12
1 Introduction ... 16
1.1 The Five Big Questions ... 17
Fourier Series ... 17
Integration ... 19
Cauchy and Riemann Integrals ... 22
The Fundamental Theorem of Calculus ... 23
A Brief History of Theorems 1.1 and 1.2^2 ... 24
Continuity and Differentiability ... 27
Term-by-term Integration ... 28
Exercises ... 28
1.2 Presumptions ... 30
Notation ... 30
Definitions ... 31
Theorems ... 33
Exercises ... 35
2 The Riemann Integral ... 38
2.1 Existence ... 39
The Darboux Integrals ... 43
Improper Integrals ... 45
Exercises ... 46
2.2 Nondifferentiable Integrals ... 48
Summary ... 53
Exercises ... 53
2.3 The Class of 1870 ... 55
Hankel's Innovations ... 57
Hankel's Types of Discontinuity ... 59
Hankel's Error ... 61
Cantor's 1872 Paper ... 61
Exercises ... 63
3 Explorations of R ... 66
3.1 Geometry of R ... 66
An Infinite Extension ... 67
Topology of R ... 68
More Definitions ... 70
Exercises ... 72
3.2 Accommodating Algebra ... 74
Implications of the Bolzano—Weierstrass Theorem ... 74
Completeness ... 76
Harnack's Mistake ... 78
Borel's Series ... 79
Compactness ... 80
Two Corollaries ... 81
How Heine's Name Got Attached to This Theorem ... 82
Exercises ... 84
3.3 Set Theory ... 86
Cardinality ... 86
The Continuum Hypothesis ... 90
Power Sets ... 91
Exercises ... 93
4 Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus ... 96
4.1 The Smith—Volterra—Cantor Sets ... 97
The Cantor Ternary Set ... 98
The Devil's Staircase ... 100
Exercises ... 103
4.2 Volterra's Function ... 104
SVC(4) ... 105
Perfect, Nowhere Dense Sets ... 109
SVC(n) ... 111
Exercises ... 111
4.3 Term-by-Term Integration ... 113
What Can Happen ... 115
Preserving Some Uniformity ... 118
Is Boundedness Sufficient? ... 119
The Arzelà—Osgood Theorem ... 121
Exercises ... 123
4.4 The Baire Category Theorem ... 124
Applications of Baire's Theorem ... 127
Baire's Big Theorem ... 129
Lebesgue's Proof of Theorem 4.11 ... 130
Discontinuities of Derivatives ... 131
Exercises ... 132
5 The Development of Measure Theory ... 135
5.1 Peano, Jordan, and Borel ... 137
Jordan Measure ... 139
Borel Measure ... 141
Borel Sets ... 142
The Limitations of Borel Measure ... 143
Exercises ... 144
5.2 Lebesgue Measure ... 146
Improving on Borel ... 149
Alternate Definition of Lebesgue Measure ... 152
Exercises ... 153
5.3 Carathéodory's Condition ... 155
Exercises ... 163
5.4 Nonmeasurable Sets ... 165
Difficulties ... 166
Pursuing the Axiom of Choice ... 169
Do Nonmeasurable Sets Exist? ... 170
Exercises ... 172
6 The Lebesgue Integral ... 174
6.1 Measurable Functions ... 174
Limits of Measurable Functions ... 176
Farewell to the Riemann Integral ... 179
Exercises ... 182
6.2 Integration ... 184
Integration of Measurable Functions ... 186
The Monotone Convergence Theorem ... 188
Exercises ... 195
6.3 Lebesgue's Dominated Convergence Theorem ... 198
Uniform Convergence ... 198
Bounded Convergence ... 199
Example 4.7 from Section 4.3 ... 199
Example 4.6 from Section 4.3 ... 201
Sufficient but Not Necessary ... 201
Fatou's Lemma ... 202
Proof of the Dominated Convergence Theorem ... 203
Exercises ... 204
6.4 Egorov's Theorem ... 206
Convergence in Measure ... 209
Limits of Step Functions ... 211
Luzin's Theorem ... 213
Exercises ... 214
7 The Fundamental Theorem of Calculus ... 218
7.1 The Dini Derivatives ... 219
Bounded Variation ... 221
Exercises ... 225
7.2 Monotonicity Implies Differentiability Almost Everywhere ... 227
Outlining the Proof ... 228
The Proof of Theorem 7.4 ... 230
The Faber—Chisholm--Young Theorem ... 233
Exercises ... 236
7.3 Absolute Continuity ... 238
The Evaluation Part ... 239
A Little History ... 241
Lebesgue Integral and Absolute Continuity ... 241
A Hierarchy of Functions ... 243
Absolute Continuity and Monotonicity ... 244
Exercises ... 245
7.4 Lebesgue's FTC ... 246
Exercises ... 253
8 Fourier Series ... 256
8.1 Pointwise Convergence ... 257
Cesàro Convergence ... 260
Exercises ... 263
8.2 Metric Spaces ... 266
L^p Spaces ... 268
Convergence ... 272
Ordering L^p Spaces ... 275
Exercises ... 275
8.3 Banach Spaces ... 278
The Riesz—Fischer Theorem ... 282
Exercises ... 285
8.4 Hubert Spaces ... 286
Complete Orthogonal Set ... 288
Complete Orthonormal Sets ... 292
Completing the Proof of the Riesz—Fischer Theorem ... 294
Exercises ... 295
9 Epilogue ... 297
Appendix A: Other Directions ... 302
A.1 The Cardinality of the Collection of Borel Sets ... 302
Connection to Baire ... 305
Exercises ... 305
A.2 The Generalized Riemann Integral ... 306
Dirichiet's Function ... 308
The Fundamental Theorem of Calculus ... 308
Comparison with the Lebesgue Integral ... 309
Final Thoughts ... 311
Exercises ... 313
Appendix B: Hints to Selected Exercises ... 314
Bibliography ... 332
Index ... 338
