An Introduction to Lebesgue Integration and Fourier Series (Applied Mathematics Series)

Titlepage......Page 1
Contents......Page 3
Preface......Page 6
1 Definition of the Riemann Integral......Page 8
2 Properties of the Riemann Integral......Page 13
3 Examples......Page 14
4 Drawbacks of the Riemann Integral......Page 15
5 Exercises......Page 16
6 In troduction......Page 20
7 Outer Measure......Page 24
8 Measurable Sets......Page 28
9 Exercises......Page 33
10 Countable Additivity......Page 36
11 Summary......Page 39
*12 Borel Sets and the Cantor Set......Page 40
*13 Necessary and Sufficient Conditions for a Set to be Measurable......Page 43
14 Lebesgue Measure for Bounded Sets......Page 45
*15 Lebesgue Measure for Unbounded Sets......Page 47
16 Exercises......Page 48
17 Definition of Measurable Functions......Page 54
18 Preservation of Measurability for Functions......Page 57
19 Simple Functions......Page 60
20 Exercises......Page 62
21 The Lebesgue Integral for Bounded Measurable Functions......Page 66
22 Simple Functions......Page 67
23 Integrability of Bounded Measurable Functions......Page 69
24 Elementary Properties of the Integral for Bounded Functions......Page 72
25 The Lebesgue Integral for Unbounded Functions......Page 75
26 Exercises......Page 80
27 Examples......Page 84
28 Convergence Theorems......Page 85
*29 A Necessary and Sufficient Condition for Riemann Integrability......Page 89
*30 Egorof\'s and Lusin\'s Theorems and an Alternative Proof of the Lebesgue Dominated Convergence Theorem......Page 92
31 Exercises......Page 95
32 Linear Spaces......Page 100
33 The Space L^2......Page 108
34 Exercises......Page 115
35 Definition and Examples......Page 120
36 Elementary Properties......Page 125
37 L^2 Convergence of Fourier Series......Page 127
38 Exercises......Page 134
39 An Application: Vibrating Strings......Page 140
40 Some Bad Examples and Good Theorems......Page 144
41 More Convergence Theorems......Page 147
42 Exercises......Page 151
 Logic and Sets......Page 154
Bounded Sets of Real Numbers......Page 155
Countable and Uncountable Sets (and discussion of the Axiom of Choice)......Page 156
Real Functions......Page 158
Sequences of Functions......Page 160
Bibliography......Page 162
Index......Page 164