Real analysis for graduate students: measure and integration theory

Title page......Page 1
Contents......Page 5
Preface......Page 9
1.1  Notation and terminology......Page 13
1.2  Some undergraduate mathematics......Page 14
2.1  Algebras and \\sigma-algebras......Page 19
2.2  The monotone class theorem......Page 22
2.3  Exercises......Page 23
3.1  Definitions and examples......Page 25
3.2  Exercises......Page 28
4  Construction of measures......Page 31
4.1  Outer measures......Page 32
4.2  Lebesgue-Stieltjes measures......Page 36
4.3  Examples and related results......Page 39
4.4  Nonmeasurable sets......Page 42
4.5  The Carath eodory extension theorem......Page 43
4.6  Exercises......Page 46
5.1  Measurability......Page 49
5.3  Lusin\'s theorem......Page 53
5.4  Exercises......Page 56
6.1  Definitions......Page 59
6.2  Exercises......Page 61
7.1  Monotone convergence theorem......Page 63
7.2  Linearity of the integral......Page 64
7.3  Fatou\'s lemma......Page 66
7.4  Dominated convergence theorem......Page 67
7.5  Exercises......Page 68
8.1  Criteria for a function to be zero a.e.......Page 75
8.2  An approximation result......Page 77
8.3  Exercises......Page 78
9.1  Comparison with the Lebesgue integral......Page 81
9.2  Exercises......Page 83
10.1  Definitions and examples......Page 87
10.2  Exercises......Page 90
11.1  Product \\sigma-algebras......Page 93
11.2  The Fubini theorem......Page 97
11.3  Examples......Page 99
11.4  Exercises......Page 100
12.1  Positive and negative sets......Page 105
12.2  Hahn decomposition theorem......Page 107
12.4  Exercises......Page 109
13.1  Absolute continuity......Page 111
13.2  The main theorem......Page 112
13.3  Lebesgue decomposition theorem......Page 115
13.4  Exercises......Page 116
14  Differentiation......Page 119
14.1  Maximal functions......Page 120
14.2  Antiderivatives......Page 124
14.3  Bounded variation......Page 125
14.4  Absolutely continuous functions......Page 130
14.5  Approach 2 - differentiability......Page 132
14.6  Approach 2 - antiderivatives......Page 136
14.7  Approach 2 - absolute continuity......Page 138
14.8  Exercises......Page 139
15.1  Norms......Page 143
15.2  Completeness......Page 146
15.3  Convolutions......Page 148
15.4  Bounded linear functionals......Page 149
15.5  Exercises......Page 153
16.1  Basic properties......Page 159
16.2  The inversion theorem......Page 162
16.3  The Plancherel theorem......Page 166
16.4  Exercises......Page 167
17  Riesz representation......Page 169
17.1  Partitions of unity......Page 170
17.2  The representation theorem......Page 171
17.3  Regularity......Page 175
17.4  Exercises......Page 176
18.1  Definitions......Page 179
18.2  The Hahn-Banach theorem......Page 180
18.3  Baire\'s theorem and consequences......Page 183
18.4  Exercises......Page 186
19.1  Inner products......Page 189
19.2  Subspaces......Page 192
19.3  Orthonormal sets......Page 194
19.4  Fourier series......Page 197
19.5  Exercises......Page 199
Bibliography......Page 203
Index......Page 204