Henstock-Kurzweil Integration on Euclidean Spaces

Contents......Page 8
Preface......Page 6
1.1 Introduction and Cousin’s Lemma......Page 12
1.2 Definition of the Henstock-Kurzweil integral......Page 15
1.3 Simple properties......Page 19
1.4 Saks-Henstock Lemma......Page 25
1.5 Notes and Remarks......Page 31
2.1 Preliminaries......Page 32
2.2 The Henstock-Kurzweil integral......Page 36
2.3 Simple properties......Page 39
2.4 Saks-Henstock Lemma......Page 46
2.5 Fubini’s Theorem......Page 54
2.6 Notes and Remarks......Page 62
3.1 Introduction......Page 64
3.2 Some convergence theorems for Lebesgue integrals......Page 67
3.3 µm-measurable sets......Page 74
3.4 A characterization of µm-measurable sets......Page 81
3.5 µm-measurable functions......Page 84
3.6 Vitali Covering Theorem......Page 90
3.7 Further properties of Lebesgue integrable functions......Page 93
3.8 The Lp spaces......Page 95
3.9 Lebesgue’s criterion for Riemann integrability......Page 99
3.10 Some characterizations of Lebesgue integrable functions......Page 102
3.11 Some results concerning one-dimensional Lebesgue integral......Page 112
3.12 Notes and Remarks......Page 115
4.1 A necessary condition for Henstock-Kurzweil integrability......Page 118
4.2 A result of Kurzweil and Jarn´ık......Page 119
4.3 Some necessary and su cient conditions for Henstock- Kurzweil integrability......Page 128
4.4 Harnack extension for one-dimensional Henstock-Kurzweil integrals......Page 130
4.5 Other results concerning one-dimensional Henstock- Kurzweil integral......Page 139
4.6 Notes and Remarks......Page 143
5.1 Lebesgue outer measure......Page 146
5.2 Basic properties of the Henstock variational measure......Page 149
5.3 Another characterization of Lebesgue integrable functions......Page 156
5.4 A result of Kurzweil and Jarn´ık revisited......Page 159
5.5 A measure-theoretic characterization of the Henstock- Kurzweil integral......Page 167
5.6 Product variational measures......Page 175
5.7 Notes and Remarks......Page 179
6.1 One-dimensional integration by parts......Page 180
6.2 On functions of bounded variation in the sense of Vitali......Page 186
6.3 The m-dimensional Riemann-Stieltjes integral......Page 191
6.4 A multiple integration by parts for the Henstock-Kurzweil integral......Page 195
6.5 Kurzweil’s multiple integration by parts formula for the Henstock-Kurzweil integral......Page 198
6.6 Riesz Representation Theorems......Page 205
6.7 Characterization of multipliers for the Henstock-Kurzweil integral......Page 209
6.8 A Banach-Steinhaus Theorem for the space of Henstock- Kurzweil integrable functions......Page 211
6.9 Notes and Remarks......Page 213
7.1 A generalized Dirichlet test......Page 216
7.2 Fourier series......Page 221
7.3 Some examples of Fourier series......Page 224
7.4 Some Lebesgue integrability theorems for trigonometric series......Page 229
7.5 Boas’ results......Page 237
7.6 On a result of Hardy and Littlewood concerning Fourier series......Page 240
7.7 Notes and Remarks......Page 243
8.1 Regularly convergent double series......Page 244
8.2 Double Fourier series......Page 251
8.3 Some examples of double Fourier series......Page 257
8.4 A Lebesgue integrability theorem for double cosine series......Page 261
8.5 A Lebesgue integrability theorem for double sine series......Page 268
8.6 A convergence theorem for Henstock-Kurzweil integrals......Page 275
8.7 Applications to double Fourier series......Page 281
8.8 Another convergence theorem for Henstock-Kurzweil integrals......Page 285
8.9 A two-dimensional analogue of Boas’ theorem......Page 289
8.10 A convergence theorem for double sine series......Page 298
8.11 Some open problems......Page 301
8.12 Notes and Remarks......Page 305
Bibliography......Page 306
Points, intervals and partitions......Page 316
Functions, integrals and function spaces......Page 318
Measures and outer measures......Page 320
Miscellaneous......Page 322
General index......Page 324