Topics in Banach space integration

Stefan Schwabik9812564284, 9789812564283, 9789812703286

The relatively new concepts of the Henstock–Kurzweil and McShane integrals based on Riemann type sums are an interesting challenge in the study of integration of Banach space-valued functions. This timely book presents an overview of the concepts developed and results achieved during the past 15 years. The Henstock–Kurzweil and McShane integrals play the central role in the book. Various forms of the integration are introduced and compared from the viewpoint of their generality. Functional analysis is the main tool for presenting the theory of summation gauge integrals.

---

Table of contents :
Cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Notation……Page 9
Contents……Page 11
1.1 Simple functions, measurability……Page 15
1.2 The integral of simple functions……Page 23
1.3 Bochner integral……Page 25
1.4 Properties of Bochner integrable functions and of the Bochner integral……Page 36
2.1 Dunford integral……Page 41
2.2 Pettis integral……Page 47
2.3 Some properties of the Pettis integral……Page 50
3.1 Systems, partitions and gauges……Page 59
3.2 Definition of the McShane and Henstock- Kurzweil integrals……Page 60
3.3 Elementary properties of the McShane and Henstock-Kurzweil integrals……Page 62
3.4 The Saks-Henstock lemma……Page 69
3.5 A convergence theorem……Page 78
3.6 The strong versions of the McShane and Henstock-Kurzweil integrals……Page 84
3.7 Integration over unbounded intervals and some remarks……Page 99
4.1 Special properties……Page 101
4.2 An equivalent definition of the McShane integral……Page 127
4.3 Another convergence theorem……Page 133
5.1 Strong McShane integrability and the Bochner integral……Page 147
5.2 The finite dimensional case……Page 164
5.3 The infinite dimensional case……Page 167
5.4 An example……Page 173
6.1 McShane integrable functions are Pettis integrable……Page 185
6.2 The problem of $mathcal{P}subsetmathcal{M}$……Page 187
6.2.1 Functions weakly equivalent to measurable ones……Page 197
6.2.2 $mathcal{P}subsetmathcal{M}$ does not hold in general……Page 202
7. Primitive of the McShane and Henstock- Kurzweil Integrals……Page 205
7.1 Absolutely continuous functions and functions of bounded variation……Page 206
7.2 Generalized absolute continuity and functions of generalized bounded variation……Page 214
7.3 Differentiability……Page 216
7.4 Primitives……Page 225
7.4.1 The strong Henstock-Kurzweil integral……Page 226
7.4.2 The McShane and the strong McShane integral……Page 232
7.4.3 The Henstock-Kurzweil integral……Page 237
7.5 Variational measures and primitives for $mathcal{SM}$ and $mathcal{SHK}$……Page 240
7.6 Controlled convergence……Page 245
8.1 Bochner integral……Page 265
8.2 Dunford and Pettis integral……Page 268
8.2.1 Denjoy approach……Page 269
8.2.2 Henstock-Kurzweil approach……Page 283
8.2.3 Some examples……Page 286
8.3 Concluding remarks……Page 287
A.1 Spaces of sequences……Page 291
A.2.2 The spaces $L_1$ and $L_infty$……Page 293
Appendix B Series in Banach Spaces……Page 297
Bibliography……Page 305
Index……Page 311