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The Bochner Integral

  • Authors
  • Jan Mikusiński

Part of the Mathematische Reihe book series (LMW, volume 55)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Jan Mikusiński
    Pages 1-7
  3. Jan Mikusiński
    Pages 8-14
  4. Jan Mikusiński
    Pages 15-22
  5. Jan Mikusiński
    Pages 23-36
  6. Jan Mikusiński
    Pages 37-39
  7. Jan Mikusiński
    Pages 40-52
  8. Jan Mikusiński
    Pages 53-64
  9. Jan Mikusiński
    Pages 83-90
  10. Jan Mikusiński
    Pages 91-105
  11. Jan Mikusiński
    Pages 114-163
  12. Jan Mikusiński
    Pages 164-182
  13. Jan Mikusiński
    Pages 183-200
  14. Jan Mikusiński
    Pages 201-210
  15. Jan Mikusiński
    Pages 234-234
  16. Back Matter
    Pages 211-233

About this book

Introduction

The theory of the Lebesgue integral is still considered as a difficult theory, no matter whether it is based the concept of measure or introduced by other methods. The primary aim of this book is to give an approach which would be as intelligible and lucid as possible. Our definition, produced in Chapter I, requires for its background only a little of the theory of absolutely convergent series so that it is understandable for students of the first undergraduate course. Nevertheless, it yields the Lebesgue integral in its full generality and, moreover, extends automatically to the Bochner integral (by replacing real coefficients of series by elements of a Banach space). It seems that our approach is simple enough as to eliminate the less useful Riemann integration theory from regular mathematics courses. Intuitively, the difference between various approaches to integration may be brought out by the following story on shoemakers. A piece of leather, like in Figure 1, is given. The task consists in measuring its area. There are three shoemakers and each of them solves the task in his own way. A B Fig. 1 The shoemaker R. divides the leather into a finite number of vertical strips and considers the strips approximately as rectangles. The sum of areas of all rectangles is taken for an approximate area of the leather (Figure 2). If he is not satisfied with the obtained exactitude, he repeats the whole procedure, by dividing the leather into thinner strips.

Keywords

Bochner integral Integral integration theory

Bibliographic information