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The Bochner Integral

  • Book
  • © 1978

Overview

Authors:
  1. Jan Mikusiński
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Part of the book series: Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften (LMW, volume 55)

Part of the book sub series: Mathematische Reihe (LMW/MA)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xii
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  2. The Lebesgue Integral

    • Jan Mikusiński
    Pages 1-7

  3. Banach Space

    • Jan Mikusiński
    Pages 8-14

  4. The Bochner Integral

    • Jan Mikusiński
    Pages 15-22

  5. Axiomatic Theory of the Integral

    • Jan Mikusiński
    Pages 23-36

  6. Applications to Set Theory

    • Jan Mikusiński
    Pages 37-39

  7. Measurable Functions

    • Jan Mikusiński
    Pages 40-52

  8. Examples and Counterexamples

    • Jan Mikusiński
    Pages 53-64

  9. The Upper Integral and Some Traditional Approaches to Integration

    • Jan Mikusiński
    Pages 65-82

  10. Defining New Integrals by Given Ones

    • Jan Mikusiński
    Pages 83-90

  11. The Fubini Theorem

    • Jan Mikusiński
    Pages 91-105

  12. Complements on Functions and Sets in the Euclidean q-Space

    • Jan Mikusiński
    Pages 106-113

  13. Changing Variables in Integrals

    • Jan Mikusiński
    Pages 114-163

  14. Integration and Derivation

    • Jan Mikusiński
    Pages 164-182

  15. Convolution

    • Jan Mikusiński
    Pages 183-200

  16. The Titchmarsh Theorem

    • Jan Mikusiński
    Pages 201-210

  17. Erratum to: Axiomatic Theory of the Integral

    • Jan Mikusiński
    Pages 234-234

  18. Back Matter

    Pages 211-233
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Keywords

About this book

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.

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