Basic Analysis IV: Measure Theory and Integration

Basic Analysis IV: Measure Theory and Integration
PDF
  • eBook:
    Basic Analysis IV: Measure Theory and Integration
  • Author:
    James K. Peterson
  • Edition:
    1 edition
  • Categories:
  • Data:
    August 13, 2020
  • ISBN:
    1138055115
  • ISBN-13:
    9781138055117
  • Language:
    English
  • Pages:
    500 pages
  • Format:
    PDF

Book Description
Basic Analysis IV: Measure Theory and Integration introduces students to concepts from measure theory and continues their training in the abstract way of looking at the world. This is a most important skill to have when your life's work will involve quantitative modeling to gain insight into the real world. This text generalizes the notion of integration to a very abstract setting in a variety of ways. We generalize the notion of the length of an interval to the measure of a set and learn how to construct the usual ideas from integration using measures. We discuss carefully the many notions of convergence that measure theory provides.
Features
• Can be used as a traditional textbook as well as for self-study
• Suitable for advanced students in mathematics and associated disciplines
• Emphasises learning how to understand the consequences of assumptions using a variety of tools to provide the proofs of propositions

Content

I - Introductory Matter
1. Introduction

II - Classical Riemann Integration
2. An Overview of Riemann Integration
3. Bounded Variation
4. Riemann Integration
5. Further Riemann Results

III - Riemann -Stieltjes Integration
6. The Riemann -Stieltjes Integral
7. Further Riemann -Stieltjes Results

IV - Abstract Measure Theory One
8. Measurability
9. Abstract Integration
10. The Lp Spaces

V - Constructing Measures
11. Building Measures
12. Lebesgue Measure
13. Cantor Sets
14. Lebesgue -Stieltjes Measure

VI - Abstract Measure Theory Two
15. Convergence Modes
16. Decomposing Measures
17. Connections to Riemann Integration
18. Fubini Type Results
19. Differentiation

VII - Summing It All Up
20. Summing It All Up

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