Basic Analysis IV Measure Theory and Integration
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.
• 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
1.Introduction. 2. An Overview Of Riemann Integration. 3. Functions Of Bounded Variation. 4.The Theory Of Riemann Integration. 5. Further Riemann Integration Results. 6. The Riemann-Stieltjes Integral. 7. Further Riemann - Stieljes Results. 8. Measurable Functions and Spaces. 9. Measure And Integration. 10. The Lp Spaces. 11. Constructing Measures. 12. Lebesgue Measure. 13. Cantor Set Experiments. 14. Lebesgue Stieljes Measure. 15. Modes Of Convergence. 16. Decomposition Of Measures. 17. Connections To Riemann Integration. 18. Fubini Type Results. 19. Differentiation. 20. Summing It All Up. References. Index. Appendix A. Appendix B. Appendix C. Appendix D.
"Mathematics is fortunate to be populated by bright practitioners. Nonetheless, amongst these we are fortunate to have rare individuals who are wise. Professor Peterson is a member of this distinguished group. His works clearly demonstrate the importance of a long career of research and teaching where he combines the two perspectives of: (1) clearly understanding the needs of diverse readers for clear exposition that scaffolds their exposure to complex material with a transparency about both where they are going and what the utility is of what they are currently reading; and, (2) the benefits of having used the mathematics under consideration in so many diverse applications. The masterly synthesis of so much complex material by a single individual is a superb achievement which will reward serious readers with insight, surprise, and breadth as well as depth."
– Professor John R. Jungck, University of Delaware
"Analysis is the bedrock of rigorous mathematical thinking and abstraction. Prof. Peterson's book does a fascinating job by taking a critical approach - highly recommended."
– Professor Nithin Nagaraj, National Institute of Advanced Studies
"Dr. Peterson's thoughtful and detailed explanations reflect his insights to a very fundamental but complex subject in Mathematics. The treatment in the book does justice to recent trends in Mathematical Analysis while staying true to the classical spirit of the subject. A thoroughly enjoyable read."
– Professor Snehanshu Saha, BITS PIlani (K K Birla Goa Campus)