Peterson J.K. Basic Analysis IV: Measure Theory and Integration

Chapman and Hall/CRC, 2020. — 500 p. — ISBN: 978-1-138-05511-7.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 propositionsIntroductory Matter
Classical Riemann Integration
An Overview of Riemann Integration
Bounded Variation
Riemann Integration
Further Riemann Results
Riemann -Stieltjes Integration
The Riemann -Stieltjes Integral
Further Riemann -Stieltjes Results
Abstract Measure Theory One
Measurability
Abstract Integration
The Lp Spaces
Constructing Measures
Building Measures
Lebesgue Measure
Cantor Sets
Lebesgue -Stieltjes Measure
Abstract Measure Theory Two
Convergence Modes
Decomposing Measures
Connections to Riemann Integration
Fubini Type Results
Differentiation
Appendixes
Undergraduate Analysis Background Check
Linear Analysis Background Check