Real Analysis

Theory of Measure and Integration

This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.

The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory.

Contents:

• Measure Spaces

• The Lebesgue Integral

• Differentiation and Integration

• The Classical Banach Spaces

• Extension of Additive Set Functions to Measures

• Measure and Integration on the Euclidean Space

• Hausdorff Measures on the Euclidean Space

• Appendices:

  • Digital Expansions of Real Numbers
  • Measurability of Limits and Derivatives
  • Lipschitz Condition and Bounded Derivatives
  • Uniform Integrability
  • Product-measurability and Factor-measurability
  • Functions of Bounded Oscillation

Readership: Mathematicians and graduate students in analysis & differential equations.
Key Features:

• The book is a very clear, incisive, thorough, didactic and encyclopedic treatise of measure and integration
• If you have this book you do not need another book on measure and integration
• The book is also suitable to graduate students in statistics and engineering