Modern Theory of Integration

Robert G. Bartle, Eastern Michigan University, Ypsilanti, MI, and University of Illinois, Urbana, IL

The book presents its subject in a pleasing, didactically surprising, and well worth reading exposition. The proofs are, as a rule, easily understandable and the significance of the theorems that are worked through is illustrated by means of numerous examples. It can be recommended as a self-study book to every student with a basic foundation in analysis. It is also very suitable as a supplementary text for a course on integration on $\mathbf{R}$. -- Translated fromJahresbericht der Deutschen Mathematiker-Vereinigung A comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral ... There is an abundant supply of exercises which serve to make this book an excellent choice for a text for a course which would contain an elementary introduction to modern integration theory. -- Zentralblatt MATH

• Part 1. Integration on compact intervalsChapter 1. Gauges and integralsChapter 2. Some examplesChapter 3. Basic properties of the integralChapter 4. The fundamental theorems of calculusChapter 5. The Saks-Henstock lemmaChapter 6. Measurable functionsChapter 7. Absolute integrabilityChapter 8. Convergence theoremsChapter 9. Integrability and mean convergenceChapter 10. Measure, measurability, and multipliersChapter 11. Modes of convergenceChapter 12. Applications to calculusChapter 13. Substitution theoremsChapter 14. Absolute continuityPart 2. Integration on infinite intervalsChapter 15. Introduction to Part 2Chapter 16. Infinite intervalsChapter 17. Further re-examinationChapter 18. Measurable setsChapter 19. Measurable functionsChapter 20. Sequences of functionsAppendixesAppendix A. Limits superior and inferiorAppendix B. Unbounded sets and sequencesAppendix C. The arctangent lemmaAppendix D. Outer measureAppendix E. Lebesgue's differentiation theoremAppendix F. Vector spacesAppendix G. Semimetric spacesAppendix H. Riemann-Stieltjes integralAppendix I. Normed linear spacesSome partial solutionsSolutions Manual