A Modern Theory of Integration

Part 1. Integration on compact intervals
Chapter 1. Gauges and integrals
Chapter 2. Some examples
Chapter 3. Basic properties of the integral
Chapter 4. The fundamental theorems of calculus
Chapter 5. The Saks-Henstock lemma
Chapter 6. Measurable functions
Chapter 7. Absolute integrability
Chapter 8. Convergence theorems
Chapter 9. Integrability and mean convergence
Chapter 10. Measure, measurability, and multipliers
Chapter 11. Modes of convergence
Chapter 12. Applications to calculus
Chapter 13. Substitution theorems
Chapter 14. Absolute continuity
Part 2. Integration on infinite intervals
Chapter 15. Introduction to Part 2
Chapter 16. Infinite intervals
Chapter 17. Further re-examination
Chapter 18. Measurable sets
Chapter 19. Measurable functions
Chapter 20. Sequences of functions
Appendixes
Appendix A. Limits superior and inferior
Appendix B. Unbounded sets and sequences
Appendix C. The arctangent lemma
Appendix D. Outer measure
Appendix E. Lebesgue's differentiation theorem
Appendix F. Vector spaces
Appendix G. Semimetric spaces
Appendix H. Riemann-Stieltjes integral
Appendix I. Normed linear spaces
Some partial solutions
Solutions Manual