Introduction to Integration

• 1. Setting the scene ; 2. Preliminaries ; 3. Intervals and step functions ; 4. Integrals of step functions ; 5. Continuous functions on compact intervals ; 6. Techniques of Integration I ; 7. Approximations ; 8. Uniform convergence and power series ; 9. Building foundations ; 10. Null sets ; 11. Linc functions ; 12. The space L of integrable functions ; 13. Non-integrable functions ; 14. Convergence Theorems: MCT and DCT ; 15. Recognizing integrable functions I ; 16. Techniques of integration II ; 17. Sums and integrals ; 18. Recognizing integrable functions II ; 19. The Continuous DCT ; 20. Differentiation of integrals ; 21. Measurable functions ; 22. Measurable sets ; 23. The character of integrable functions ; 24. Integration VS. differentiation ; 25. Integrable functions of Rk ; 26. Fubini's Theorem and Tonelli's Theorem ; 27. Transformations of Rk ; 28. The spaces L1, L2 and Lp ; 29. Fourier series: pointwise convergence ; 30. Fourier series: convergence re-assessed ; 31. L2-spaces: orthogonal sequences ; 32. L2-spaces as Hilbert spaces ; 33. The Fourier transform ; 34. Integration in probability theory ; Appendix I ; Appendix II ; Bibliography ; Notation index ; Subject index

Delightful book. Those who know Hilary Priestley will recognise at once the impish sense of fun which permeates this book (even down to the selection of notation): she has a real gift for the memorable phrase and the agonies oand ecstasies of teaching 25 years worth of Oxford undergraduates are etched in the motivational and orientational remarks, helpful reiterations of key points, local stock-taking' susummaries and tight internal sign-posting. By its very nature integration theory cannot be made easy, but Professor Priestley will rapidly earn the gratitude of a new generation of students for making it as pleasantly palatable as one could wish for.