Book:H.A. Priestley/Introduction to Integration

H.A. Priestley: Introduction to Complex Analysis

Published $\text {1997}$, Introduction to Integration

ISBN 0-19-850123-4

Subject Matter

• Integral Calculus

Contents

Preface

Notation

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. $\text L^{\text{inc} }$ functions

12. The class $\text L$ 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 on $\R^k$

26. Fubini's Theorem and Tonelli's Theorem

27. Transformations of $\R^k$

28. The spaces $\text L^1$, $\text L^2$, and $\text L^p$

29. Fourier series: pointwise convergence

30. Fourier series: convergence reassessed

31. $\text L^2$-spaces: orthogonal sequences

32. $\text L^2$-spaces as Hilbert spaces

33. Fourier transforms

34. Integration in probability theory

Appendix I: historical remarks

Appendix II: reference

Bibliography

Notation index

Subject index