Book:Marshall E. Munroe/Introduction to Measure and Integration

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Marshall E. Munroe: Introduction to Measure and Integration

Published $1953$.

Subject Matter


Chapter I. Point Set Theory
1. Sets and functions
2. Algebra of sets
3. Cardinal numbers
4. Metric spaces
5. Limits and continuity
6. Function spaces
7. Linear spaces
8. Additive classes and Borel sets
9. Category
Chapter II. Measure — General Theory
10. Additive set functions
11. Outer measures
12. Regular outer measures
13. Metric outer measures
Chapter III. Measure — Specific Examples
14. Lebesgue-Stieltjes measures
*15. Probability
*16. Hausdorff measures
*17. Haar measure
18. Non-measurable sets
Chapter IV. Measurable Functions
19. Definitions and basic properties
20. Operations on measurable functions
21. Approximation theorems
*22. Stochastic variables
Chapter V. Integration
23. The integral of a simple function
24. Integrable functions
25. Elementary properties of the integral
26. Additivity of the integral
27. Absolute continuity
28. Fubini's theorem
*29. Expectation of a stochastic variable
Chapter VI. Convergence Theorems
30. Uniform and almost everywhere convergence
31. Convergence in measure
32. Mean convergence
33. The Hölder and Minkowski inequalities
34. The $L_p$ spaces
*35. Linear functionals on Banach spaces
*36. Orthogonal expansions in Hilbert space
*37. The mean ergodic theorem
Chapter VII. Differentiation
38. Summary of the problem
39. Vitali coverings
40. Differentiation of additive set functions
41. The Lebesgue decomposition
*42. Metric density and approximate continuity
*43. Differentiation with respect to nets

Index of Postulates
Index of Symbols