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

Subject Matter

 * Measure Theory

Contents

 * 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 &mdash; General Theory
 * 10. Additive set functions
 * 11. Outer measures
 * 12. Regular outer measures
 * 13. Metric outer measures


 * Chapter III. Measure &mdash; 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


 * Bibliography
 * Index of Postulates
 * Index of Symbols
 * Index