Measure, Lebesgue Integrals, and Hilbert Space

Translator's Note......Page 5
Foreword......Page 7
Contents......Page 9
List of Symbols......Page 11
1. Measure Of Plane Sets......Page 13
2. Systems Of Sets......Page 31
3. Measures On Semirings. Continuation Of A Measure From A Semiring To The Minimal Ring Over It......Page 38
4. Continuations Of Jordan Measures......Page 41
5. Countable Additivity. General Problem Of Continuation Of Measures......Page 47
6. Lebesgue Continuation Of Measure, Defined On A Semiring With A Unit......Page 51
7. Lebesgue Continuation Of Measures In The General Case......Page 57
8. Definition And Basic Properties Of Measurable Functions......Page 60
9. Sequences Of Measurable Functions. Different Types Of Convergence......Page 66
10. The Lebesgue Integral For Simple Functions......Page 73
11. General Definition And Basic Properties Of The Lebesgue Integral......Page 75
12. Limiting Processes Under The Lebesgue Integral Sign......Page 81
13. Comparison Of The Lebesgue Integral And The Riemann Integral......Page 87
14. Direct Products Of Systems Of Sets And Measures......Page 90
15. Expressing The Plane Measure By The Integral Of A Linear Measure And The Geometric Definition Of The Lebesgue Integral......Page 94
16. Fubini's Theorem......Page 98
17. The Integral As A Set Function......Page 102
18. The L_2 Space......Page 104
19. Mean Convergence. Sets In L_2 Which Are Everywhere Complete......Page 109
20, L_2 Spaces With A Countable Basis......Page 112
21. Orthogonal Systems Of Functions. Orthogonalisation......Page 116
22. Fourier Series On Orthogonal Systems. Riesz-fischer Theorem......Page 121
23. The Isomorphism Of The Spaces L_2 And L_2......Page 127
24. Abstract Hilbert Space......Page 130
25. Subspaces. Orthogonal Complements. Direct Sum......Page 133
26. Linear And Bilinear Functionals In Hilbert Space......Page 138
27. Completely Continous Self-adjoint Operators In H......Page 141
28. Linear Equations With Completely Continuous Operators......Page 146
29. Integral Equations With A Symmetric Kernel......Page 147
Additions And Corrections Tovolume I*......Page 150
Subject Index......Page 155