Measure, Lebesgue Integrals, and Hilbert Space

Cover......Page 1
Title: Measure, Lebesgue Integrals, and Hilbert Space......Page 2
LCCN 61-12279......Page 3
Translator's Note......Page 6
Foreword......Page 8
Contents......Page 10
List of Symbols......Page 12
1. Measure of Plane Sets......Page 14
2. Systems of Sets......Page 32
3. Measures on Semirings. Continuation of a Measure from a Semiring to the Minimal Ring over it......Page 39
4. Continuations of Jordan Measures......Page 42
5. Countable Additivity. General Problem of Continuation of Measures......Page 48
6. Lebesgue Continuation of Measure, Defined on a Semiring with a Unit......Page 52
7. Lebesgue Continuation of Measures In the General Case......Page 58
8. Definition and Basic Properties of Measurable Functions......Page 61
9. Sequences of Measurable Functions. Different Types of Convergence.......Page 67
CHAPTER III: THE LEBESGUE INTEGRAL......Page 73
10. The Lebesgue Integral·For Simple Functions......Page 74
11. General Definition and Basic Properties of the Lebesgue Integral......Page 76
12. Limiting Processes Under the Lebesgue Integral Sign......Page 82
13. Comparison of the Lebesgue Integral and the Riemann Integral......Page 88
14. Direct Products of Systems of Sets and Measures......Page 91
15. Expressing the Plane Measure by the Integral of a Linear Measure and the Geometric Definition of the Lebesgue Integral......Page 95
16. Fubini's Theorem......Page 99
17. The Integral as a Set Function......Page 103
18. The L2 Space......Page 105
19. Mean Convergence. Sets in L2 which are Everywhere Complete.......Page 110
20. L2 Spaces with a Countable Basis......Page 113
21. Orthogonal Systems of Functions. Orthogonalisation.......Page 117
22. Fourier Series on Orthogonal Systems. Riesz-Fischer Theorem.......Page 122
23. The Isomorphism of the Spaces L2 and l2......Page 128
24. Abstract Hilbert Space......Page 131
25. Subspaces. Orthogonal Complements. Direct Sum.......Page 134
26. Linear and Bilinear Functionals in Hilbert Space......Page 139
27. Completely Continuous Self-Adjoint Operators in H......Page 142
29. Integral Equations with a Symmetric Kernel......Page 148
ADDITIONS AND CORRECTIONS TO VOLUME 1*......Page 151
SUGGESTED REFERENCES......Page 156
INDEX......Page 158