## Measure, Lebesgue Integrals and Hilbert Space – Kolmogorov and Fomin

In this post, we will see the book Measure, Lebesgue Integrals and Hilbert Space by A. N. Kolmogorov and S. V. Fomin. This publication is the second book. of the “Elements of the Theory of Functions and Functional Analysis,” the first book of which (“Metric and Normed Spaces”) appeared in 1954. In this second book the main role is played by measure theory and the Lebesgue integral. These concepts, in particular the concept of measure, are discussed with a sufficient degree of generality; however, for greater clarity we start with the concept of a Lebesgue measure for plane sets. If the reader so desires he can, having read §1, proceed immediately to Chapter II and then to the Lebesgue integral, taking as the measure, with respect to which the integral is being taken, the usual Lebesgue measure on the line or on the plane.

The theory of measure and of the Lebesgue integral as set forth in this book is based on lectures by A.N. Kolmogorov given by him repeatedly in the Mechanics-Mathematics Faculty of the Moscow State University. The final preparation of the text for publication was carried out by S. V. Fomin.

The two books correspond to the program of the course “Analysis III” which was given for the mathematics students by A. N. Kolmogorov. At the end of this volume the reader will find corrections pertaining to the text of the first volume.

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## Contents

Translator’s  Note. v

Foreword  vii

List of Symbols. xi

CHAPTER I Measure Theory

1. Measure of Plane Sets. 1

2. Systems of Sets. 19

3. Measures on Semirings. Continuation of a Measure from a Semiring to the Minimal Ring over it. 26

4. Continuations of Jordan Measures. 29

5. Countable Additivity. General Problem of Continuation of Measures. 35

6. Lebesgue Continuation of Measure, Defined on a Semiring with a Unit. 39

7. Lebesgue Continuation of Measures in the General Case. 45

CHAPTER II Measurable Functions

8. Definition and Basic Properties of Measurable Functions. 48

9. Sequences of Measurable Functions. Different Types of Convergence. 54

CHAPTER III

The Lebesgue Integral

10. The Lebesgue Integral for Simple Functions. 61

11. General Definition and Basic Properties of the Lebesgue Integral. 63

12. Limiting Processes Under the Lebesgue Integral Sign. 69

13. Comparison of the Lebesgue Integral and the Riemann Integral. 75

14. Direct Products of Systems of Sets and Measures. 78

15. Expressing the Plane Measure by the Integral of a Linear Measure and the Geometric Definition of the Lebesgue Integral. 82

16. Fubini’s Theorem. 86

17. The Integral as a Set Function. 90

CHAPTER IV

Functions Which Are Square Integrable

18. The L_2 Space. 92

19. Mean Convergence. Sets in L2 which are Everywhere Complete. 97

20. L_2 Spaces with a Countable Basis. 100

21. Orthogonal Systems of Functions. Orthogonalisation. 104

22. Fourier Series on Orthogonal Systems. Riesz-Fischer Theorem.109

23. The Isomorphism of the Spaces L_2 and l_2. 115

CHAPTER V

The Abstract Hilbert Space. Integral Equations with a Symmetric Kernel

24. Abstract Hilbert Space. 118

25. Subspaces. Orthogonal Complements. Direct Sum. 121

26. Linear and Bilinear Functionals in Hilbert Space. 126

27. Completely Continuous Self-Adjoint Operators in H. 129

28. Linear Equations with Completely Continuous Operators. 134

29. Integral Equations with a Symmetric Kernel. 135

Additions and Corrections to Volume I 138

Subject Index 143 