Book:A.N. Kolmogorov/Elements of the Theory of Functions and Functional Analysis/Volume 2

A.N. Kolmogorov and S.V. Fomin: Elements of the Theory of Functions and Functional Analysis, Volume $\text { 2 }$

Published $\text {1961}$, Graylock Press (translated by Leo F. Boron)

Subject Matter

• Functional Analysis

Contents

Preface

Translator's Note

Chapter V: Measure Theory

33. The measure of plane sets

34. Collections of sets

35. Measures of semi-rings. Extension of a measure on a semi-ring to the minimal ring over the semi-ring

36. Extension of Jordan curve

37. Complete additivity. The general problem of the extension of measures

38. The Lebesgue extension of a measure defined on a semi-ring with unity

39. Extension of Lebesgue measures in the general case

Chapter VI: Measurable Functions

40. Definition and fundamental properties of measurable functions

41. Sequences of measurable functions. Various types of convergence

Chapter VII: The Lebesgue integral

42. The Lebesgue integral of simple functions

43. The general definition and fundamental properties of the Lebesgue integral

44. Passage to the limit under the Lebesgue integral

45. Comparison of the Lebesgue and Riemann integrals

46. Products of sets and measures

47. The representation of plane measure in terms of the linear measure of sections and the geometric definition of the Lebesgue integral

48. Fubini's theorem

49. The integral as a set function

Chapter VIII: Square Integrable Functions

50. The space $L_2$

51. Mean convergence. Dense subsets $L_2$

52. $L_2$ spaces with countable bases

53. Orthogonal sets of functions. Orthogonalization

54. Fourier series over orthogonal sets. The Riesz-Fischer theorem

55. Isomorphism of the spaces $L_2$ and $l_2$

Chapter IX: Abstract Hilbert Space. Integral Equations with Symmetric Kernel

56. Abstract Hilbert space

57. Subspaces. Orthogonal complements. Direct sums

58. Linear and bilinear functionals in Hilbert space

59. Completely continuous self adjoint-operators in $H$

60. Linear operator equations with completely continuous operators

61. Integral equations with symmetric kernel

Supplement and Corrections to Volume 1

Index

Cited by

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.)