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

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