Elements of the Theory of Functions and Functional Analysis, Volume 2, Measure. The Lebesgue Integral. Hilbert Space

Cover......Page 1 S Title......Page 2 OTHER GRAYLOCK PUBLICATIONS......Page 3 Title: Elements of the Theory of Functionsand Functional Analysis, VOLUME 2, MEASURE. THE LEBESGLTE INTEGRAL. HILBERT SPACE......Page 4 LCCN 5704134......Page 5 CONTENTS......Page 6 PREFACE......Page 8 TRANSLATORS' NOTE......Page 10 §33. The measure of plane sets......Page 11 §34. Collections of sets......Page 25 §35. Measures on semi-rings. Extension of a measure on a semi-ring to the minimal ring over the semi-ring......Page 30 EXERCISES......Page 32 §36. Extension of the Jordan measure......Page 33 EXERCISES......Page 37 §37. Complete additivity. The general problem of the extension of measures......Page 38 EXERCISES......Page 40 §38. The Lebesgue extension of a measure defined on a semi-ring with unity......Page 41 EXERCISES......Page 45 §39. Extension of Lebesgue measures in the general case......Page 46 EXERCISES......Page 47 §40. Definition and fundamental properties of measurable functions......Page 48 §41. Sequences of measurable functions. Various types of convergence......Page 52 EXERCISES......Page 57 §42. The Lebesgue integral of simple functions......Page 58 EXERCISES......Page 60 §43. The general definition and fundamental properties of the Lebesgue integral......Page 61 EXERCISES......Page 65 §44. Passage to the limit under the Lebesgue integral......Page 66 EXERCISES......Page 71 §45. Comparison of the Lebesgue and Riemann integrals......Page 72 EXERCISES......Page 74 §46. Products of sets and measures......Page 75 §47. The representation of plane measure in terms of the linear measure of sections, and the geometric definition of the Lebesgue integral......Page 78 EXERCISES......Page 81 §48. Fubini's theorem......Page 82 EXERCISES......Page 85 §49. The integral as a set function......Page 87 EXERCISES......Page 88 §50. The space L2......Page 89 EXERCISES......Page 92 §51. Mean convergence. Dense subsets of L2......Page 94 EXERCISES......Page 97 §52. L2 spaces with countable bases......Page 98 EXERCISES......Page 100 §53. Orthogonal sets of functions. Orthogonalization......Page 101 EXERCISES......Page 105 §54. Fourier series over orthogonal sets. The Riesz-Fisher theorem......Page 106 EXERCISES......Page 110 §55. Isomorphism of the spaces L2 and 12......Page 111 EXERCISES......Page 112 §56. Abstract Hubert space......Page 113 EXERCISES......Page 115 §57. Subspaces. Orthogonal complements. Direct sums......Page 116 EXERCISES......Page 119 §58. Linear and bilinear functionals in Hubert space......Page 120 EXERCISES......Page 123 §59. Completely continuous seif-adjoint operators in H......Page 125 EXERCISES......Page 128 §60. Linear equations in completely continuous operators......Page 129 §61. Integral equations with symmetric kernel......Page 130 EXERCISES......Page 132 SUPPLEMENT AND CORRECTIONS TO VOLUME 1......Page 133 INDEX......Page 137