Kolmogorov A.N., Fomin S.V. — Measure, Lebesgue Integrals, and Hilbert Space :: Электронная библиотека попечительского совета мехмата МГУ

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Kolmogorov A.N., Fomin S.V. — Measure, Lebesgue Integrals, and Hilbert Space

Kolmogorov A.N., Fomin S.V. — Measure, Lebesgue Integrals, and Hilbert Space

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Название: Measure, Lebesgue Integrals, and Hilbert Space

Авторы: Kolmogorov A.N., Fomin S.V.

Аннотация:

Measure Theory.
Measure of Plane Sets.
Systems of Sets.
Measures on Semirings. Continuation of a Measure from a Semiring to the Minimal Ring over it.
Continuations of Jordan Measures.
Countable Additivity. General Problem of Continuation of Measures.
Lebesgue Continuation of Measure, Defined on a Semiring with a Unit.
Lebesgue Continuation of Measures in the General Case.
Measurable Functions.
Definition and Basic Properties of Measurable Functions.
Sequences of Measurable Functions. Different Types of Convergence.
The Lebesgue Integral.
The Lebesgue Integral for Simple Functions.
General Definition and Basic Properties of the Lebesgue Integral.
Limiting Processes Under the Lebesgue Integral Sign.
Comparison of the Lebesgue Integral and the Riemann Integral.
Direct Products of Systems of Sets and Measures.
Expressing the Plane Measure by the Integral of a.
Linear Measure and the Geometric Definition of the Lebesgue Integral.
Fubini's Theorem.
The Integral as a Set Function.
Functions Which Are Square Integrahie.
The L2 Space.
Mean Convergence. Sets in L2 which are Everywhere Complete.
L2 Spaces with a Countable Basis.
Orthogonal Systems of Functions. Orthogonalisation.
Fourier Series on Orthogonal Systems.
Riesz-Fischer Theorem.
The Isomorphism of the Spaces L2 and l2.
The Abstract Hilbert Space. Integral Equations with a Symmetric Kernel.
Abstract Hilbert Space.
Subspaces. Orthogonal Complements. Direct Sum.
Linear and Bilinear Functionals in Hilbert Space.
Completely Continous Self-Adjoint Operators in H.
Linear Equations with Completely Continuous Operators.
Integral Equations with a Symmetric Kernel.

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Рубрика: Математика/

Серия: Посвящена 110-летию со дня рождения Колмогорова Андрея Николаевича

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1961

Количество страниц: 147

Добавлена в каталог: 12.04.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$\delta$ -ring      24
$\sigma$ -additivity      15 35
$\sigma$ -ring      19
Additive measure      2
Additivity- $\sigma$       15 35
Algebra of sets      20
Algebra, Borel      25
Basis, countable      101
Bessel’s inequality      111
Bilinear functional      128
Bilinear functional, (symmetric)      128
Borel algebra (B-algebra)      25
Borel closure of a system of sets      25
Borel measurable functions      48
Borel set      24
Cauchy — Bunjakovskii (Schwarz) inequality      95
Closed system      112
Closure, linear      104
Complete system of functions      105
Completely continuous operator      130
Completeness of the $L_{2}$ space      95
Continuation of a measure      28
Continuation of a measure, (Jordan)      29
Continuation of a measure, (Lebesgue)      43
Continuity of measure      15
Convergence of a sequence of functions in the mean (square)      97 98 100
Convergence, almost everywhere      54 100
Convergence, uniform with the exception of a set of small measure      55
Countable additivity of measure      15 35 43 101
Countable basis      101
Countable dimension      100
Dependence, linear      104
Dimension, countable      100
Dirichlet function      77
Eigenfunctions      132 133
Eigenvectors      132 133
Equivalent      52 94
Fatou’s theorem      73
Fourier coefficients      110
Fourier series      111
Fubini’s Theorem      87
Functional, bilinear      128
Functional, quadratic      129
Functions, complete system of      105
Functions, simple      50
General form of a linear functional      126
Geometric definition of a Lebesgue integral      82
Heisenberg’s matrix mechanics      117
Hilbert space, $L_{2}$       92
Hilbert space, abstract      118
Independence, linear      104
Inner measure      6 31 40
Integrable      61 63
Integral equation with symmetric kernel      135
Integral operator      123
Integral, as a set function      90
Isomorphic spaces      115
Isomorphism of $L_{2}$ and $l_{2}$ spaces      115
Lebesgue integral      63 76 106
Lebesgue measurable set      7 40 45
Lebesgue non-measurable set      18
Lebesgue — Stieltjes measure      17
Limiting process under Lebesgue integral sign      69

Linear closure      105
Linear manifold      121
Linear self-adjoint operator      129
Linearly independent closed system of functions      112
Linearly independent complete system of functions      105
Linearly independent normalised system of functions      106
Linearly independent orthogonal system of functions      106 107
Linearly independent system of functions      104
Luzin’s theorem      53
Manifold, linear      121
Measurable function      48 49
Measure of a set      26
Measure, absolutely convergent      26
Measure, complete      46
Measure, continuity of      15
Measure, discrete      18
Measure, inner      6 31 40
Measure, Jordan      30
Measure, Lebesgue      7 40
Measure, Lebesgue — Stieltjes      17
Measure, singular      17
Measure, with countable basis      101
Minimal ring      21
Mutually orthogonal      106
Nonseparable Hilbert space      119
Normalised system      106
Orthogonal complement to subspace      123
Orthogonal, mutually      106
Orthogonalisation of a system of vectors (functions)      109
Orthonormal system      106
Outer measure      6 31 40
Parseval’s equation      112
Plane measure and linear measure      82
Power, nth of a set      78
Product of measures      82
Product, vector      94
Quadratic functional      129
Radon’s theorem      90
Riemann integral (comparison)      75
Riesz — Fischer theorem      113
Ring of sets      19
Ring, its unit      19
Ring, minimal      21 24
Scalar product      94
Schr $\ddot{o}$ dinger’s wave mechanics      117
Schwarz inequality      95
Self-adjoint linear operator      129
Semiring      21
Set, algebra of      20
Set, B-closure      25
Set, function      90
Set, measurable      40
Set, of unique measure      44
Simple functions      50
Square integrable      93
Subspace of Hilbert space      121
Summable      93
Tchebishev’s inequality      68
Uniqueness, set of      38
Unit of system of sets      19 39
Unit sphere, weak compactness      127
Vector product      94
Weak compactness of unit sphere      127

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