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Bichteler K. — Integration - a functional approach
Bichteler K. — Integration - a functional approach



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Название: Integration - a functional approach

Автор: Bichteler K.

Аннотация:

This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory.
From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level.
The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of S-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations - such as stochastic integration - where the set-measuring approach to integration does not work.


Язык: en

Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 193

Добавлена в каталог: 02.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Integral of Lebesgue      64
Integral, elementary      33 35
Integral, extended      25 57 65
Integral, positive elementary      35
Integral, product of elementary      123
Integrand, elementary      35
Integrand, elementary Banach space valued      77
Interpolation      99 133
Intersection      4
Inverse      4
Isometric      116 120 122
Iterated integration      124
Iterated upper integral      125
Jensen's inequality      154
JORDAN      29
Jordan mean      22 46 57
Laplace transform      14
Largely $\mathcal{E}$-uniformly continuous      71
Largely from $\overline{\mathcal{E}}$      68
Largely uniform convergence      69 72
Lattice      107
Lattice algebra      6
Lattice ring      6 7 11 14 18 23 35 37 39 40 47 54 56 65 86 89 97 99 125 134 138 141 147 149 151 171
Lattice ring, generated by functions      83
Lattice, generated by functions      83
Least upper bound      106
Lebesgue      93 97
Lebesgue, inner measure      94
Lebesgue, integrable      59 69
Lebesgue, integral      64
Lebesgue, integral, elementary      35
Lebesgue, measurable      70
Lebesgue, measure      19 49 64
Lebesgue, negligible      26 49
Less than $(\leq)$      3
Limit inferior      3
Limit superior      3
Limit, uniform      6 10
Linear functional on a real vector space      110
Linear functional, positive      114
Linear operator      105
Linear space      6
Linearly generate      19
Lipschitz function      56
Little $\ell^p$-spaces      28
Littlewood's Principles      67 68
Localizable      109
Locally compact      37
Locally integrable      149
Lorentz space      132
Lower semicontinuous      47 88
Lower, envelope, integrable      60
Majorize      19
Majorize by a mean      64 78 90 125 135
Majorize by a seminorm      29 32 46 48
Majorize by a step function      2
Majorize by an integrable function      60
Majorize, the elementary integral      22 32 46 57 64 78 90 125 135
Majorize, the integral      19 22 23 48
Maximal mean      74 90 99 102
Maximal operator of Hardy — Littlewood      155
Maximum of elements in an ordered set      106
MCT      104
Mean      32 46
Mean closure      52
Mean on a ring of functions      46
Mean, $\sigma$-finite      62
Mean, (totally) finite      62
Mean, canonical      90
Mean, continuous along arbitrary increasing sequences      91 99
Mean, convergence      45
Mean, Daniell      45 91 92
Mean, Daniell — Stone      48 56 78 92
Mean, maximal      74 90 99 102
Mean, order-continuous      48 88
Mean, solidity of a      135
Mean, strictly increasing      92
Mean-convergence      22
Mean-dense      56 74
Measurability      67 70
Measurable      67 70 97
Measurable for $(\mathcal{E},{\| \ \ \|}^\ast)$      70
Measurable function      70 97
Measurable function, simple      79
Measurable on a $\sigma$-algebra      87
Measurable set      78 97
Measurable space      90
Measurable, function on a set      70
Measurable, Lebesgue      70
Measure of finite variation      139
Measure on a ring of sets      38
Measure zero      27
Measure, $\sigma$-finite      62 129
Measure, image of a      153
Measure, inner      94
Measure, outer      64
Measure, positive, on a ring of sets      38
Measure, product of      124
Measure, signed      135
Measure, totally finite      62
Metric      4
Metric space      4 15 47 71 73 88
Minimal representation      19
Minimum of elements in an ordered set      106
Minkowski functional      115
Minkowski's inequality      101 119 120
Monotone class      89
Monotone Convergence Theorem      104
Natural number      3
Negative $(\leq 0)$      3
Negative part      108
Negative, strictly (<0)      3
Negative-homogeneity      20
Negligible      49
Norm      10 27
Norm of a linear functional      110
Norm, uniform      10
Normal homomorphism of vector lattices      148
Normed vector lattice      107
Normed vector space      27
Numerical function      4
Observable      41
Operator of strong type      133
Operator of weak type      133
Operator, linear      105
Order      106
Order bound      106
Order complete      107
Order cone      108 142
Order interval      56
Order on $\mathrm{L}^p$      106
Order preserving map      116
Order, quotient, for vector lattices      148
Order-completeness      107
Order-completeness of $\mathrm{L}^p$      109
Order-completeness of $\mathrm{L}^\infty$      110
Order-completeness of the dual of a vector lattice      116
Order-completeness, non- of $\mathcal{L}^p$      109
Order-continuous, elementary integral      37 138
Order-continuous, mean      48 88
Orthogonal      121
Orthogonal elements of a vector lattice      145
Outer measure      49 64 93
Outer regularity      65
p-integrable      103
p-mean      100
p-mean, convergence      100
p-Norm      33 100
Pairing      116
Parseval's identity      122
Partition      18
Permanence properties      23 67 79
Permanence properties of $\mathcal{L}^p$      104
Permanence properties of Baire and Borel functions      83
Permanence properties of integrable functions      52 65
Permanence properties of integrable sets      61
Permanence properties of measurability      72 76 77 97 129
Permanence properties of negligibility      49
Permanence properties of the integral      135
Permanence properties of the Riemann integral      23 29
Perpendicular      121
Piecewise continuous      163
Pointed cone      108
Polarization      12
Positive      120
Positive $(\geq 0)$      3
Positive bilinear form      120
Positive bounded linear functional      114
Positive elementary integral      18 29 35
Positive measure on a ring of sets      38
Positive part      107
Positive Radon measure      37
Positive, strictly (>0)      3
Positive-homogeneous      19
Power set      164
probability      41 153
Product of elementary integrals      123
Product of measures on rings of sets      124
Product of Radon measures      125
Product of rings of sets      124
Pseudometric      15
Pseudometric of convergence in measure      82
Quotient norm      28 106
Quotient order for vector lattices      148
Radon measure      138
Radon measure, positive      37 138
Radon measure, product of      125
Radon — Nikodym derivative      152 153
Rational number      3
Real Hilbert space      121
Real line      3
Real numbers      3
rectangle      124
Reflexive      99 114
Reflexive, relation      106
Regularity inner and outer      65
Relation, antisymmetric      106
Relation, reflexive      106
Relation, transitive      106
Relative complement      4
Riemann      1 17
Riemann integrable on (a,b]      26
Riemann, integrable      1 17 20 21 22 24 25 26 162
Riemann, integral      1 17 23 37
Riemann, lower sum      17
Riemann, squeeze      1 93 97
Riemann, upper sum      17
Riesz representation theorem      64
Riesz space      107
Riesz space of functions      6
Right translate      65
Right-continuity of a distribution function      141
Right-continuous version of a function      141
Ring of functions      6 7 11 13 14 16 46 52 57 68 71 86 90 123
Ring of sets      19 38 40 61 78 85 93 95 96 98 124 137 147
Ring of sets, generated      38
Ring, generated by functions      83
Self-adjoint      134
Self-confined      8 29;
Self-confining      8 14 16 18 56
Seminorm      2 10 27
Seminorm solid      29 32 46 48 49 54 68 87 92 102 103 107 115
Seminorm, the, of an inner product      120
Seminormed vector lattice      107
Seminormed vector space      27
Separable      28 60 88 105 122
Separating the points      12 16 73 162
Sequential closure or span      84
Sequentially closed family of functions      84
Set function      93
Set, difference of      4
Set, identified with indicator function      5
Set, integrable      61
Set, measurable      78 97
Signed elementary integral      134
Signed measure      135
Simple $\mathcal{F}$-measurable function      87
Simple function      40
Simple integrable function      61 64 97
Simple measurable function      79
Solid, seminorm      29 32 46 48 49 54 68 87 92 102 103 107 115
Solid, seminorm on an ordered vector space      107
Solid, subset of a vector lattice      145
Solidity      11 22 29 32 46 48 54 68 87 92 102 103 107 146
Solidity of a mean      135
SQUEEZE      1 2 20 93 97
Step function      38 40
Step function on the line      18
Strictly increasing      109
Strictly increasing mean      92 102
Strictly, positive or negative      3
Strong type operator      133
Subadditivity      11 19 22 27 54 120 133 165 172 173
Subadditivity, countable      33 44 46 48 66 68 72 90 94 102 103 126
Submultiplicative      105
Summable sequence in a seminormed space      28
Superadditivity      20 45 94 136 173
Superadditivity, countable      47
Support      7
Support of ${\| \ \ \|}^\bullet$ on an integrable set      78
Support of a function      61 65 125 138 156
Support, function of compact      7 26 37 47
Surjection      4
Symmetric bilinear form      120
Symmetric, difference      6 19
Tensor product of function spaces      8
Totally finite, elementary integral      62
Totally finite, mean      62
Totally finite, measure      62
Transitive relation      106
Translation invariance      60 105
Triangle inequality      2 15 27 168
Triangle inequality on a vector lattice      144
Trigonometric polynomial      31
Uniform closure      10 13 15 165
Uniform convergence on arbitrarily large sets      72
Uniform convergence, largely      69 72
Uniform distance      10
Uniform integrability      56 57
Uniform limit      6 10
Uniform norm      10
Uniform, dominated convergence      25
Uniformity      15
Uniformly continuous      15 69;
Uniformly continuous, largely      71
union      4
Union-and-down-procedure      128
Upper integral      42
Upper integral, iterated      125
Upper Riemann integral      1 19
Upper Riemann sum      17
Upper, Envelope, $\mathcal{E}$-Baire      91 131 172
Upper, envelope, integrable      60
Upper, integral, Daniell — Stone      48
Variation of an element in a vector lattice      107
Variation of an elementary integral      135
Variation, finite      135
Variation, function of finite      140
Vector lattice      107 108 123 142
Vector lattice of functions      6 128
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