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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
$(\mathcal{E},{\| \ \ \|}^\ast)$-measurable      70 79 89
$\delta$-: the prefix $\delta$-      35
$\delta$-continuity      35 38
$\delta$-ring      61
$\mathcal{E}$-Baire function      84 91 102 104
$\mathcal{E}$-dominated $\mathcal{E}$-Baire functions      147
$\mathcal{E}$-uniformity      15 70
$\mathcal{E}$-uniformly continuous      15 70
$\mu$-integrable      52
$\sigma$-: the prefix $\sigma$-      35
$\sigma$-additivity      35 38
$\sigma$-additivity of a measure on a ring of sets      38
$\sigma$-additivity of an elementary integral      36
$\sigma$-additivity of Radon measures      37
$\sigma$-algebra      62 80 85 86 87 90 97
$\sigma$-algebra of $(\mathcal{E},{\| \ \ \|}^\ast)$-measurable sets      79
$\sigma$-algebra, Baire      88
$\sigma$-algebra, Borel      88
$\sigma$-algebra, function measurable on a      87
$\sigma$-algebra, generated by a set of functions      87 89
$\sigma$-compact      47
$\sigma$-continuity      35 38
$\sigma$-finite      62 81 109 110
$\sigma$-finite, carrier      73
$\sigma$-finite, elementary integral      62 129
$\sigma$-finite, family of functions      85
$\sigma$-finite, mean      62 117
$\sigma$-finite, measure      62 149
$\sigma$-finite, set      62
$\sigma$-ring      85
(M)      46 102
a.e.      49
Absolute continuity in a vector lattice      146
Absolute continuity of $\sigma$-additive integrals      147
Absolute continuity of a measure with density      149
Absolute continuity of means      75
Absolute value in a vector lattice      107
Absolute-homogeneity      11 22 27 46 102 120
Absolutely summable sequence in a seminormed space      28
Adequate cover      76
Aesthetic      15 27 95
Algebra of functions      6
Algebra of sets      38 41 62
Algebra, generated by functions      83
Almost everywhere      49
Almost everywhere, function, defined      50
Ambient set      5 35 38 42
Antilinear      120
Antisymmetric relation      106
arctan metric      4 73 162
atomic      148
Axiom of Choice      66
Baire, $\sigma$-algebra      88
Baire, class      84
Baire, function      85 88
Baire, measurable      88 131
Baire, set      85
Banach lattice      107 116
Banach space      27 29 106
Banach space, classical      99
Band in a vector lattice      145 146
Basis of a Hilbert space      121
Bessel's inequality      121
Bigger than $(\geq)$      3 106
Bijection      4
Bilinear      120
Bochner integrable      77
Borel, $\sigma$-algebra      88
Borel, function      88
Borel, measurable      88
Borel, set      88
Bounded linear functional      110
Bourbaki      42
C*-algebra      41
C-space      122
Canonical mean      90
Canonical representation of an open set      64
Caratheodory      93 96 97
Caratheodory, cut condition      96
Carrier      7 8 34 49
Carrier, function of $\sigma$-finite      73 84
Carrier, functions of finite      47 89
Cauchy sequence      27
Cauchy sequence in measure      82
Cauchy — Schwarz inequality      120
Chebyshev's inequality      47 132 156
Chopped function      6
Chopping      6 57 62
Closed under chopping      6 13 50 56 62 84 103
Closed under composition with a Baire function      89
Closed under composition with continuous functions      13 26 56 72 85
Closed under limits of sequences      72 83
Closure      28
Closure in mean      52
Closure sequential      84
Closure under algebraic operations      82
Closure under limits of sequences      83
Closure uniform      10 13 15 165
Compactness      4
Complement      5
Complement, relative      4
Complete in measure      82
Complete metric space      73
Complete normed space      27
Complete seminormed space      27
Completeness of $\mathcal{F}^p$      103
Completeness of $\mathcal{F}^\ast$      50
Completeness of $\mathcal{L}^0$      82
Completeness of $\mathcal{L}^1$      52
Completeness of $\mathcal{L}^1_E$      77
Completeness of $\mathcal{L}^1_\mathbb{C}$      76
Completeness of $\mathcal{L}^p$      104
Completeness of $\mathcal{L}^\infty$      105
Completeness of $\mathcal{L}^\natural$      28
Completeness of $\mathrm{L}^p$      109
Completeness of $\overline{\mathbb{R}}$      4
Completeness of a Hilbert space      121
Completeness of a quotient      28
Completeness of a seminormed space      27
Completeness of some spaces      28
Completeness of the dual      111
Completeness, order, of $\mathrm{L}^p$      109
Completeness, order, of a vector lattice      107
Complex number      3
composition      7 13 26 56 72 89
Conditional expectation      153
Cone      108
Cone, convex      108
Cone, order      108
Cone, pointed      108
Confine(d)      8 14 16 18 25 29 37 56
Conjugate exponents      100 116 120 161 172
Continuity along arbitrary increasing sequences      47 91 92 102 129
Continuity along decreasing sequences      47
Continuity along increasing sequences of $\mathcal{E}^\uparrow$      43
Continuity along increasingly directed subsets of $\mathcal{E}^\Uparrow$      48
Continuity functions are a.e. defined      50
Continuity of a map      4
Continuity sets are functions      5
Contractive      27
Convention about $\pm \infty$      4
Convention about the set [statement]      5
Convergence in mean      45
Convergence in measure      56 81
Convergence in p-mean      100
Convergence, dominated      55
Convergence, dominated theorem      32 55 104
Convergence, dominated uniform      25
Convex, cone      108
Convex, function      171
Convex, set      108 114
Convolution      131
Countable subadditivity      91 103
Countable, subadditivity      66 72 94 102 103 126
Countable, superadditivity      47
Counting measure      42
Cut Condition of Caratheodory      96
Daniell      2
Daniell lower integral      45
Daniell mean for the product measure      128
Daniell — Stone mean      48 56 78 92
Daniell — Stone upper integral      48
Daniell's up-down procedure      42
Daniell, mean      2 24 32 45 59 70 78 91 92
Daniell, upper integral      42
DCT: the Dominated Convergence Theorem      32 55 104 118
Decreasingly directed      56 106
Dense      28
Dense, uniformly      8 14
density      132 149 152
Difference of sets      4
Difference, symmetric      6 19
Dirac measure      148 171
Directed decreasingly      56 106
Directed increasingly      8 37 56 106
Disjoint $\sigma$-additive integrals      147
Disjoint elements of a vector lattice      145
Disjoint, measure from Lebesgue measure      155
Distance uniform      10
Distribution function      41 140 154
Distribution function, Right-continuity of a      141
Dominated Convergence Theorem      55 104
Dominated uniform convergence      25
Double dual      114
Dual of a seminormed space      110
Elementary Banach space valued integrand      77
Elementary complex-valued      76
Elementary complex-valued integrand      76
Elementary function      35 46
Elementary integral      35 97
Elementary integral, $\sigma$-finite      62 129
Elementary integral, order-continuous      37 138
Elementary integral, positive      29
Elementary integral, signed      134
Elementary integral, totally finite      62
Elementary integrand      35
Elementary Lebesgue integral      35
Elementary sets      94
Elementary, integrals, product of      123
Elementary, positive integral      35
Envelope, $\mathcal{E}$-Baire      91 131 172
Envelope, integral      47
Envelope, lower $\mathcal{E}$-Baire      91 131 172
Envelope, Riemann integral      1
Envelope, Riemann sum      17
Envelope, upper $\mathcal{E}$-Baire      91 131 172
Essential supremum norm      100
Euclidean distance      165
Euclidean norm      120
Euclidean Space $\mathbb{R}^n$      121
Event      41
Exceed      3 106
Expectation      41
Exponents conjugate      100 116 120 161 172
Extended integral      25 57 65
Extended reals      4
Extension by continuity      2 78 90
Extension by linearity      19 38 41 124 138 139
Extension of a linear functional      112
Extension of a measure      2
Extension of an integral      32 35 144
Extension under a seminorm      29 78 110
Finite in mean      50
Finite mean      62
Finite variation      135
Finite variation, measure of      139
Forest      7 142
Fourier, coefficients      99
Fourier, series      31
Fourier, transform      29 77
Fubini's theorem      52 124 128
Function      50
Function of finite variation      140
Function, $\mathcal{E}$-Baire      84 91 102 104
Function, $\mu$-integrable      52
Function, Baire      88
Function, Borel      88
Function, chopped      6
Function, defined almost everywhere      50
Function, elementary      35
Function, integrable      52 98
Function, integrable for product measure      128
Function, Lebesgue integrable      69
Function, lower semicontinuous      47 88
Function, meaning a.e. defined function      50
Function, measurable      70 97
Function, measurable on a $\sigma$-algebra      87 154
Function, measurable on a set      70
Function, numerical      4
Function, right-continuous version of a      141
Function, simple      40
Function, simple $\mathcal{F}$-measurable      87
Functional      22 27 46
Functional, bounded linear      110
Functional, extension of a linear      112
Fundamental theorem of calculus      154
Gauge      25 112
Generator of a $\sigma$-algebra      87
Generator of a Hilbert space      122
Gram — Schmid      122
Greatest lower bound      106
Hardy — Littlewood maximal operator      132 155
Hausdorff      37
Hausdorff, completion      16
Hausdorff, maximal principle      113
Hilbert space (real)      121
Hoelder's inequality      99 100 101 116 119 161 164 171
Homeomorphism      4
Homomorphism of vector lattices      148
Ideal in a vector lattice      145
Idempotent      5
Identity, Parseval's      122
Image of a measure      153
Increasing, nature of $\int^\ast$      43
Increasing, nature of $\int^\natural$      19
Increasingly directed      8 37 56 106
Indicator function      5 61 78
Inequality, Bessel's      121
Inequality, Cauchy — Schwarz      120
Inequality, Chebyshev's      47 132 156
Inequality, Hoelder's      99 101 116 119 161 164 171
Inequality, Jensen's      154
Inequality, Minkowski's      101 119
Inequality, of Hoelder      100
Inequality, of Minkowski      120
Inequality, triangle      2 27 144 168
Infinity norm      100
Injection      4 114 116
Inner measure      94
Inner product      120
Inner product on $L^2$      121
Inner regularity      65
INTEGER      3
Integrable      52 98
Integrable function      52 98
Integrable function, simple      61 64
Integrable set      61
Integrable, Bochner      77
Integrable, locally      149
Integrable, uniformly      57
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