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Kurtz D.S., Swartz C.W. — Theories of Integration
Kurtz D.S., Swartz C.W. — Theories of Integration

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Название: Theories of Integration

Авторы: Kurtz D.S., Swartz C.W.

Аннотация:

Kurtz and Swartz (both New Mexico State U.) introduce a broad selection of integration theories focusing on the integrals named in the title. They present classical problems in integration theory in historical order to show how new theories were developed to solve problems that earlier ones could not handle. The detail of discussion varies from integral to integral. The four chapters are independent, and each contains 30-60 exercises to make it usable as a text in an introductory real analysis course


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 268

Добавлена в каталог: 19.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
#      74
$B(x_0, r)$      80
$C(f, \mathcal{P})$      6
$d_i$      123 244
$D_{\epsilon}(f)$      39
$E^c$      68
$E^y$      120
$E_1 \Delta E_2$      80
$E_x$      120
$f^-$      28
$f^y$      117
$f^|$      28
$f_x$      117
$I^0$      33
$J (S , \mathcal{P})$      38
$L (f \mathcal{P})$      21
$l([x_{I - 1}, x_l])$      11
$lim_i inf x_i$      90
$lim_i sup x_i$      90
$L^1 (E)$      123
$m^* (E)$      60
$M^1 ( I )$      244
$m_* (E)$      64
$m_i$      20
$m_n$      84
$m_n^* ( E)$      81
$pv \int_a^b f$      44
$pv \int_{\infty}^{\infty} f$      45
$S(f, \mathcal{D})$      141 207 224
$S(f, \mathcal{P}, \{t_i \}^n_{I = 1}$      7 11
$U(f, \mathcal{P})$      20
$v(\varphi, \mathcal{P})$      172
$Var (\varphi, [a, b])$      172
$\bar{c}(S)$      39
$\bar{D} f$      136
$\bar{S}$      38
$\bar{\int}^b_a f$      22
$\chi_I$      12
$\gamma$-fine free tagged partition      224
$\gamma$-fine tagged partition      140 156 207
$\Gamma(x)$      51
$\inf _I f$      12 141 208 225
$\inf_E f$      31 99
$\int \varphi$      97
$\int^b_a f$      12 141
$\mathbb{R}^*$      85
$\mathbb{R}^n$      80
$\mathcal{B} \mathcal{V} ([a, b])$      172
$\mathcal{B} \mathcal{V} (\mathbb{R})$      218
$\mathcal{B}(X)$      72
$\mathcal{B}(\mathbb{R})$      73
$\mathcal{B}(\mathbb{R}^n)$      84
$\mathcal{F}_{\sigma}$      73
$\mathcal{G}_{\delta}$      73
$\mathcal{H} \mathcal{L} (I)$      205
$\mathcal{M}$      68
$\mathcal{M}_I$      240
$\mathcal{M}_n$      84
$\mu (\mathcal{P})$      7 11
$\omega (f, x)$      38
$\parallel \parallel_1$      111 1237
$\Sigma$      8060
$\sigma$-algebra      69
$\underline{Df}$      136
$\underline{\int}^b_a f$      22
$\vee$      28
$\wedge$      28
a.e.      42
Abel's Test      217
Absolutely continuous      106 202
Absolutely convergent      115
Absolutely integrable      see Riemann
Absolutely integrable, integrable      see Lebesgue integrable see see
Additivity condition      33
Alexiewicz semi-norm      205
Algebra      69
Almost all      89
Almost every      89
Almost everywhere      42 89
Archimedes      2
Ball      80
Borel measure      77
Borel sets      72
Bounded      80
Bounded convergence theorem      110 188
Bounded variation      172
Brick      80
Canonical form      86
Cantor set      79
Cantor set, generalized      79
Caratheodory      64
Cauchy      6
Cauchy criterion      19 150 228
Cauchy principal value      44 45
Cauchy sequence      123
Cauchy sum      6
Cauchy — Riemann integrable      42 44
Cauchy — Riemann integrable, conditionally      46
Cauchy — Schwarz inequality      127
Change of variables      37
Characteristic function      12
Closed      80
Closure      38
Compact      80
Comparison Test      45 170
complete      123
Conditionally integrable      161 see
Converge      42 80 123
Countable additivity      60
Countably additive      73 74
Countably subadditive      61
Counting measure      74
d(F)      40
d(I,J)      64
Darboux      20
Darboux integrable      22
Darboux sum, lower      20
Darboux sum, upper      20
Dedekind's Test      179
Denjoy      135
Dense      131
Derivative, lower      136
Derivative, upper      136
Dirichlet function      14
Discrete metric      122
Distance      80
Distance from I to J      64
Distance-1 metric      122
diverge      42
Dominated Convergence Theorem      110 187 239
E + h      61
Egoroff      90
Even function      47
Extended real numbers      85
Extended real-valued function      86
Fatou      109
Fatou's lemma      109 186 239
Fischer      123
Free tagged partition      224
Free tagged subpartition      228
Fubini      117
Fubini’s Theorem      118 213 256
Fundamental Theorem of Calculus: Part I      34 134 143 190 232 see
Fundamental Theorem of Calculus: Part II      35 191 193 233
Gamma function      51
Gauge      140 156 207
Gauge integral      141
Generalized Fundamental Theorem of Calculus: Part I      148
Generalized Riemann integral      141
Henstock — Kurzweil integrable      141 156 208
Henstock — Kurzweil integrable, absolutely      147
Henstock — Kurzweil integral, indefinite      164 175 190 195
Henstock’s Lemma      163 229
Improper integral      42
Indefinite integral      see Riemann integral see see
Inner measure      64
Integrable Darboux      see Darboux integrable
Integrable Henstock — Kurzweil      see Henstock — Kurzweil integrable
Integrable Lebesgue      see Lebesgue integrable
Integrable McShane      see McShane integrable
Integrable over E, Henstock — Kurzweil      194
Integrable over E, Lebesgue      103
Integrable over E, McShane      240
Integrable over E, Riemann      31
Integrable Riemann      see Riemann integrable
Integration by parts      37 149
Integration by substitution      37
Interior      33
interval      80 207
Jordan content, outer      38
Lebesgue      56
Lebesgue integrable      103
Lebesgue integrable, absolutely      104
Lebesgue integral      97 99
Lebesgue integral, indefinite      203
Lebesgue measurable      68 83
Lebesgue measure      68 84
Lebesgue measure 0      41
LIMIT      123
Limit, inferior      90
Limit, superior      90
Linearity      15
Lipschitz condition      35
Lipschitz constant      35
Littlewood      92
Littlewood’s three principles      92
Lusin      93
M      68
Major function      136 245
Maximum      28
McShane integrable      224 254
McShane integrable, absolutely      229
McShane integral      254
Mean value theorem      48
Measurable function      86
Measurable set      64
Measure      74
mesh      7 11
Metric      122
Metric space      122
Mikusinski      113
Minimum      28
Minor function      136 245
Monotone      61
Monotone Convergence Theorem      100 104 181 184 236 238
Multiplier      171
Norm      80 122
Null set      41 68
Odd function      47
open      80
Operator      15
Oscillation      38
Outer measure      60 81
Partition      6 11 154 155 207
Perron      135
Perron integrable      136
Positivity      15 16 146 228
Probkme de la mesure des ensembles      60
Probkme d’intkgration      56
refinement      22
Regular inner      126
Regular outer      77
Riemann      7
Riemann integrable      12
Riemann integrable, absolutely      46
Riemann integral, indefinite      35 204
Riemann integral, lower      22
Riemann integral, upper      22
Riemann sum      7 141 207 224
Riemann — Lebesgue lemma      131
Riesz      123
Riesz — Fischer theorem      123 244
Sampling point      7
Semi-metric      122
Semi-norm      122
sgm      36
Signum function      36
Simple function      59 86
Step function      27 94
Straddle Lemma      138
Subpartit ion      163
Symmetric difference      80
Tag      139 155 207 224
Tagged partition      139 155 207
Tagged partition, free      224
Tagged subpartition      163
Tagged subpartition, free      228
Tchebyshev      102
Tchebyshev’s inequality      102
Test set      68
Tonelli      119
Tonelli’s Theorem      119 213 257
Translation      61
Translation invariant      61
v(I)      80
Variation      172
Variation, negligible      196
Vector lattice      29
Vector space      29
Vitali cover      198
Vitali Covering Lemma      199
Volume      80 207
x-section      120
y-section      120
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