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Saks S. — Theory of the integral,

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Название: Theory of the integral,

Автор: Saks S.

Аннотация:

Covering all the standard topics, the author begins with a discussion of the integral in an abstract space, additive classes of sets, measurable functions, and integration of sequences of functions. Succeeding chapters cover Carath?odory measure; functions of bounded variation and the Lebesgue-Stieltjes integral; the derivation of additive functions of a set and of an interval; more.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-d edition

Год издания: 1937

Количество страниц: 354

Добавлена в каталог: 25.11.2012

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

Предметный указатель

Almost      16
angle      310
Area, Elementary, of a polyhedron      165
Area, of a continuous surface      165
Area, of an interval      59
Boundary of a set      40
Centre, of a dosed sphere      40
Centre, of an open sphere      40
Chain of points      122
Classes of sets, Additive      7
Classes of sets, Additive in the weak sense      7
Classes of sets, Complete with respect to a measure      86
Classes of sets, Completely additive      7
Closure of a set      40
Coefficients of an approximate differential      300
Complement of a set      6
Condition, ( $l^{+}_{\alpha}$ )      192
Condition, ( $l^{-}_{\alpha}$ )      192
Condition, ( $l_{\alpha}$ )      192
Condition, ( $T_{1}$ )      277
Condition, ( $T_{2}$ )      277
Condition, (D)      290
Condition, (S)      282
Condition, Luain's, (N)      224
Condition, of Lipschitz for functions of a real variable on a Bet      264
Condition, of Lipschitz for functions of two variables on a set      304
Condition, of Lipschitz for functions of two variables on an interval      169
Contingent      263
Coordinates of a point      56
Covering in the sense of Vitali      109
Cubes      57
Cubes, Half open      57
Cubes, Open      57
CURVES      121
Curves, rectifiable      122
Curves, rectifiable on an interval      122
Decomposition, Jordan, of a function of a set      11
Decomposition, Jordan, of a function of an interval      62
Decomposition, Lebesgue, of a function of a set      35
Decomposition, Lebesgue, of a function of an interval      120
Densities, Outer upper and lower      128
Densities, Right-hand and left-hand      215
Densities, Upper and lower      129
Derivates, (Ordinary) upper and lower      106
Derivates, Approximate bilateral, upper and lower      220
Derivates, Approximate right-hand and left-hand, upper and lower      220
Derivates, Bilateral      108
Derivates, Dini      108
Derivates, Extreme      106
Derivates, General upper and lower      100
Derivates, Intermediate      108
Derivates, Opposite      269
Derivates, relative to a set      108
Derivates, Strong upper and lower      106
Derivates, Symmetrical, upper and lower      149
Derivates, Unilateral (right- and left-hand)      108
Derivates, Unique      106
Derivates, with respect to a function      108
Derivates, with respect to a measure and a sequence of nets in a metrical apace      154
Derivates, with respect to a sequence of nets of intervals      188
Derivative, approximate      220
Derivative, General      106
Derivative, of a complex function      196
Derivative, ordinary      106
Derivative, relative to a set      108
Derivative, Roussel      210
Derivative, strong      106
Derivative, Unilateral      108
Derivative, with respect to a function      109
Derivative, with respect to a measure and a sequence of nets in a metrical space      154
Derivative, with respect to a sequence of nets of intervals      188
Diameter      40
Difference of sets      5
Differential, Approximate      300
Differential, Extreme      309
Differential, total      300
Differential, Upper and lower      309
Direction of a half-line      262
Distance, of a point and a set      40
Distance, of points      39
Distance, of points in an Euclidean space      56
Distance, of points in the torus space      157
Distance, of sets      40
Element of a set      4
Expressions of de Geocze      171
Faces of a polyhedron      164
Faces of an interval      57
Families of sets      see "Clashes of sets"
figure      58
Figure, Elementary      58
Function (of a point)      see also "Functions (of an interval)" "Functions
Function (of a point), Approximately continuous      132
Function (of a point), Characteristic, of sets      6
Function (of a point), Continuous      42
Function (of a point), Cylindrical      159
Function (of a point), Equimeasurable      143
Function (of a point), Finite      6
Function (of a point), Integrable      20
Function (of a point), integrable in the Lebesgue sense      65
Function (of a point), integrable in the Lebesgue — Stieltjes sense      65
Function (of a point), Measurable      12
Function (of a point), Simple      7
Function (of a point), Summable      65
Function (of a point), Upper and lower semi-continuous      42
Function of singularities, of a function of a set      35
Function of singularities, of a function of an interval      120
Functions (of a complex variable), Complex      195
Functions (of a complex variable), Derivable      196
Functions (of a complex variable), Holomorphic      195
Functions (of a real variable)      96 see "Functions
Functions (of a real variable), $ACG_{*}$       231
Functions (of a real variable), $AC_{*}$       231
Functions (of a real variable), $VBG_{*}$       228
Functions (of a real variable), $VB_{*}$       228
Functions (of a real variable), $\mathscr{D}$ -integrable      241
Functions (of a real variable), $\mathscr{D}_{*}$ -integrable      241
Functions (of a real variable), $\mathscr{P}$ -integrable      201
Functions (of a real variable), Absolutely continuous      96
Functions (of a real variable), absolutely continuous (in the wide sense)      223
Functions (of a real variable), absolutely continuous in the restricted sense      231
Functions (of a real variable), AC      223
Functions (of a real variable), ACG      223
Functions (of a real variable), Approximately derivable      220
Functions (of a real variable), continuous in the sense of Darboux      272
Functions (of a real variable), generalized absolutely continuous (in the wide sense)      223
Functions (of a real variable), generalized absolutely continuous in the restricted sense      231
Functions (of a real variable), Inner and outer, of a superposition      286
Functions (of a real variable), integrable in the sense of Newton      156
Functions (of a real variable), integrable in the sense of Perron      201
Functions (of a real variable), Inverse      286
Functions (of a real variable), of bounded variation      96
Functions (of a real variable), of bounded variation (in the wide sense)      221
Functions (of a real variable), of generalized bounded variation (in the wide sense)      221
Functions (of a real variable), of generalized bounded variation in the restricted sense      228
Functions (of a real variable), of hounded variation in the restricted sense      228
Functions (of a real variable), Regular      97
Functions (of a real variable), Saltus      97
Functions (of a real variable), Singular      96
Functions (of a real variable), VB      221
Functions (of a real variable), VBG      221
Functions (of a set), Absolutely continuous      30 66
Functions (of a set), Additive      8
Functions (of a set), Monotone      8
Functions (of a set), Non-decreasing and non-increasing      5
Functions (of a set), Singular      30 66
Functions (of an interval)      59 see
Functions (of an interval) of bounded variation      61
Functions (of an interval), Absolutely continuous      93
Functions (of an interval), Additive      61
Functions (of an interval), Continuous      59

Functions (of an interval), continuous and discontinuous at a hyperplane      60
Functions (of an interval), Major and minor      191
Functions (of an interval), Major and minor, with respect to a function      207
Functions (of an interval), Monotone non-decreasing and non -increasing      61
Functions (of an interval), Singular      93
Functions (of two real variables), Absolutely continuous      169
Functions (of two real variables), Approximately different]able      300
Functions (of two real variables), Differentiable      300
Functions (of two real variables), of bounded variation      169
Functions (of two real variables), Totally differentiate      300
Graph, of a curve      121
Graph, of a function      88
Half-plane      263 264
Half-space      263 306
Half-tangent, Intermediate      262
Hyperplane      57
Hyperplane, Extreme tangent      263
Hyperplane, Intermediate tangent      263
Hyperplane, orthogonal to an axis      57
Hyperplane, tangent      263
Hyperplane, Unique tangent      263
Increment of a function      96
Integrals      241
Integrals, $\mathscr{D_{*}}$ -      241
Integrals, $\mathscr{L}$ -      241
Integrals, $\mathscr{PS}$ -      207
Integrals, $\mathscr{P}$ -      201
Integrals, $\mathscr{S}$ -      244
Integrals, Compatible      254
Integrals, definite      19 20 46 254
Integrals, Definite and indefinite Burkill      166
Integrals, Denjoy — Khintchine      241
Integrals, Denjoy — Perron      241
Integrals, Denjoy, in the restricted sense      241
Integrals, Denjoy, in the wide House      241
Integrals, indefinite      29 254
Integrals, Lebesgue      65
Integrals, Lebesgue — Stieltjes      65
Integrals, Mean value      178
Integrals, Perron      201
Integrals, Perron — Stieltjes      207
Integrals, Upper and lower Burkill      166
Interior of a set      40
Intervals      57
Intervals, closed      57
Intervals, Degenerate      57
Intervals, Half-open      57
Intervals, in the torus space      157
Intervals, open      57
Length of a curve      122
Length of a set      54
Length of an interval      59
Length of arc of a curve      122
Length, Outer, of a set      54
Limit, approximate      219
Limit, Approximate bilateral and unilateral      220
Limit, Approximate extreme      219
Limit, Approximate, upper and lower      218
Limit, of a sequence of points      39
Limit, of a sequence of seta      5
Limit, Upper and lower, of a sequence of Bets      5
Lines, straight      56 57
Maximum, of a function at a point      42
Maximum, Strict      261
Measure      16
Measure, function      117
Measure, in the torus space      158
Measure, Lebesgue      65
Measure, of a Bet      46
Measure, Outer Caratheodory      43
Measure, Outer Lebesgue      65
Measure, Outer linear, of a, set on a line      297
Measure, Outer, in the torus space      157
Measure, Outer, of a set      46
Measure, Regular outer      50
Mesh of a net      153
Minimum, of a function at a point      42
Minimum, strict      261
Neighbourhood of a point      40
Net, in a metrical spate      153
Net, Normal, of intervals      188
Net, of closed intervals      57
Net, of half open intervals      57
Nucleus of a determining system      47
Number, Characteristic, of a family of sets      40
Operation (A)      47
Ordinate-set of a function      88
Oscillation, of a function of a point at a point      42
Oscillation, of a function of an interval at a set      60
Oscillation, of a function of an interval on a set      60
Parameter, of a curve      121
Parameter, of regularity of a set      106
Part, Absolutely continuous, of a function of an interval      120
Part, Common, of sets      5
Part, Non-negative and non-positive, of a function      13
Part, Real and imaginary, of a complex function      195
Plane      56 57 see
Plane, Whole      264
Points, internal      40
Points, isolated      40
Points, isolated on the right or the left      215
Points, of a curve      121
Points, of a space      6
Points, of accumulation      40
Points, of density      129
Points, of dispersion      128
Points, of linear density in the direction nf an axis      298
Points, of outer density      128
Points, Right- and left-hand, of accumulation      215
Points, Unilateral (right- and left-hand), of outer density      215
Polyhedron      164
Portion of a set      41
Primitive of Newton      186
Product of sets      5
Product of sets, Cartesian      82
Product of sets, Combinatory      82
Radius, of a closed sphere      40
Radius, of a neighbourhood      40
Radius, of an open sphere      40
Relations, Denjoy      269
Sequences, Ascending and descending, of sets      5
Sequences, Binary, of nets of intervals      191
Sequences, Convergent, of points      40
Sequences, Convergent, of sets      5
Sequences, Monotone, of sets      5
Sequences, Non-decreasing and non-increasing, of sets      5
Sequences, Normal, of nets of intervals      188
Sequences, of mean value integrals      178
Sequences, Regular, of nets in a metrical space      153
Sequences, Regular, of nets of intervals      57
Sets, ( $\mathfrak{B}$ )      41
Sets, ( $\mathfrak{F}$ )      40
Sets, ( $\mathfrak{G}$ )      40
Sets, Borel      41
Sets, bounded      40
Sets, closed      40
Sets, Cylindrical, in the torus space      159
Sets, Derived      40
Sets, dosed in a set      41
Sets, empty      4
Sets, Enumerable      4
Sets, everywhere dense in a set      41
Sets, Isolated      40
Sets, Linear      56
Sets, measurable      7
Sets, measurable ( $\mathfrak{B}$ )      41
Sets, measurable with respect to an outer Caratheodory measure      44
Sets, Measurable, in the torus space      159
Sets, non-dense in a set      41
Sets, Non-overlapping      40

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