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| Bartle R.G. — The Elements of Integration |
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| Предметный указатель |
 -measurable set 100
 -algebra 6
-algebra generated by a family of sets 7
-field 6
-finite measure 20
Absolute continuity 35 84
Absolute integrativity 42
Additivity, countable 4 19
Algebra of sets 96
Algebra, Borel 7
Algebra, extended Borel 8
Almost everywhere 22
Almost everywhere convergence 22 65
Almost uniform convergence 72 ff.
Archimedes 1
Banach space 60
Borel algebra 7
Borel measurability of continuous and monotone functions 9
Borel measure 20 104
Borel set 7
Borel — Stieltjes measure 20 105
Caratheodory extension theorem 101
Caratheodory, C. 100
Cauchy sequence, in  58
Cauchy — Bunyakovskii — Schwarz inequality 57
Characteristic function 3 9
Charge 23
Combinations of measurable functions 9
Complete measure space 103
Complete normed linear space 58
Completeness theorem 59
Completion of a measurable space 25
Completion of Borel measure 110
Completion of measure 25
Complex-valued functions, integral of 49
Complex-valued functions, measurability of 13
Conjugate indices 57
Continuity, absolute 35 84
Convergence in 58 66
Convergence in mean 66 ff.
Convergence, almost everywhere 22 65
Convergence, almost uniform 72 ff.
Convergence, dominated 44 75
Convergence, pointwise 65
Convergence, uniform 65
Countably additive 19
Countably subadditive 100
Counting measure 20
De Morgan laws 7
Decomposition of measures 80—95
Dominated Convergence Theorem 44 75
Egoroff's theorem 74
Endpoint of an interval 2
Equivalence of functions 54
Essentially bounded function 60
Eudoxos 1
Extended real numbers 5
Extension of measures 96—112
Fatou's lemma 33 39
Field 96
Frechet, M. 4
Fubini's theorem 119
Function measure 19
Function, characteristic 3 9
Function, complex valued 13
Function, continuous 9
Function, essentially bounded 60
Function, imaginary part of 14
Function, integrable 41
Function, measurable 8 10
Function, monotone 9
Function, negative part of 10
Function, positive part of 10
Function, real part of 14
Function, simple 3 27
Function, step 3 38
Function, summable 41
Functional, linear 59
Functional, representation of 90 ff. 106
Hahn decomposition theorem 81—82
Hahn extension theorem 103
Hahn, F. 103
Hoelder's inequality 56
Inequality, Cauchy — Bunyakovskii — Schwartz 57
Inequality, Hoelder 56
Inequality, Minkowski 57
Infimum 5
Integrable function 41
Integral of complex-valued functions 49
Integral of nonnegative measurable functions 30
Integral of nonnegative simple functions 28
Integral of real-valued integrable functions 41
Integral, indefinite 42
Integral, Riemann 3 38
interval 2
Joichi, T.J. 78—79
Jordan decomposition theorem 83
Lebesgue decomposition theorem 88
Lebesgue dominated convergence theorem 44
Lebesgue measure 20 104
Lebesgue spaces 52—64
Lebesgue — Stieltjes measure 105
Lebesgue, H. 1
| Leibniz, G.W. 1
Length of an interval 3 96
Levi, B. 31
Limit of a sequence of extended real numbers 5
Limit of a sequence of sets 16
Linear functional 89 ff. 106
Mean convergence 66 ff.
Measurable function 8 10
Measurable set 7
Measurable space 6
Measure 19
Measure generated by a monotone function 20 105
Measure, -finite 20
Measure, absolute continuous 84
Measure, Borel 20 104
Measure, Borel — Stieltjes 20 105
Measure, construction of 96—112
Measure, counting 20
Measure, finite 20
Measure, Lebesgue 20 105
Measure, Lebesgue — Stieltjes 105
Measure, outer 99
Measure, singular 88
Measure, space 22
Measure, unit 20
Minkowski's inequality 57
Monotone class 17 18 116
Monotone class lemma 116
Monotone Convergence Theorem 31
Monotone function 9
Negative part 10
Negative set 80
Negative variation 82
Newton, I. 1
Nonmeasurable set 104
Norm 52
Norm of a linear functional 89
Normed linear convergence 68
Null set 80
Outer measure 99
Pointwise convergence 65
Positive part 10
Positive set 80
Positive variation 82
Product of two measures 115
Pseudo-norm 52
Radon — Nikodym derivative 87 94
Radon — Nikodym, theorem of 85 94 95
rectangle 113
Representation theorems of linear functionals 90 106
Riemann integral 3 38
Riemann, B. 1
Riesz representation theorem, on  90 ff.
Riesz representation theorem, on 91 ff.
Riesz representation theorem, on C 106 ff.
Riesz, F. 58 69 90—92 106—108
Rota, G.-C. 129
Schwarz inequality 57
Section 115
Semi-norm 52
Set, -measurable 100
Set, Borel 7
Set, negative 80
Set, null 80
Set, positive 80
Simple function 3 27
Singular measure 6
Space measure 22
Space product 113
Space, measurable 6
Standard representation 27
Step function 3 38
Stieltjes — Lebesgue measure 105
Subadditivity, countable 100
Supremum 5
Theodore, R. 23
Theorem, Caratheodory extensions 101
Theorem, completeness 59
Theorem, dominated convergence 44
Theorem, Egoroff 74
Theorem, Fatou 33
Theorem, Fubini 119
Theorem, Hahn decomposition 81
Theorem, Hahn extension 103
Theorem, Jordan decomposition 83
Theorem, Lebesgue decomposition 88
Theorem, Lebesgue dominated convergence 44
Theorem, Levi monotone convergence 31
Theorem, monotone class 116
Theorem, monotone convergence 31
Theorem, product measure 114
Theorem, Radon — Nikodym 85
Theorem, Riesz representation 90 91 106
Theorem, Riesz — Fischer 59
Theorem, Tonelli 118
Theorem, Vitali convergence 76
Tonelli's theorem 118
Total variation 83
Truncation 16
Uniform convergence 65 ff.
Unit measure 20
Variation 25 82—83
Vitali convergence theorem 118
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