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- Abel-Dirichlet test
- for convergence of improper integrals, 400
- for uniform convergence of parametrized C-integrals, 408
- Absolute
- extrema, 82
- maxima, 82
- minima, 82
- Absolute continuity of the integral, 275
- Absolute convergence of improper integrals, 393
- Absolutely continuous functions on E1, 378
- and L-integrals, 380
- Absolutely continuous with respect to a set function t, 197
- Additive extensions of set functions, 129
- Additive set functions, 126, 126
- Additivity of the integral, 260, 290
- Additivity of volume
- countable, 104
- of intervals, 101
- sigma-additivity, 104
- Almost everywhere (a.e.), 231
- convergence of functions, 231
- Almost measurable functions, 231, 231
- Almost uniform convergence of functions, 239
- Egorov's theorem, 240, 283
- Antiderivatives, 357
- and L-integrals, 357
- and R-integrals, 362
- change of variable in, 363
- primitives, 359
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- Baire categories (of sets), 70
- sets of Category I, 71
- sets of Category II, 71
- Baire's theorem, 71
- Banach spaces, 76
- integration of functions with values in, 285-291 , 305
- open map principle, 75
- uniform boundedness principle, 75
- Banach-Steinhaus uniform boundedness principle, 75
- Basic covering of a set, 138
- Basic covering value of a set, 138
- Basis of a vector space, 16
- Bicontinuous maps, 70
- Bijective
- functions, 52
- linear maps, 53
- Borel
- fields, 162
- measurable functions, 222
- measures, 162
- restrictions of measures, 162
- sets, 162
- Boundedness, linear, 9
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- Csigma-sets, 104
- volume of, 107
- C-simple sets, 99
- C's, family of C-simple sets, 99
- C-integrals, see Improper integrals
- parametrized, 402; it see also Parametrized C-integrals
- Cantor's set, 76
- Caratheodory property (CP), 145, 146, 157
- Cauchy criterion
- for convergence of improper integrals, 391
- for uniform convergence of parametrized C-integrals, 403
- Cauchy integrals (C-integrals), see Improper integrals
- parametrized, 402; see also Parametrized C-integrals
- Cauchy principal value (CPV), 402
- Chain rule
- classical notation for, 31
- for differentiable functions, 28
- on En and Cn, 30
- Change of measure in generalized integrals, 332
- Change of variable
- in antiderivatives, 363
- in Lebesgue integration, 386
- Characteristic functions, 246
- Clopen maps, 61
- Closed maps, 59
- Closed sets in topologies, 161
- Compact regular (CR) set functions on topological spaces, 209
- Comparison test
- for improper integrals, 393, 399
- for uniform convergence of parametrized C-integrals,405
- Complete measures, 148
- Complete normed spaces, see Banach spaces
- Completions of measures, 159
- completions of generalized measures, 205
- Completely additive set functions, see sigma-additive set functions
- Continuous
- functions between topological spaces, 161
- linear map, 13
- set functions, 131, 147
- with respect to a set function t (t-continuous), 197
- Continously differentiable functions, 38, 57
- Convergence of functions
- almost everywhere, 231
- almost uniform, 239
- Egorov's theorem, 240, 283
- in measure, 239, 280
- Lebesgue's theorem, 240, 283
- Riesz' theorem, 280
- Convergence of improper integrals, 388
- absolute, 393
- Cauchy criterion for, 391
- comparison test for, 393, 399
- conditional, 391
- Abel-Dirichlet test for, 400
- Convergent sequences of sets, 180
- Countably-additive set functions, see sigma-additive set functions
- Coverings of sets, 137
- basic, 138
- M-coverings of a set, 137
- Omega-coverings of a set, 213
- Vitali, 180; see also Vitali coverings
- CP, the Caratheodory property, 145
- Critical points, 82
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- Darboux sums (upper and lower), 307
- Decompositions
- Lebesgue, 342
- of generalized measures, 344
- Derivates
- of point functions, 373
- of set functions187
- Derivatives
- directional, see Directional derivatives
- of set functions, 210
- Radon-Nikodym, 338, 351
- partial, seePartial derivatives
- Determinants
- functional, 49
- of matrices, 47, 96
- Differentiable functions, 17
- and directional derivatives, 19
- chain rule for, 28
- continuously, 38, 57
- differentials of, 17
- and partial derivatives, 19, 22
- in a normed space, 17
- m times differentiable, 38
- Differentiable set functions, 210
- Differentials, 17
- chain rule for, 28
- of functions in a normed space, 17
- of order m, 39
- Differentiation of set functions, 210-216
- K-differentiation, 211
- Lebesgue differentiation, 211, 351
- Omega-differentiation, 211, 353
- Directional derivatives, 1
- differentiable functions and, 19
- Finite Increments Law for, 7
- higher order, 35
- of linear maps, 15
- Discriminant of a quadratic polynomial, 80
- Disjoint set families,99
- Dominated convergence theorem, 273, 327
- Dot products, linear functionals on En and Cn as, 10
- Double series, 110, 115
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- En
- intervals in, 97
- volume of open sets in, 108
- Elementary functions, 218
- integrable, 241
- integrals of, 241
- integration of, 241-250
- Euler's theorem for homogeneous functions, 34
- Extended-real functions
- integration of, 251-267; see also Integration of extended-real functions
- integrable, 252
- lower integrals of, 251
- upper integrals of, 251
- Extremum, extrema
- absolute, 82
- conditional, 88
- local, 79, 89
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- Fatou's lemma, 272
- Fields of sets, 116
- generated by a set family, 117
- Finite Increments Law for directional derivatives 7
- Finite set functions, 125
- Finite with respect to a set function t (t-finite), 197
- Finitely additive set functions, 126, 126
- Frechet's theorem, 237
- Fubini
- map, 294
- theorem, 298, 301, 305, 334
- Functional determinants, 49
- Functionals, linear, see Linear functionals
- Functions. See also Maps
- bijective, 52
- continuous, 161
- differentiable, see Differentiable functions
- homeomorphisms, 70
- homogeneous, 34
- implicit function theorem, 64
- inverse function theorem, 61
- partially derived, 2
- Fundamental theorem of calculus, 360
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- Generalized integration, 323ff.
- change of measure, 332
- dominated convergence theorem, 327
- Fubini property in, 334
- indefinite integrals in, 330
- Generalized measure spaces, 194
- integration in, 323ff.
- Generalized measures, 194
- completion of, 205
- decomposition of, 344
- signed measures, 194, 199
- Gradient of a function, 20
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- Hadamard's theorem, 96
- Hahn decomposition theorem, 201
- Hereditary set families, 123
- Homeomorphisms, 70
- Homogeneous functions, 34
- Euler's theorem for, 34
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- Implicit
- differentiation, 66, 87
- function theorem, 64
- Improper integrals, 388
- absolute convergence of, 393
- Cauchy criterion for, 391
- Cauchy principal value (CPV) of, 402
- comparison test for, 393, 399
- conditional convergence of, 391
- Abel-Dirichelet test for convergence of, 400
- iterated, 394
- convergence of, 388
- singularities of, 387
- Indefinite integrals, 263, 293, 330
- indefinite L-integrals, 366
- Independence, linear, 16
- Inner products representing linear functionals on En and Cn, 10
- Integrable functions
- elementary, 241
- extended-real, 252
- with values in complete normed spaces, 285
- Riemann, 307, 317
- Integrals
- Cauchy (C-integrals), 388; see also Improper integrals
- in generalized measure spaces, 323ff.
- indefinite, 263, 293, 330
- improper, 388; see also Improper integrals
- iterated, 294
- Lebesgue, 357
- Lebesgue integrals and Riemann integrals, 313
- lower, 251
- of elementary functions, 241
- orthodox, 247
- parametrized C-integrals, 402; see also Parametrized C-integrals
- Riemann (R-integrals), 308ff.; see also Riemann integrals
- Riemann-Stieltjes, 318
- Stieltjes, 319, 321ff.
- unorthodox, 247
- upper, 251
- with respect to Lebesgue measure (L-integrals), 357
- Integration
- absolute continuity of the integral, 275
- additivity of the integral, 260, 290
- by parts, 321
- dominated convergence theorem, 273, 327
- Fatou's lemma, 272
- in generalized measure spaces, 323ff.
- of elementary functions, 241-250
- of extended-real functions, 251-267
- of functions with values in Banach spaces, 285-291, 305
- linearity of the integral, 267, 288
- monotone convergence theorem, 271
- weighted law of the mean, 269
- Intervals in En, 97
- additivity of volume of, 101
- simple step functions on, 218
- step functions on, 218
- Inverse function theorem, 61
- Iterated integrals, 294
- iterated improper integrals, 394
- Fubini map, 294
- Fubini theorem, 298, 301, 305, 334
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- Jacobian matrix, 18
- Jacobians, 49
- Jordan components, 203
- Jordan decompositions, 202
- Jordan components, 203
- Jordan outer content, 140
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- K (the set of all cubes in En), 186, 210
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- L-measurable, see Lebesgue-measurable.
- L-integrable, see Lebesgue-integrable.
- L-integrals, 357
- and absolutely continuous functions, 380
- indefinite, 366
- L-primitive, 366
- Lagrange form of the remainder in Taylor's Theorem, 42
- Lagrange multipliers, 89
- Lebesgue
- decompositions, 342
- extensions, 154, 168
- Lebesgue-integrable functions, 241
- Lebesgue-measurable functions, 222
- Lebesgue-measurable sets, 168
- measure, 168-175
- outer measure, 138
- nonmeasurable sets under Lebesgue measure, 173
- points of functions, 382
- premeasure, 126, 138, 168
- premeasure space, 138
- sets of functions, 382
- Lebesgues-Stieltjes
- measurable functions, 222
- measures, 176
- measures in En, 179
- outer measures, 146, 176
- premeasures, 176
- set functions, 127, 135, 176
- signed Lebesgues-Stieltjes measures, 206, 335
- Linear boundededness, 9
- Left-continuous set functions, 131
- Linear functionals, 7
- on En and Cn as dot products, 10
- Linear maps, 7
- as a normed linear space, 13
- bijective, 53
- bounded, 9
- continuous, 9, 13
- directional derivatives of, 15
- matrix representation of composite, 12
- matrix representation of, 11
- norm of, 13
- uniformly continuous on En or Cn, 10
- Linear subspaces of a vector space, 16
- Linear independence, 16
- Linearity of the integral
- of extended-real functions, 267
- of functions with values in Banach spaces, 288
- Lipschitz condition, 25, 384
- Local
- extremum, extrema, 79, 89
- maximum, maxima, 79
- minimum, minimima, 79
- Lower
- Darboux sums, 307
- integrals, 251
- Riemann integrals, 308
- LS, see Lebesgues-Stieltjes.
- Luzin's theorem, 234
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- M-test for uniform convergence of parametrized C-integrals, 405
- Maps. See also Functions
- bicontinuous, 70
- clopen, 61
- closed, 59
- linear, see Linear maps
- open, 59
- open map principle, 75
- Matrix, matrices
- as elements of a vector space, 15
- determinants of, 47, 96
- Jacobian, 18
- n x n matrices as a noncommutative ring with identity, 15
- of composite linear maps, 12
- representation of a linear map, 11
- Maximum, maxima
- absolute, 82
- conditional, 88
- local, 79
- Meagre sets, 71
- Measurable covers of sets, 156
- Measurable functions
- almost, 231
- Borel, 222
- Frechet's theorem, 237
- Lebesgue (L), 222
- Lebesgues-Stieltjes (LS), 222
- Luzin's theorem, 234
- M-measurable functions, 218
- m-measurable functions, 231
- Tietze's theorem, 236
- Measurable sets, 147
- nonmeasurable sets under Lebesgue measure, 173
- outer, 149
- Measurable spaces, 217
- Measure spaces, 147
- almost measurable functions on sets in, 231
- probability spaces as, 148
- topological, 162
- Measures, 147, 194. See also Set functions
- Borel restrictions of, 162
- as extensions of premeasures, 154
- Borel, 162
- complete, 148
- completions of, 159
- constructed from outer measures, 152
- generalized, 194
- Lebesgue, 168-175
- Lebesgue extensions, 154
- Lebesgues-Stieltjes, 176
- Lebesgues-Stieltjes measures in En, 179
- outer, 138, 139; see also Outer measures
- product, 293
- regular, 162
- rotation-invariant, 192
- signed, 194, 199
- signed Lebesgue-Stieljes, 206, 335
- strongly regular, 162
- totally sigma-finite, 169
- translation-invariant, 171
- Metric spaces
- as topological spaces, 161
- networks of sets in, 212
- Minimum, minima
- absolute, 82
- conditiona, 88
- local, 79
- Monotone convergence theorem, 271
- Monotone set functions, 136, 117
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- Networks of sets in metric spaces, 212
- Nonmeasurable sets under Lebesgue measure, 173
- Norm of a linear map, 13
- Normal Vitali coverings, 192
- Nowhere-dense sets, 70
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- Omega-coverings of a set, 213
- Omega-differentiation, 211
- and Radon-Nikodym derivative, 353
- Open map principle, 75
- Open maps, 59
- Open sets
- in topologies, 161
- volume of, 108
- Operator, linear, 7
- Orthodox integrals, 247
- Outer content, 140
- Jordan, 140
- Outer measurable sets, 149
- Outer measure spaces, 149
- Outer measures, 138, 139
- Caratheodory property~(CP), 145
- constructing measures from, 152
- Lebesgue outer measure, 138, 146, 176
- Lebesgues-Stieltjes, 146
- outer measurable sets, 149
- regular, 155, 156
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- P(S), the power set of S, 116
- Parametrized C-integrals, 402
- Abel-Dirichlet test for uniform convergence of, 408
- Cauchy criterion for uniform convergence of, 403
- comparison test for uniform convergence of, 405
- M-test for uniform convergence of, 405
- Partial derivatives, 3
- differentiable functions and, 19, 22
- higher order, 35
- Partially derived function, 2
- Partitions of sets, 195, 217
- elementary functions on, 218
- refinements of, 218, 308
- simple functions on, 218
- Permutable series, 110
- Polar coordinates, 46, 50, 55, 306, 395
- Positive series, 111
- Power set P(S), 116
- Premeasures, 137, 147
- measures as extensions of, 154
- induced outer measures from, 138
- Lebesgue, 126, 138, 168
- Lebesgues-Stieltjes, 176
- Premeasure spaces, 138
- Lebesgue, 138
- Primitives, see Antiderivatives
- Probability spaces, 148
- Product measures, 293
- Products of set families, 120
- Pseudometric spaces, 165
- Pseudometrics, 165
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- Quadratic forms, symmetric, 80
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- R-integrals, see Riemann integrals
- Radon-Nikodym derivatives, 338
- and Lebesgue differentiation, 351
- and Omega-differentiation, 353
- Radon-Nikodym theorem, 338
- Refinements of partitions of sets, 218, 308
- Regular measures, 162
- Regular set functions, 140
- compact, 209
- outer measures as, 155, 156
- Regulated functions, 312
- Residual sets, 71
- Riemann-integrable functions, 307, 317
- Riemann integrals, 308ff.
- Darboux sums (lower and upper), 307
- Lebesgue integrals and, 313
- lower, 307
- regulated functions, 312
- Riemann sums, 321
- upper, 307
- Riemann sums, 321
- Riemann-Stieltjes integrals, 318
- Right-continuous set functions, 131
- Ring
- n x n matrices as a noncommutative ring with identity, 15
- Rings of sets, 101, 115
- generated by a set family, 117
- Rotation-invariant measures, 192
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- sigma-additive set functions, 126, 147
- sigma-additivity of volume, 104
- sigma-algebras of sets, 116. See also sigma-field.
- sigma-fields of sets, 116
- Borel fields, 162
- genereated by a set family M, 117
- sigma-finite set functions, 140
- totally, 140, 169
- sigma-rings of sets, 116, 147
- Borel fields, 162
- generated by a semiring, 119
- generated by a set family, 117
- sigma-subadditive set functions, 137, 147
- sigma0-finiteness, 167
- Semifinite set functions, 126
- Semirings of sets, 98
- Separable sets, 223
- Series
- double, 110, 115
- permutable, 110
- positive, 111
- Sets
- Borel, 162
- Csigma, 104
- C-simple, 99
- Cantor's set, 76
- convergent sequences of, 180
- families of, see Set families
- Lebesgue-measurable, 168
- meagre, 71
- measurable, 147
- nonmeagre, 71
- nowhere dense, 70
- of Category I, 71
- of Category II, 71
- outer measurable, 149
- partitions of, 195
- residual, 71
- rings of, 101, 115
- sigma-algebras of, 116
- sigma-fields of, 116
- sigma-rings of, 116
- semirings of, 98
- separable, 223
- symmetric difference of, 122
- Vitali coverings of, 180
- volume of, see Volume
- Set algebras, 116. See also Set fields.
- Set families, 98
- set algebras, 116
- C-simple sets C's, 99
- disjoint, 99
- fields, 116
- hereditary, 123
- products of, 120
- rings, 101, 115
- sigma-algebras, 116
- sigma-fields, 116
- sigma-rings, 116
- semirings, 98
- Set fields, 116
- generated by a set family, 117
- Set functions, 125
- absolutely continuous with respect to a set function t (absolutely t-continuous), 197
- additive, 126, 137
- additive extension of, 129
- compact regular (CR) set functions on topological spaces, 209
- continuous, 131, 147
- continuous with respect to a set function t (t-continuous), 197
- derivates of, 187
- derivatives of, 210
- differentiable, 210
- finite, 125
- finite with respect to a set function t (t-finite), 197
- finitely additive, 126, 126
- generalized measures, 194
- Lebesgue premeasure, 126
- Lebesgues-Stieltjes, 127, 135, 176
- left-continuous, 131
- monotone, 136, 147
- outer measures, 138; see also Outer measures
- premeasures, 137
- regular, 140, 155
- right-continuous, 131
- rotation-invariant, 192
- sigma-additive, 126
- sigma-finite, 140
- sigma-subadditive, 137
- semifinite, 126
- signed measures, 194, 199
- signed Lebesgues-Stieltjes measures, 206, 335
- singular with respect to a set function t (t-singular), 341
- total variation of, 194
- totally sigma-finite, 140, 169
- translation-invariant, 171
- volume of sets, see Volume
- Set rings, 101, 115
- generated by a set family, 117
- Signed Lebesgues-Stieltjes measure spaces, 206
- induced by a function of bounded variation, 206
- integration in, 335
- Signed measure spaces, 194, 199
- Hahn decomposition theorem, 201
- Jordan components, 203
- Jordan decompositions, 202
- negative sets in, 199
- positive sets in, 199
- Simple functions, 218
- simple step functions, 218
- Singular with respect to a set function t (t-singular), 341
- Singularities of improper integrals, 387
- Span of vectors in a vector space, 16
- Step functions, 218
- simple, 218
- Stieltjes integrals, 319, 321ff.
- integration by parts, 321
- laws of the mean, 322
- Strongly regular measures, 162, 234, 237, 347
- Sylvester's theorem, 80
- Symmetric difference of sets, 122
- Symmetric quadratic forms, 80
- Sylvester's theorem, 80
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- Taylor polynomial, 43
- Taylor's Theorem, 40
- generalized, 45
- Lagrange form of remainder, 42
- Taylor polynomial, 43
- Tietze's theorem, 236
- Topological measure spaces, 162
- Topological spaces, 161
- compact regular (CR) set functions on, 209
- continuous functions between, 161
- metric spaces as, 161
- pseudometric spaces as, 165
- Topologies, 161
- closed sets in, 161
- open sets in, 161
- Total variation of set functions, 194
- Totally sigma-finite set functions, 140, 169
- Translation-invariant set functions, 171
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- Uniform boundedness principle of Banach and Steinhaus, 75
- Uniformly normal Vitali coverings, 192
- Universal Vitali coverings, 192
- Unorthodox integrals, 247
- Upper
- Darboux sums, 307
- integrals, 251
- Riemann integrals, 307
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- V-coverings, see Vitali coverings
- Vectors
- span of a set of, 16
- Vector spaces
- basis of, 16
- dimension of, 16
- linear subspaces of, 16
- matrices as elements of, 15
- span of vectors in, 16
- Vitali coverings, 180
- normal, 192
- uniformly normal, 192
- universal, 192
- Volume
- additivity of volume of intervals, 101
- monotinicity of, 109
- of Csigma-sets in En, 107
- of open sets in En, 108
- of sets, 125
- sigma-subadditivity of, 109
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- Weighted law of the mean, 269
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