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 AbelDirichlet test
 for convergence of improper integrals, 400
 for uniform convergence of parametrized Cintegrals, 408
 Absolute
 extrema, 82
 maxima, 82
 minima, 82
 Absolute continuity of the integral, 275
 Absolute convergence of improper integrals, 393
 Absolutely continuous functions on E^{1}, 378
 and Lintegrals, 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
 sigmaadditivity, 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 Lintegrals, 357
 and Rintegrals, 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, 285291 , 305
 open map principle, 75
 uniform boundedness principle, 75
 BanachSteinhaus 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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 C_{sigma}sets, 104
 volume of, 107
 Csimple sets, 99
 C'_{s}, family of Csimple sets, 99
 Cintegrals, see Improper integrals
 parametrized, 402; it see also Parametrized Cintegrals
 Cantor's set, 76
 Caratheodory property (CP), 145, 146, 157
 Cauchy criterion
 for convergence of improper integrals, 391
 for uniform convergence of parametrized Cintegrals, 403
 Cauchy integrals (Cintegrals), see Improper integrals
 parametrized, 402; see also Parametrized Cintegrals
 Cauchy principal value (CPV), 402
 Chain rule
 classical notation for, 31
 for differentiable functions, 28
 on E^{n} and C^{n}, 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 Cintegrals,405
 Complete measures, 148
 Complete normed spaces, see Banach spaces
 Completions of measures, 159
 completions of generalized measures, 205
 Completely additive set functions, see sigmaadditive set functions
 Continuous
 functions between topological spaces, 161
 linear map, 13
 set functions, 131, 147
 with respect to a set function t (tcontinuous), 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
 AbelDirichlet test for, 400
 Convergent sequences of sets, 180
 Countablyadditive set functions, see sigmaadditive set functions
 Coverings of sets, 137
 basic, 138
 Mcoverings of a set, 137
 Omegacoverings 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
 RadonNikodym, 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, 210216
 Kdifferentiation, 211
 Lebesgue differentiation, 211, 351
 Omegadifferentiation, 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 E^{n} and C^{n} as, 10
 Double series, 110, 115

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 E^{n}
 intervals in, 97
 volume of open sets in, 108
 Elementary functions, 218
 integrable, 241
 integrals of, 241
 integration of, 241250
 Euler's theorem for homogeneous functions, 34
 Extendedreal functions
 integration of, 251267; see also Integration of extendedreal 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 (tfinite), 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
 AbelDirichelet test for convergence of, 400
 iterated, 394
 convergence of, 388
 singularities of, 387
 Indefinite integrals, 263, 293, 330
 indefinite Lintegrals, 366
 Independence, linear, 16
 Inner products representing linear functionals on E^{n} and C^{n}, 10
 Integrable functions
 elementary, 241
 extendedreal, 252
 with values in complete normed spaces, 285
 Riemann, 307, 317
 Integrals
 Cauchy (Cintegrals), 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 Cintegrals, 402; see also Parametrized Cintegrals
 Riemann (Rintegrals), 308ff.; see also Riemann integrals
 RiemannStieltjes, 318
 Stieltjes, 319, 321ff.
 unorthodox, 247
 upper, 251
 with respect to Lebesgue measure (Lintegrals), 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, 241250
 of extendedreal functions, 251267
 of functions with values in Banach spaces, 285291, 305
 linearity of the integral, 267, 288
 monotone convergence theorem, 271
 weighted law of the mean, 269
 Intervals in E^{n}, 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 E^{n}), 186, 210

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 Lmeasurable, see Lebesguemeasurable.
 Lintegrable, see Lebesgueintegrable.
 Lintegrals, 357
 and absolutely continuous functions, 380
 indefinite, 366
 Lprimitive, 366
 Lagrange form of the remainder in Taylor's Theorem, 42
 Lagrange multipliers, 89
 Lebesgue
 decompositions, 342
 extensions, 154, 168
 Lebesgueintegrable functions, 241
 Lebesguemeasurable functions, 222
 Lebesguemeasurable sets, 168
 measure, 168175
 outer measure, 138
 nonmeasurable sets under Lebesgue measure, 173
 points of functions, 382
 premeasure, 126, 138, 168
 premeasure space, 138
 sets of functions, 382
 LebesguesStieltjes
 measurable functions, 222
 measures, 176
 measures in E^{n}, 179
 outer measures, 146, 176
 premeasures, 176
 set functions, 127, 135, 176
 signed LebesguesStieltjes measures, 206, 335
 Linear boundededness, 9
 Leftcontinuous set functions, 131
 Linear functionals, 7
 on E^{n} and C^{n} 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 E^{n} or C^{n}, 10
 Linear subspaces of a vector space, 16
 Linear independence, 16
 Linearity of the integral
 of extendedreal 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 LebesguesStieltjes.
 Luzin's theorem, 234


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 Mtest for uniform convergence of parametrized Cintegrals, 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
 LebesguesStieltjes (LS), 222
 Luzin's theorem, 234
 Mmeasurable functions, 218
 mmeasurable 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, 168175
 Lebesgue extensions, 154
 LebesguesStieltjes, 176
 LebesguesStieltjes measures in E^{n}, 179
 outer, 138, 139; see also Outer measures
 product, 293
 regular, 162
 rotationinvariant, 192
 signed, 194, 199
 signed LebesgueStieljes, 206, 335
 strongly regular, 162
 totally sigmafinite, 169
 translationinvariant, 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
 Nowheredense sets, 70

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 Omegacoverings of a set, 213
 Omegadifferentiation, 211
 and RadonNikodym 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
 LebesguesStieltjes, 146
 outer measurable sets, 149
 regular, 155, 156

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 P(S), the power set of S, 116
 Parametrized Cintegrals, 402
 AbelDirichlet test for uniform convergence of, 408
 Cauchy criterion for uniform convergence of, 403
 comparison test for uniform convergence of, 405
 Mtest 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
 LebesguesStieltjes, 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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 Rintegrals, see Riemann integrals
 RadonNikodym derivatives, 338
 and Lebesgue differentiation, 351
 and Omegadifferentiation, 353
 RadonNikodym 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
 Riemannintegrable 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
 RiemannStieltjes integrals, 318
 Rightcontinuous 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
 Rotationinvariant measures, 192

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 sigmaadditive set functions, 126, 147
 sigmaadditivity of volume, 104
 sigmaalgebras of sets, 116. See also sigmafield.
 sigmafields of sets, 116
 Borel fields, 162
 genereated by a set family M, 117
 sigmafinite set functions, 140
 totally, 140, 169
 sigmarings of sets, 116, 147
 Borel fields, 162
 generated by a semiring, 119
 generated by a set family, 117
 sigmasubadditive set functions, 137, 147
 sigma^{0}finiteness, 167
 Semifinite set functions, 126
 Semirings of sets, 98
 Separable sets, 223
 Series
 double, 110, 115
 permutable, 110
 positive, 111
 Sets
 Borel, 162
 C_{sigma}, 104
 Csimple, 99
 Cantor's set, 76
 convergent sequences of, 180
 families of, see Set families
 Lebesguemeasurable, 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
 sigmaalgebras of, 116
 sigmafields of, 116
 sigmarings 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
 Csimple sets C'_{s}, 99
 disjoint, 99
 fields, 116
 hereditary, 123
 products of, 120
 rings, 101, 115
 sigmaalgebras, 116
 sigmafields, 116
 sigmarings, 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 tcontinuous), 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 (tcontinuous), 197
 derivates of, 187
 derivatives of, 210
 differentiable, 210
 finite, 125
 finite with respect to a set function t (tfinite), 197
 finitely additive, 126, 126
 generalized measures, 194
 Lebesgue premeasure, 126
 LebesguesStieltjes, 127, 135, 176
 leftcontinuous, 131
 monotone, 136, 147
 outer measures, 138; see also Outer measures
 premeasures, 137
 regular, 140, 155
 rightcontinuous, 131
 rotationinvariant, 192
 sigmaadditive, 126
 sigmafinite, 140
 sigmasubadditive, 137
 semifinite, 126
 signed measures, 194, 199
 signed LebesguesStieltjes measures, 206, 335
 singular with respect to a set function t (tsingular), 341
 total variation of, 194
 totally sigmafinite, 140, 169
 translationinvariant, 171
 volume of sets, see Volume
 Set rings, 101, 115
 generated by a set family, 117
 Signed LebesguesStieltjes 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 (tsingular), 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 sigmafinite set functions, 140, 169
 Translationinvariant 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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 Vcoverings, 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 C_{sigma}sets in E^{n}, 107
 of open sets in E^{n}, 108
 of sets, 125
 sigmasubadditivity of, 109

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 Weighted law of the mean, 269
