Bibliography - Springer978-1-4612-0549-4/1.pdf · Bibliography ABIAN, A., The theory of sets and...
Transcript of Bibliography - Springer978-1-4612-0549-4/1.pdf · Bibliography ABIAN, A., The theory of sets and...
Bibliography
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BEAR, H.S., A primer of Lebesgue integration, Academic, New York, 1995.
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BERBERIAN, S.K., Lectures on functional analysis and operator theory, SpringerVerlag, New York, 1974.
BERBERIAN, S.K., A first course in real analysis, Springer-Verlag, New York, 1994.
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Index of Notations
SYMBOL PAGE SYMBOL PAGE
xEA 2 lim sup an 79 lP',N,Z,Q,JR,<C 2 liminf an 80
AcB 3 limn--->oo an 82 AUB,AnB 4 .X*(A) 87 A',A-B 4 M(.X*) 96
{a} 8 .Xi(A), .Ae(A) 97 f:X---->Y 9 r 99
CfJA 10 S(£), A(£) 101 gof 10 (X,S,f.L) 103 P(X) 12 A~B 107
uiEI Ai , niEI Ai 13 BIR(T), Bc(T) 119
TiiEI Xi 13 llxlloo, doo(x, y) 119 prj 14 llxiiP 119
Xjrv 19 dp(x, y) 122 x~Y 23 limn--->oo Xn 124 ErvF 46 Xn----'> X 124 E;:SF 47 Br(c) 126
cardE 49 Ur(c) 127
~0' c 49 Sr(c) 129
card E :S: card F 50 A 134
F(E,F)' FE 56 Ao 137 ordE 59 intA, extA 137
w 60 8A, bdA 137 ord E :S: ord F 62 limx--->c,xEA f(x) 142
~1, n 70 limx--->c f(x) 142
iR 73 f(c+ ), f(c-) 144
L.::iEI ai 74 f'(c), f{(c), J;(c) 145
2::~ 1 ak 76 B(X) 149
Xi i X, Xi 1 X 78 fUg,fng 151
471
472 Index of Notations
SYMBOL PAGE SYMBOL PAGE
j+, J-, 1/1 152 diamA 278 SUPn fn, infn fn 152 B(T,Y) 289
limsupfn 153 D(f,g) 290 liminf fn 153 C(X, Y) 292
N(f), ISF 160 CJR(X), Cc(X) 292
.Ci(JL) 167 .ct c 312
I fd~t 167 II Jilt 314
IE fdJL 172 Ll c 317
I: Jd>.. 172 .C'e' 11/lloo 323
!·JL 184 Lao c 324
V«.JL 192 .Ct' II/IlP 332
V~f 202 v c 336 a-<r 202 Ref, lmf 349
(I), IJ(l)l 205 llJ 362
BV[a,b] 212 Co(X) 363
AC[a,b] 213 JL*(A) 365 limsupx-c,xEA f(x) 216 SxT 371
liminfx-c, xEA f(x) 216 JLXV 378 o+,o+,o-,o_ 220 ff hdvdJL 384 limsupx-c f(x) 233 JLA 389
liminfx-c f(x) 233 I vi 431
D,D 236 JLj_V 444
J .. , r 267 f*g 458
Index
Absolute continuity of measures, 192, 322, 440
of signed measures, 440, 445 Absolutely continuous function, 206,
213 Adherence of a subset of a topological
space, 134 Adherent point, 133 a.e., 157 a.e. primitive, 248 a.e. unique, 157 algebra of sets, 101 Algebra, 212 Almost everywhere, 157 Arzela-Young theorem, 113 Ascoli's theorem, 401, 407 Axiom of Choice, 13
Baire category theorem, 304 Baire space, 304 Banach algebra, 331, 352 Banach space, 318, 461 Base for a topology, 279 Basis of a vector space, 43 Bijection, 10 Bijective function, 10 Bilateral sequence, 11 Bilinear mapping, 344 Boolean ring, 432
of sets, 107 Borel function, 151, 453 Borel sets of R, 102
of a topological space, 149 Borel space, 149 Boundary of a subset of a topological
space, 137 Boundary point, 137
Bounded above, 22 Bounded below, 22 Bounded function, 119, 288 Bounded set, in a metric space, 298
in a pre-ordered set, 22 Bounded variation, function of,
202, 264
c· -algebra, 352 Canonical factorization of a function,
20 Cantor set, 99 Cantor's theorem, 50 Cardinal arithmetic, 52 Cardinal numbers, 49
countable, 52 finite, 52 infinite, 52 uncountable, 52
Cardinality, 49 finite, 37 of an ordinal number, 68
Cartesian product, 5 Cauchy sequence, 278 Cauchy's criterion for convergence, 31
for uniform convergence, 287, 288 Cavalieri's principle, 397 Chain in an ordered set, 21 Characteristic function of a subset, 10 Choice function, 14 Closed ball in a metric space, 126 Closed set in a metric space, 126
in a topological space, 131 Closure of a subset of a topological
space, 134 Commutative diagram, 20 Compact linear mapping, 461
473
474
Compact space, 275 subset, 276
Complement of a subset, 4 Complete measure space, 108, 156
metric space, 282 ordered field, 26, 33 seminormed space, 461
Completion of a metric space, 309 Complex measure, 319
numbers, 2, 118, 350 Composite function, 10 Composition of functions, 10
of relations, 8 Conclusion of a proposition, 3 Congruence of integers, 17 Continuity at a point, 139 Continuous function, 139
linear mapping, 461 Continuum Hypothesis, 58 Continuum, cardinal of, 49, 57 Contraction of a measure to a subset,
389 Contradiction, law of, 5 Contrapositive form of a theorem, 3 Convergent sequence in R, 82
in a metric space, 124 sequence in a topological space, 137
Convergent sequence of functions, pointwise, 153, 286
Converse of a proposition, 3 Convolution of functions, 458 Coordinate, 13 Countable set, 38 Countably infinite set, 39 Covering, 274
De Morgan's formulas, 13 Decreasing net, 77 Dedekind cut, 32 Dense subset of a topological space,
136 Denumerable set, 39 Denumerably infinite set, 39 Derivates of a function, 220, 236 Derivative of a function, 145 Diagonal, 8, 17 Diameter of a subset, 130, 278,
298, 299 Difference of two subsets, 4
Index
Differentiable function, 145 Differentiation, term-by-term, 296 Dini derivates of a function, 220 Dini's theorem, 293 Dirac measure, 106 Discrete measure, 106
metric, 118 metric space, 118 topology on a set, 131
Distance function, 118 Domain of a relation, 8 Dominated convergence theorem,
176 Dual of an order relation, 21 Dual space, 343
Empty function, 17 Equi-uniformly continuous set of
functions, 400 Equicontinuous set of functions, 400
at a point, 399 Equipollent sets, 46 Equivalence class, 18 Equivalence relation, 9, 17
Deduced from a function, 17 Equivalent metrics, 131
propositions, 3 signed measures, 448
Essential bound, 323 Essential supremum of a function,
157, 323 Essentially bounded function,
157, 323 Essentially unique, 157 Euclidean metric, 122
space, 122 Excluded middle, law of, 5 Extended real numbers, 73
algebraic operations on, 7 4 Exterior measure, 97 Exterior of a subset of a topological
space, 137
Family indexed by a set, 11 Fatou's lemma, 177 Final set of a function, 9 Finite character, a set of sets of, 43 Finite covering, 274 Finite diameter, subset of, 130, 278
Index
Finite intersection property, 275 Finite set, 34, 38 Finite signed measure, 190, 319 Finite-dimensional normed spaces,
F. Riesz's characterization, 468 Finitely additive set function, 112 First category, subset of the, 303 First countable topological space, 133 Frequently, 80 Fubini's theorem, 387, 395 Fubini-Tonelli theorem, 385, 392 Function, 9 Function of bounded variation,
202, 264 Fundamental sequence of
neighborhoods, 133 Fundamental theorem of calculus,
Lebesgue's, 252
Generated algebra, 101 a-algebra, 101
Graph of a function, 9
Hahn decomposition of a finite signed measure, 192
Hahn-Kolmogorov extension theorem, 367
Hausdorff space, 146, 275 Hausdorff's maximality principle,
after 42 Reine-Borel theorem, 273 Hilbert space, 345 Holder's inequality, 120, 332 Homeomorphism, 140 Hypothesis of a proposition, 3
Ideal, order, 63 Identity indexing, 11
mapping, 10 relation, 8
Image, direct, 7 Imaginary part of a complex-valued
function, 349 Increasing net, 77 Indefinite integral, 184, 207, 213, 248 Index set, 11 Induction, principle of mathematical,
23 Inductive partially ordered set, 42
Infimum, 73 Infinite set, 34, 38
475
Initial ordinal of a cardinal number, 69
Initial segment of a well-ordered set, 60
Initial set of a function, 9 Injection, 10 Injective function, 10 Inner product, 344 Inner regularity of Lebesgue measure,
105 Insertion mapping, 10 Integers, 2 Integrable function, 167, 178, 312
simple function, 160 Integral, 167, 177, 312 Interior measure, 97 Interior of a subset of a topological
space, 137 Interior point, 127, 132 Intersection of subsets, 4, 13 Inverse function, 16
image, 7, 11 Isometric mapping, 307 Isometry, 307 Iterated integral, 384 Iterated limits theorem, 294
Jordan decomposition of a function of bounded variation, 205
Jordan-Hahn decomposition of a signed measure, 431
Kernel function, 463
l.s.c., 229, 232 .C1-norm, 314 Lebesgue decomposition of a function
of bounded variation, 264 of signed measures, 446, 450
Lebesgue measure, 96 Lebesgue number of a covering, 401 Lebesgue outer measure, 87 Lebesgue's criterion for Riemann
integrability, 268 Fundamental theorem of calculus,
252 singular function, 210
476
Lebesgue-integrable function, 167 Lebesgue-measurable function, 150
set, 93 Left limit of a function, 144 Left-derivative of a function, 145 Left-differentiable function, 145 Lemma on monotone classes, 180 Lexicographic ordering of a product of
ordered sets, 22, 25, 45 reverse of, 71
Limit inferior of a function, 216, 233 of a sequence, 80
Limit of a function at a point, 142 Limit of a sequence in R, 82
in a metric space, 124 Limit of a sequence of functions,
pointwise, 153, 286 Limit ordinal, 71 Limit superior of a function, 216,
233 of a sequence, 79
Lindelof's theorem, 280 Linear mapping, compact, 461
continuous, 461 Linear ordering, 21 Lipschitz condition, 206, 270, 328, 410 Locally compact space, 363 Lower bound, 22 Lower derivate of a function, 236 Lower semicontinuous function, 229,
232 Lower semicontinuous regularization
of a function, 268 £P-norm, 332
Majorant, 22 Mapping, 9 Mathematical induction, principle of,
23 Maximal element in a pre-ordered set,
22 Meager subset, 303 Mean, convergence in, 315
of order p, 461 Measurable function, 149, 311
rectangle, 372 space, 149 subset, 93,95, 149,366
Measure on a 0'-algebra, 103
on an algebra of sets, 365 Measure space, 103 Metric, 117 Metric space, 118 Metric subspace, 138
Index
Metrizable topological space, 131 Minimal element in a pre-ordered set,
23 Minkowski p-metric, 122
p-norm, 119 space, 122
Minkowski's inequality, 120, 333 Minorant, 22 Monotone class of sets, 179
lemma on, 180 Monotone convergence theorem, 176
set function, 112 Monotonic functions, 222, 253 Mutually singular signed measures,
444
Negation of a proposition, 2 Negative part of a function, 152 Negligent function, 213 Negligible set, 87, 95, 156 Neighborhood of a point, 128, 132 Nested closed sets property, 299 Nested intervals, theorem on, 32 Net, 77 Nonmeager subset, 304 Nonmeasurable set, Vitali's example,
110 Nonnegative integers, 2 Nonoverlapping intervals, 206 Nonzero set of a function, 160 Norm, 314 Normed space, 314
finite-dimensional, 468 Null functions on a measure space, 314 Null set, 156
One-one correspondence, 10 function, 10
Onto function, 10 Open ball in a metric space, 127 Open covering, 27 4 Open set in a metric space, 128
in a topological space, 131 Operator, 9
Index
Order ideal, 63 isomorphism, 23 monomorphism, 23 morphism, 23 relation, 20
Ordinal numbers, 60 countable, 67 denumerable, 67 finite, 67 infinite, 67 natural ordering of, 62 sum of, 68 uncountable, 67
Ordinality, 59 of a cardinal number, 69
Outer measure, 95, 365 Lebesgue, 87
Outer regularity of Lebesgue measure, 105
p-norm, 332 Partial ordering, 20 Partially ordered set, 21 Partition of a set, 18 Peano's existence theorem, 416 Picard's existence theorem, 410 Planar Lebesgue measure, 379 Point mass, 106 Pointwise Cauchy sequence of
functions, 286 convergent sequence of functions.
153, 286 limit of a sequence of functions,
153, 286 totally bounded set of functions,
399 Polar decomposition of a signed
measure, 444 Positive integers, 2 Positive linear form, 162, 168 Positive part of a function, 152 Power series, 84, 288, 297, 353 Power set, 12 Pre-ordered set, 21 Pre-ordering relation, 21 Primitive, a.e., 248 Principle of mathematical induction,
23 of transfinite induction, 42
Product measure space, 378 Product of sets, 13
real or complex measures, 379 a-finite measures, 378 order relations, 22, 24 a-algebras, 371 topological spaces, 352
Projection mapping, 14 Proper subset, 4 Proposition, 2 Pseudometric, 118, 123
space, 118, 123
477
pth-power integrable function, 332 Purely negative subset, 428 Purely positive subset, 428
Quasicompact space, 27 4 subset, 274
Quotient mapping, 19 Quotient of a pre-order relation, 22 Quotient set, 19
Radon-Nikodym theorem, 195, 441 Range of a relation, 8 Rare subset, 303 Rational numbers, 2 Real measure, 319 Real numbers, 2
extended, 73 Real part of a complex-valued
function, 349 Real variable, function of, 33 Real-valued function, 33 Reflexive relation, 17, 20 Regularization of a function, lower
semicontinuous, 268 upper semicontinuous, 268
Relation, 6 Relative topology induced on a
subset, 138, 146, 276 Restriction of a function, 11 Reverse of a relation, 8
of an order relation, 21 Riemann integrability, Lebesgue's
criterion, 268 Riesz representation theorem,
327, 341 Riesz-Fischer theorem, 341 Right limit of a function, 144
478
Right-derivative of a function, 145 Right-differentiable function, 145 Ring of subsets, 432 Rising sun lemma, 244 Russell's paradox, 5
Schroder-Bernstein theorem, 47 Second category, subset of the, 304 Sections of a function on a product
space, 383 of a subset of a product space, 372
Self-adjoint linear mapping, 468 Semicontinuous approximations of a
Lebesgue-integrable function, 239 Semicontinuous regularization of a
function, lower, 268 upper, 268
Seminorm, 314 Seminormed space, 460
complete, 461 Separable metric space, 279 Separated space, 146, 275 Sequence, 11 Sesquilinear form, 344 u-algebra of subsets, 101 u-finite measure on an algebra of
sets, 369 u-ring of subsets, 432 Signed measure, 424
finite, 190 Similar pre-ordered sets, 23 Simple function, 154 Simple ordering, 21 Simply ordered set, 21 Singleton, 8 Singular function, 264
Lebesgue's, 210 Sphere in a metric space, 129 Stone-Weierstrass theorem, 359, 361 Subcovering, 274 Subset, 3 Subtractible functions, 423
measures, 423 Subtractive set function, 320 Sup-metric, 119, 290 Sup-norm, 119 Superset, 3 Supremum, 73
Surjection, 10 Surjective function, 10 Symmetric difference of sets, 107 Symmetric relation, 17
Index
Term-by-term differentiation, 296 Theorem on nested intervals, 32 Theorem, 3 Topological space, 131 Topology, 131
of uniform convergence, 399 Total ordering, 21 Total variation of a finite signed
measure, 198 of a function, 202 of a signed measure, 431
Totally bounded metric space, 278 Totally bounded set of functions, 399 Transfinite induction, principle of, 42 Transformation, 9 Transitive relation, 17, 20 Trichotomy, law of, 21
for cardinal numbers, 51 for ordinal numbers, 63
Trigonometric polynomials, 363 Trivial measure space, 106
relation, 18 topology on a set, 131
Tukey's lemma, 43
u.s.c., 231, 232 Ultimately, 80 Uncountable set, 38 Uniform boundedness principle, 305 Uniform convergence, 286
topology of, 399 Uniform limit, 286 Uniformly Cauchy, 286 Uniformly continuous function, 300 Union of subsets, 4, 13 Unique extension theorem, 370 Unitary space, 122 Upper bound, 22 Upper derivate of a function, 236 Upper envelope of a family of
functions, 230 Upper semicontinuous function,
231, 232
Index
Upper semicontinuous regularization of a function, 268
Upward directed partially ordered set, 77
Weierstrass M -test, 288 Weierstrass approximation
theorem, 361
Weierstrass-Bolzano property, 277 Weierstrass-Bolzano theorem, 276 Well-ordered set, 41, 59 Well-ordering of a set, 41 Well-ordering theorem, 42
Zermelo's theorem, 42 Zorn's lemma, 42
479