Bibliography

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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