The theory of functions of a real variable and the theory of Fourier’s series

Ernest William Hobson9780548969373, 054896937X

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Table of contents :
Title……Page 1
CAMBRIDGE UNIVERSITY PRESS……Page 2
Title page……Page 3
PREFACE……Page 4
PREFACE TO THE FIRST EDITION……Page 5
CONTENTS……Page 10
CORRIGENDA……Page 15
1 Introduction……Page 16
2-4 Ordinal numbers……Page 18
5 Mathematical induction……Page 22
6,7 Cardinal numbers……Page 23
8-10 The operations on integral numbers……Page 26
11-13 Fractional numbers……Page 28
14,15 Negative numbers, and the number zero……Page 32
16,17 Irrational numbers……Page 34
18 Kronecker’s scheme of arithmetization……Page 37
19-22 The Dedekind theory of irrational numbers……Page 38
23-28 The Cantor theory of irrational numbers……Page 42
29 Convergent sequences of real numbers……Page 49
30-32 The arithmetical theory of limits……Page 51
33,34 Equivalence of the definitions of Dedekind and Cantor……Page 54
35 The non-existence of infinitesimals……Page 56
36-38 The theory of indices……Page 57
39,40 The representation of real numbers……Page 60
41,42 The continuum of real numbers……Page 65
43 The continuum given by intuition……Page 68
44,45 The straight line as a continuum……Page 69
46 Introduction……Page 73
47,48 The upper and lower boundaries of a linear set of points……Page 74
49 Non-linear sets of points……Page 76
50 Limiting point of a convergent sequence of intervals, or cells……Page 78
51 Systems of nets……Page 80
52-54 The limiting points and the derivatives of a set……Page 82
55 Descriptive terminology……Page 87
56 Properties of closed and open sets……Page 91
58-60 Enumerable aggregates……Page 93
61,62 The power, or cardinal number, of an aggregate……Page 98
63 The arithmetic continuum……Page 100
64-66 Transfinite ordinal numbers……Page 103
67 Properties of aggregates of closed sets……Page 109
68 The transfinite derivatives of a set of points……Page 110
69-72 Sets of intervals or cells……Page 111
73-77 The Heine-Borel theorem……Page 117
78,79 The Lebesgue chain of intervals……Page 124
80-83 Closed and perfect linear sets……Page 127
85-87 Closed sets in two or more dimensions……Page 135
88-91 The analysis of sets in general……Page 138
92 Inner and outer limiting sets……Page 143
93-96 Sets of the first, and of the second, category……Page 144
97-101 Ordinary inner limiting sets……Page 149
102-113 Plane sets of points……Page 155
114 The classification of a family of sets of points……Page 166
115 Sets of sequences of integers……Page 167
117-119 The content of a set of points……Page 169
120 The problem of measure……Page 173
121-123 The measures of open and closed sets……Page 174
124,125 The content or measure of a closed set……Page 177
126,127 The exterior and interior measures of a set……Page 180
128-131 Measurable sets of points……Page 181
132 Sets that are measurable ($B$)……Page 186
133 Congruent sets……Page 187
134 The measure of unbounded sets……Page 188
135,136 The measure of sets related to a system of sets……Page 189
137-140 The metric density of a set of points……Page 193
141 The resolution of sets of points in accordance with metrical properties……Page 197
142 Jordan’s measure of a set of points……Page 198
143 The sections of a closed set……Page 199
144 Introduction……Page 203
145 The cardinal number of an aggregate……Page 204
146,147 The relative order of cardinal numbers……Page 206
148,149 The addition and multiplication of cardinal numbers……Page 207
150 Cardinal numbers as exponents……Page 208
151,152 The smallest transfinite cardinal number……Page 209
153-155 The equivalence theorem……Page 211
156 Division of cardinal numbers by finite numbers……Page 215
157 The order-type of simply ordered aggregates……Page 218
158 The addition and multiplication of order-types……Page 219
160,161 The structure of simply ordered aggregates……Page 220
162-164 The order-types $eta$, $theta$, $pi$……Page 223
165-168 Normally ordered aggregates……Page 226
169-171 The theory of ordinal numbers……Page 231
172-174 The ordinal numbers of the second class……Page 233
175 The cardinal number of the second class of ordinals……Page 236
176,177 The general theory of aleph-numbers……Page 238
178-180 The arithmetic of ordinal numbers of the second class……Page 240
181 The theory of order-functions……Page 242
182-186 The cardinal number of the continuum……Page 243
187-193 General discussion of the theory……Page 250
194-196 The paradoxes of Burali-Forti and Russell……Page 259
197-200 The multiplicative axiom……Page 263
202 The normal ordering of an aggregate……Page 267
203, 204 The comparability of aggregates……Page 268
205 Introduction……Page 271
206, 207 The functional relation……Page 272
208 Functions of a variable aggregate……Page 277
209, 210 The upper and lower boundaries and limits of functions……Page 278
211-213 The continuity of functions……Page 281
214 Continuous functions defined for a continuous interval……Page 285
215,216 Continuous functions defined at points of a set……Page 287
217 Uniform continuity……Page 289
219 The continuity of unbounded functions……Page 291
220-223 The limits of a function at a point……Page 293
224-226 The discontinuities of functions……Page 298
227 Ordinary discontinuities……Page 301
228 The symmetry of functional limits……Page 302
229 Functions continuous in an open interval……Page 303
230-234 Semi-continuous functions……Page 305
235 Approximate continuity……Page 310
236,237 The classification of discontinuous functions……Page 311
238-240 Point-wise discontinuous functions……Page 314
241,242 Definition of point-wise discontinuous functions by extension……Page 319
243 Functions of bounded variation……Page 322
244 Function of bounded variation expressed as the difference of two monotone functions……Page 325
245-248 Functions of bounded total fluctuation……Page 326
249 Resolution of a function of bounded variation……Page 332
250,251 Rectifiable curves……Page 333
252 The variation of a function of bounded variation over a linear set of points……Page 336
253,254 Functions of two variables that are of bounded variation……Page 337
255 Quasi-monotone functions……Page 340
256-258 The maxima, minima, and lines of invariability of continuous functions……Page 341
259,260 The derivatives of functions……Page 345
261-268 The differential coefficients of continuous functions……Page 349
269 Functions with lines of invariability……Page 359
270-274 The successive differential coefficients of a continuous function……Page 360
275,276 Oscillating continuous functions……Page 366
277,278 Properties of incrementary ratios……Page 369
279-285 Properties of the derivatives of continuous functions……Page 371
286 Functions with one derivative assigned……Page 378
287-290 The construction of continuous functions……Page 379
291-300 General properties of derivatives……Page 383
301 Functions of two variables……Page 395
302-306 Double and repeated limits……Page 396
307,308 The limits of monotone functions of two variables……Page 405
309,310 Partial differential coefficients……Page 407
311-315 Higher partial differential coefficients……Page 412
316-319 Functions defined implicitly……Page 422
320,321 Maxima and minima of a function of two variables……Page 431
322-324 Properties of a function continuous with respect to each variable……Page 434
325-328 The representation of a square on a linear interval……Page 438
329 Introduction……Page 446
330 The Riemann integral in a linear interval……Page 447
331-334 The upper and lower Riemann integrals……Page 448
335 Particular cases of functions that are integrable ($R$)……Page 454
336 Geometrical interpretation of Riemann integration……Page 455
337 Properties of the definite Riemann integral……Page 457
338-340 $R$-integrals of functions of two or more variables……Page 461
341,342 Integrable null-functions and equivalent integrals……Page 466
343-349 The fundamental theorem of the Integral Calculus……Page 467
351 Integration by parts……Page 476
352,353 Cauchy’s definition of an improper integral……Page 479
354-359 Riemann integrals over an unbounded interval……Page 483
360,361 Change of the variable in a single integral……Page 491
362-365 Repeated integrals……Page 494
366-368 Improper double integrals……Page 503
369-371 The double integral over an infinite domain……Page 510
372-375 The transformation of double integrals……Page 514
376 The Riemann-Stieltjes integral……Page 521
377-381 The upper and lower Riemann-Stieltjes integrals……Page 523
383,384 Measurable functions……Page 532
385-388 The Lebesgue integral of a measurable function……Page 534
389 Other definitions of an integral……Page 541
390 The $L$-integral as the measure of a set of points……Page 543
391 The $R$-integral as an $L$-integral……Page 544
392, 393 The Lebesgue integral as a function of a set of points……Page 545
394 Equivalent $L$-integrals……Page 547
395-399 Properties of the Lebesgue integral……Page 548
400 The limits of a sequence of measurable functions……Page 552
401-403 The derivatives of a function……Page 553
404-409 Indefinite integrals……Page 555
410-414 The fundamental theorem of the Integral Calculus for a Lebesgue integral……Page 563
415 The total variation of an indefinite integral……Page 572
416,417 The generalized indefinite integral……Page 574
418,419 The indefinite integral of a function of two variables……Page 575
420 Integration by parts for the $L$-integral……Page 578
421-426 Mean value theorems……Page 579
427-429 Repeated Lebesgue integrals……Page 588
430-433 A fundamental approximation theorem……Page 593
434-436 Approximate representation of an $L$-integral as a Riemann sum……Page 600
437-439 Lebesgue integrals over an unbounded field……Page 604
440-442 Change of the independent variable in a Lebesgue integral……Page 607
443, 444 Harnack’s definition of an integral……Page 614
445-448 The Lebesgue-Stieltjes integral……Page 620
449-452 Hellinger’s integrals……Page 624
453-455 Harnack-Lebesgue integrals……Page 631
456 The $HL$-integral over a finite set of intervals……Page 635
457-461 The conditions for the existence of an $HL$-integral……Page 636
463 Integration by parts for the Harnack-Lebesgue integral……Page 643
464-466 The Denjoy integral……Page 644
467-471 The fundamental theorem of the Integral Calculus for the Denjoy integral……Page 651
472-475 Properties of the Denjoy integral……Page 662
476-478 Extensions of the definition of the Denjoy integral……Page 668
479-482 The Young integral……Page 671
LIST OF AUTHORS QUOTED……Page 680
GENERAL INDEX……Page 683