Principles of analysis: measure, integration, functional analysis, and applications
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| Format: | Book |
| Language: | English |
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Boca Raton
CRC Press, Taylor & Francis Group
2018
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XX, 520 Seiten |
| ISBN: | 9781498773287 |
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| 100 | 1 | |a Junghenn, Hugo D. |d 1939- |0 (DE-588)1089835477 |4 aut | |
| 245 | 1 | 0 | |a Principles of analysis |b measure, integration, functional analysis, and applications |c Hugo D. Junghenn |
| 246 | 1 | 3 | |a Principles of real analysis |
| 264 | 1 | |a Boca Raton |b CRC Press, Taylor & Francis Group |c 2018 | |
| 300 | |a XX, 520 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Mathematical analysis |v Textbooks | |
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| _version_ | 1804178408214626304 |
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| adam_text | Contents
Preface xix
0 Preliminaries 1
0.1 Sets ......................................................................... 1
Set Operations............................................................ 1
Number Systems............................................................ 2
Relations................................................................. 3
Functions................................................................. 4
Cardinality............................................................... 6
0.2 Algebraic Structures........................................................ 7
Semigroups and Groups................................................... 7
Linear Spaces............................................................. 8
Linear rPransformations................................................... 9
Quotient Linear Spaces................................................... 10
Algebras................................................................. 10
0.3 Metric Spaces ............................................................... 10
Open and Closed Sets ................................................... 11
Interior, Closure, and Boundary......................................... 12
Sequential Convergence. Completeness .................................... 12
Continuity . .......................................................... 13
Category............................................................... 14
0.4 Normed Linear Spaces......................................................... 15
Norms and Seminorms...................................................... 15
Banach Spaces ........................................................... 15
Completion of a Normed Space ............................................ 16
Infinite Series in Normed Spaces ........................................ 16
Unordered Sums in Normed Spaces.......................................... 17
Bounded Linear Transformations........................................... 18
Banach Algebras.......................................................... 18
0.5 Topological Spaces........................................................... 19
Open and Closed Sets .................................................... 19
Neighborhood Systems..................................................... 20
Neighborhood Bases ...................................................... 20
Relative Topology........................................................ 21
Nets................................................................... 21
0.6 Continuity in Topological Spaces............................................. 23
Definition and General Properties...................................... 23
Initial Topologies....................................................... 24
Product Topology......................................................... 24
Final Topologies ........................................................ 24
Quotient Topology........................................................ 25
vu
viii Contents
The Space of Continuous Functions...................................... 25
F-sigma and G-delta Sets............................................... 25
0.7 Normal Topological Spaces ................................................ 26
Urysohn’s Lemma ....................................................... 26
Tietze Extension Theorem............................................... 27
0.8 Compact Topological Spaces ............................................... 27
Convergence in Compact Spaces.......................................... 28
Compactness of Cartesian Products...................................... 29
Continuity and Compactness............................................. 29
0.9 Totally Bounded Metric Spaces............................................. 30
0.10 Equicontinuity ........................................................... 31
0.11 The Stone-Weierstrass Theorem ............................................ 32
0.12 Locally Compact Topological Spaces........................................ 33
General Properties..................................................... 33
Functions with Compact Support ........................................ 34
Functions That Vanish at Infinity...................................... 35
The One-Point Compactification......................................... 35
0.13 Spaces of Differentiable Functions ....................................... 36
0.14 Partitions of Unity....................................................... 37
0.15 Connectedness ............................................................ 39
I Measure and Integration 41
1 Measurable Sets 43
1.1 Introduction ........................................................... 43
1.2 Measurable Spaces......................................................... 44
Fields and Sigma Fields................................................ 44
Generated Sigma Fields................................................. 45
Borel Sets............................................................. 45
Extended Borel Sets.................................................... 46
Product Sigma Fields................................................... 46
Pi-Systems and Lambda-Systems ......................................... 47
Exercises.............................................................. 48
1.3 Measures ................................................................. 50
Set Functions.......................................................... 50
Properties and Examples of Measures.................................... 51
Exercises.............................................................. 52
1.4 Complete Measure Spaces................................................... 54
Completion Theorem..................................................... 54
Null Sets.............................................................. 55
Exercises............................................................ 55
1.5 Outer Measure and Measurability .......................................... 55
Construction of an Outer Measure ...................................... 56
Caratheodory}s Theorem................................................. 56
Exercises.............................................................. 58
1.6 Extension of a Measure ................................................... 58
The Measure Extension Theorem.......................................... 59
Approximation Property of the Extension................................ 61
Completeness of the Extension.......................................... 61
Uniqueness of the Extension............................................ 62
Exercises.............................................................. 63
Contents
IX
1.7 Lebesgue Measure ........................................................... 63
The Volume Set Function.................................................. 63
Construction of the Measure . ........................................... 65
Exercises................................................................ 65
1.8 Lebesgue-Stieltjes Measures................................................. 66
Regularity............................................................... 66
One-Dimensional Distribution Functions................................... 67
* Higher Dimensional Distribution Functions.........,.................. 69
Exercises................................................................ 70
*1.9 Some Special Sets .......................................................... 71
An Uncountable Set with Lebesgue Measure Zero............................ 71
Non-Lebesgue-Measurable Sets........................................... 71
A Lebesgue Measurable, Non-Borel Set..................................... 72
Exercises................................................................ 73
2 Measurable Functions 75
2.1 Measurable Transformations ................................................ 75
General Properties....................................................... 75
Exercises................................................................ 77
2.2 Measurable Numerical Functions.............................................. 78
Criteria for Measurability............................................... 78
Almost Everywhere Properties........................................... 79
Combinatorial and Limit Properties of Measurable Functions............... 79
Exercises................................................................ 81
2.3 Simple Functions............................................................ 82
A Fundamental Convergence Theorem........................................ 82
Applications ............................................................ 83
Exercises................................................................ 84
2.4 Convergence of Measurable Functions ........................................ 85
Modes of Convergence..................................................... 85
Relationships Among the Modes of Convergence............................. 86
Exercises.............................................................. 87
3 Integration 89
3.1 Construction of the Integral................................................ 89
Integral of a Nonnegative Simple Function................................ 89
Integral of a Real-Valued Function....................................... 90
Integral of a Complex-Valued Function.................................... 91
Integral over a Measurable Set........................................... 91
3.2 Basic Properties of the Integral............................................ 92
Almost Everywhere Properties............................................. 92
Monotone Convergence Theorem ............................................ 93
Linearity of the Integral................................................ 93
Integration Against an Image Measure .................................... 96
Integration Against a Measure with Density............................... 96
Change of Variables Theorem.............................................. 97
Exercises.............................................................. 97
3.3 Connections with the Riemann Integral on .................................. 100
The Darboux Integral.................................................... 101
The Riemann Integral.................................................... 103
Measure Zero Criterion for Riemann Integrability........................ 104
X
Contents
Improper Riemann Integrals.............................................. 106
Exercises............................................................... 107
3.4 Convergence Theorems ...................................................... 108
The General Monotone Convergence Theorem................................ 108
Fatou’s Lemma........................................................... 109
The Dominated Convergence Theorem ...................................... 109
Exercises............................................................... 110
3.5 Integration against a Product Measure ..................................... Ill
Construction of the Product of Two Measures ............................ Ill
Fubini’s Theorem........................................................ 112
The d-Dimensional Case.................................................. 114
Exercises............................................................... 115
3.6 Applications of Fubini’s Theorem .......................................... 116
Gaussian Density........................................................ 116
Integration by Parts.................................................... 116
Spherical Coordinates .................................................. 118
Volume of a d-Dimensional Ball.......................................... 118
Integration of Radial Functions......................................... 119
Surface Area of a d-Dimensional Ball.................................... 120
Exercises............................................................... 121
4 Lp Spaces 123
4.1 Definition and General Properties ......................................... 123
The Case 1 p oo..................................................... 123
The Case p — oo......................................................... 126
The Case 0 p 1 127
Iv -Spaces.............................................................. 127
Exercises............................................................... 128
4.2 Lv Approximation........................................................... 129
Approximation by Simple Functions....................................... 129
Approximation by Continuous Functions................................... 130
Approximation by Step Functions......................................... 131
Exercises............................................................... 131
4.3 Lp Convergence ............................................................ 131
Exercises............................................................... 133
*4.4 Uniform Integrability ..................................................... 133
Exercises............................................................... 135
*4.5 Convex Functions and Jensen’s Inequality................................... 136
Exercises............................................................... 138
5 Differentiation 139
5.1 Signed Measures ........................................................... 139
Definition and a Fundamental Example.................................... 139
The Hahn-Jordan Decomposition........................................... 140
Exercises............................................................... 142
5.2 Complex Measures........................................................... 143
The Total Variation Measure ............................................ 144
The Vitali-Hahn-Saks Theorem............................................ 145
The Banach Space of Complex Measures.................................... 146
Integration against a Signed or Complex Measure......................... 147
Exercises............................................................... 147
Contents
xi
5.3 Absolute Continuity of Measures............................................. 148
General Properties of Absolute Continuity ............................... 148
The Radon-Nikodym Theorem ............................................... 149
Lebesgue-Decomposition of a Measure.................................... 152
Exercises................................................................ 153
5.4 Differentiation of Measures ................................................ 154
Definition and Properties of the Derivative.............................. 154
Connections with the Classical Derivative................................ 156
Existence of the Measure Derivative...................................... 157
Exercises................................................................ 159
5.5 Functions of Bounded Variation ............................................. 159
Definition and Basic Properties.......................................... 159
The Total Variation Function............................................. 161
Differentiation of Functions of Bounded Variation........................ 162
Exercises................................................................ 163
5.6 Absolutely Continuous Functions............................................ 164
Definition and Basic Properties.......................................... 164
Fundamental Theorems of Calculus......................................... 165
Exercises................................................................ 167
6 Fourier Analysis on Rd 169
6.1 Convolution of Functions ................................................... 169
Definition and Basic Properties.......................................... 169
Approximate Identities................................................. . 170
Exercises................................................................ 171
6.2 The Fourier Transform ...................................................... 171
Definition and Basic Properties.......................................... 171
The Fourier Inversion Theorem............................................ 172
Exercises................................................................ 174
6.3 Rapidly Decreasing Functions ............................................... 174
Definition and Basic Properties.......................................... 174
The Plancherel Theorem................................................... 176
Exercises................................................................ 177
6.4 Fourier Analysis of Measures on Rd ......................................... 178
Convolution of Measures.................................................. 178
The Fourier-Stieltjes Transform.......................................... 179
Exercises................................................................ 180
7 Measures on Locally Compact Spaces 181
7.1 Radon Measures ............................................................. 181
Definition and Basic Properties.......................................... 181
Consequences of Regularity .............................................. 182
The Space of Complex Radon Measures...................................... 182
The Support of a Radon Measure........................................... 183
Exercises................................................................ 184
7.2 The Riesz Representation Theorem ........................................... 184
Exercises................................................................ 187
7.3 Products of Radon Measures ................................................. 188
Finitely Many Measures................................................... 188
Infinitely Many Measures ................................................ 189
Exercises................................................................ 190
Contents
Xll
7.4 Vague Convergence .............................................. 191
Exercises............................................................. 192
*7.5 The Daniell-Stone Representation Theorem................................... 193
II Functional Analysis 197
8 Banach Spaces 199
8.1 General Properties of Normed Spaces ..................................... 199
Topology and Geometry................................................. 200
Separable Spaces.................................................... 201
Equivalent Norms ..................................................... 201
Finite Dimensional Spaces............................................. 202
* Strictly Convex Spaces............................................... 203
Exercises............................................................. 205
8.2 Bounded Linear Transformations........................................... 206
The Operator Norm..................................................... 207
The Banach Algebra 8 (9C)............................................. 208
The Dual Space SC*.................................................... 208
Bilinear Transformations ............................................ 208
Exercises............................................................. 209
8.3 Concrete Representations of Dual Spaces ................................. 210
The Dual of Co ... *.............................................. 210
The Dual of C..................................................... 210
The Dual of Lp.................................................... 211
The Dual of Co(X)................................................... 212
Exercises ....................................................... 213
8.4 Some Constructions ...................................................... 214
Product Spaces........................................................ 214
Direct Sums ........................................................ 215
Quotient Spaces ...................................................... 216
Exercises............................................................ 217
8.5 Hahn-Banach Extension Theorems........................................... 218
Real Version.......................................................... 218
Complex Version....................................................... 219
Normed Space Version.................................................. 220
The Bidual of a Normed Space ......................................... 221
* Invariant Versions................................................... 221
Exercises............................................................. 222
*8.6 Applications of the Hahn-Banach Theorem ................................... 222
The Moment Problem.................................................... 222
Invariant Means..................................................... 223
Banach Limits......................................................... 224
Invariant Set Functions............................................... 224
Exercises............................................................. 225
8.7 Baire Category in Banach Spaces....................................... 225
The Uniform Boundedness Principle .................................... 225
The Open Mapping Theorem.............................................. 226
The Closed Graph Theorem.............................................. 228
Exercises............................................................. 228
*8.8 Applications ............................................... 229
Divergent Fourier Series.............................................. 229
Contents
Vector-Valued Analytic Functions....................................... 231
Summability ........................................................... 231
Schauder Bases......................................................... 232
Exercises.............................................................. 233
8.9 The Dual Operator . . . .................................................. 234
Definition and Properties ............................................. 234
Annihilators ........................................................ 234
Duals of Quotient Spaces and Subspaces................................. 235
Exercises.............................................................. 236
8.10 Compact Operators ........................................................ 236
* Fredholm Alternative for Compact Operators........................... 238
Exercises ............................................................. 240
9 Locally Convex Spaces 241
9.1 General Properties........................................................ 241
Geometry and Topology.................................................. 241
Seminormed Spaces ..................................................... 242
Frechet Spaces......................................................... 244
Exercises.............................................................. 246
9.2 Continuous Linear Functionals ............................................ 246
Continuity on Topological Vector Spaces ............................... 246
Continuity on Locally Convex Spaces.................................... 248
Continuity on Finite Dimensional Spaces................................ 248
Exercises............................................................ 248
9.3 Hahn-Banach Separation Theorems .......................................... 249
Weak Separation in a TVS .............................................. 249
Strict Separation in a LCS............................................. 249
Some Consequences of the Separation Theorems........................... 250
The Bipolar Theorem ................................................... 252
Exercises.............................................................. 252
*9.4 Some Constructions ....................................................... 252
Product Spaces......................................................... 252
Quotient Spaces ....................................................... 253
Strict Inductive Limits................................................ 254
Exercises.............................................................. 255
10 Weak Topologies on Normed Spaces 257
10.1 The Weak Topology......................................................... 257
Definition and General Properties...................................... 257
Weak Sequential Convergence............................................ 258
Convexity and Closure.................................................. 259
* Application: Weak Bases.............................................. 260
Exercises.............................................................. 261
10.2 The Weak* Topology ....................................................... 262
Definition and General Properties...................................... 262
The Dual of ........................................................... 262
The Banach-Alaoglu Theorem............................................. 263
* Application: Means on Function Spaces................................ 263
Weak* Continuity....................................................... 264
* The Closed Range Theorem............................................. 265
Exercises.............................................................. 266
xiv Contents
10.3 Reflexive Spaces .......................................................... 267
Examples and Basic Properties .......................................... 267
Weak Compactness and Reflexivity........................................ 268
Exercises.................,........................................... 268
*10.4 Uniformly Convex Spaces................................................... 269
Definition and General Properties....................................... 269
Connections with Strict Convexity....................................... 270
Weak and Strong Convergence............................................. 270
Connection with Reflexivity............................................. 271
Exercises............................................................... 271
11 Hilbert Spaces 273
11.1 General Principles ........................................................ 273
Sesquilinear Forms...................................................... 273
Semi-Inner-Product Spaces............................................... 274
Inner Product Spaces. Hilbert Spaces.................................... 275
Isomorphisms of Hilbert Spaces.......................................... 277
Exercises............................................................... 278
11.2 Orthogonality ............................................................. 278
Orthogonal Complements ................................................. 278
The Riesz Representation Theorem........................................ 280
Exercises............................................................... 280
11.3 Orthonormal Bases ........................................................ 281
The Dimension of a Hilbert Space........................................ 283
The Gram-Schmidt Process ............................................. 283
Fourier Series ......................................................... 284
Exercises............................................................... 285
11.4 The Hilbert Space Adjoint ............................................... 286
Bounded Sesquilinear Functionals........................................ 286
The Lax-Milgram Theorem................................................. 287
Definition and Properties of the Adjoint................................ 287
as a C*-algebra.................................................. 288
Exercises............................................................... 288
12 Operator Theory 289
12.1 Classes of Operators....................................................... 289
Normal Operators ....................................................... 289
Self-Adjoint Operators.................................................. 289
Positive Operators...................................................... 291
Orthogonal Projections and Idempotents ................................. 292
Unitary Operators....................................................... 293
* Partial Isometries..................................................... 294
Exercises........................................................... 295
12.2 Compact Operators and Operators of Finite Rank............................. 296
Rank One Operators...................................................... 297
An Approximation Theorem.............................................. 297
Exercises.............................................................. 298
12.3 The Spectral Theorem for Compact Normal Operators ......................... 299
Eigenvalues and Eigenvectors.......................................... 299
Diagonalizable Operators................................................ 299
The Spectral Theorem.................................................... 301
Contents
xv
Exercises............................................................... 303
*12.4 Hilbert-Schmidt Operators ................................................. 303
The Hilbert-Schmidt Norm................................................ 303
The Hilbert-Schmidt inner Product....................................... 304
The Hilbert-Schmidt Operator A® B..................................... 306
Hilbert-Schmidt Integral Operators...................................... 307
Exercises............................................................. 309
*12.5 Trace Class Operators.................................................... 309
The Trace Norm........................................................ . 309
The Trace............................................................... 311
The Dual Spaces SoW and 313
Exercises............................................................. 314
13 Banach Algebras 315
13.1 Introduction .............................................................. 315
Definitions and Examples................................................ 315
The Group of Invertible Elements........................................ 316
The Cauchy Product of Series............................................ 317
Exercises............................................................... 318
13.2 Spectral Theory ........................................................... 319
The Spectrum of an Element............................................ 319
The Spectral Radius Formula ............................................ 320
Normal Elements in a C* -Algebra........................................ 321
Exercises............................................................... 323
13.3 The Spectrum of an Algebra ................................................ 324
Characters.............................................................. 324
Maximal Ideals.......................................................... 324
Exercises............................................................... 326
13.4 Gelfand Theory ............................................................ 326
The Representation Theorem ............................................. 326
Application: The Stone-Ôech Compactification.......................... 327
Application: Wiener’s Theorem........................................... 328
Exercises............................................................... 329
*13.5 The Non-unital Case........................................................ 329
The Unitization of a Banach Algebra . .................................. 329
The Non-unital Representation Theorem................................... 330
The Spectrum of Cq(X)................................................... 330
The Spectrum of Ll(Rd).................................................. 331
Exercises............................................................... 332
13.6 Operator Calculus ......................................................... 333
The Continuous Functional Calculus..................................... 333
Applications to Operators on Hilbert Space.............................. 334
The Borel Functional Calculus........................................... 336
The Spectral Theorem for Normal Operators............................... 338
Exercises . ............................................................ 339
14 Miscellaneous Topics 341
14.1 Weak Sequential Compactness ............................................... 341
The Eberlein-Smulian Theorem............................................ 342
14.2 Weak Compactness in L1 .................................................... 344
Weak Convergence and Uniform Integrability.............................. 344
xvi Contents
The Dunford-Pettis Theorem ........................................... 345
14.3 Convexity and Compactness ................................................. 346
The Krein-Smulian Theorem............................................... 346
Mazur’s Theorem......................................................... 347
The Finite Dimensional Case........................................... 347
14.4 Extreme Points............................................................. 348
Definitions and Examples................................................ 348
The Krein-Milman Theorem................................................ 350
14.5 Applications of the Krein-Milman Theorem................................... 352
Existence of Ergodic Measures.......................................... 352
The Stone-Weierstrass Theorem........................................... 353
The Banach-Stone Theorem................................................ 354
The Lyapunov Convexity Theorem.......................................... 355
The Ryll-Nardzewski Fixed Point Theorem............................... 356
14.6 Vector-Valued Integrals ................................................... 357
Weak Integrals in Banach Spaces......................................... 358
Weak Integrals in Locally Convex Spaces.............................. 360
The Bochner Integral.................................................... 361
14.7 Choquet’s Theorem ....................................................... 364
III Applications 367
15 Distributions 369
15.1 General Theory ............................................................ 369
The Frechet Space C%(U)................................................. 369
The Spaces 1 (U) andDf (U).............................................. 370
Examples of Distributions............................................... 370
15.2 Operations on Distributions................................................ 371
Derivative of a Locally Integrable Function............................. 371
Derivative of a Distribution............................................ 371
Multiplication by a Smooth Function .................................... 372
Composition with Linear Maps ........................................... 372
15.3 Distributions with Compact Support......................................... 372
15.4 Convolution of Distributions .............................................. 374
15.5 Tempered Distributions..................................................... 377
The Fourier Transform of a Tempered Distribution........................ 379
15.6 Sobolev Theory ............................................................ 380
Sobolev Spaces.......................................................... 380
Application: Elliptic PDEs.............................................. 381
Sobolev Inequalities.................................................... 382
16 Analysis on Locally Compact Groups 385
16.1 Topological Groups ........................................................ 385
Definitions and Basic Properties........................................ 385
Translation and Uniform Continuity ..................................... 386
16.2 Haar Measure .............................................................. 387
Definition and Basic Properties......................................... 387
Existence of Haar Measure............................................... 389
Essential Uniqueness of Haar Measure.................................... 391
The Modular Function.................................................... 392
16.3 Some Constructions ........................................................ 394
Contents xvii
Haar Measure on Direct Products........................................ 394
Haar Measure on Semidirect Products.................................... 394
Haar Measure on Quotient Groups........................................ 396
16.4 The Lx-Group Algebra ..................................................... 397
Convolution and Involution............................................. 397
Approximate Identities................................................. 399
The Measure Algebra.................................................... 400
16.5 Representations .......................................................... 401
Positive-Definite Functions ........................................... 401
Functions of Positive Type............................................. 402
Unitary Representations................................................ 403
Irreducible Representations ........................................... 406
Unitary Representations of Compact Groups.............................. 408
16.6 Locally Compact Abelian Groups ........................................... 411
The Dual Group......................................................... 411
Bochner’s Theorem...................................................... 415
The Inversion Theorem ................................................. 416
The Plancherel Theorem................................................. 419
The Pontrjagin Duality Theorem......................................... 420
17 Analysis on Semigroups 423
17.1 Semigroups with Topology ................................................. 423
17.2 Weakly Almost Periodic Functions.......................................... 424
Definition and Basic Properties...................................... 424
The Dual of the Space of Weakly Almost Periodic Functions............. 424
The Weakly Almost Periodic Compactification............................ 426
Invariant Means on Weakly Almost Periodic Functions ................... 428
17.3 Almost Periodic Functions ................................................ 429
Definition and Basic Properties........................................ 429
The Almost Periodic Compactification .................................. 430
17.4 The Structure of Compact Semigroups....................................... 431
Ellis’s Theorem........................................................ 431
Existence of Idempotents............................................... 432
Ideal Structure........................................................ 432
17.5 Strongly Almost Periodic Functions ....................................... 433
Definition and Basic Properties........................................ 433
The Strongly Almost Periodic Compactification.......................... 435
17.6 Semigroups of Operators .................................................. 437
Definitions and Basic Properties....................................... 437
Dynamical Properties of Semigroups of Operators ....................... 438
Ergodic Properties of Semigroups of Operators.......................... 441
18 Probability Theory 443
18.1 Random Variables ......................................................... 443
Expectation and Variance............................................... 443
Probability Distributions............................................ 444
18.2 Independence.............................................................. 446
Independent Events..................................................... 446
Independent Random Variables........................................... 446
18.3 Conditional Expectation .................................................. 448
18.4 Sequences of Independent Random Variables ................................ 449
xviii Contents
Infinite Product Measures.............................................. 450
The Distribution of a Sequence of Random Variables..................... 452
Zero-One Laws.......................................................... 453
Laws of Large Numbers ................................................. 455
The Central Limit Theorem.............................................. 458
The Individual Ergodic Theorem......................................... 459
Stationary Processes................................................... 462
18.5 Discrete-Time Martingales ................................................ 463
Filtrations ........................................................... 464
Definition and General Properties of Martingales....................... 464
Stopping Times. Optional Sampling...................................... 466
Upcrossings............................................................ 468
Convergence of Martingales............................................. 469
Reversed Martingales................................................... 471
18.6 General Stochastic Processes ............................................. 472
The Consistency Conditions............................................. 472
The Product of Measurable Spaces....................................... 473
The Kolmogorov Extension Theorem ...................................... 474
18.7 Brownian Motion .......................................................... 476
Construction of Brownian Motion........................................ 477
Non-Differentiability of Brownian Paths ............................... 481
Variation of Brownian Paths ........................................... 482
Brownian Motion as a Martingale........................................ 483
18.8 Stochastic Integration.................................................... 484
The Ito Integral of a Step Process..................................... 484
The General Ito Integral............................................... 486
The Ito Integral as a Martingale....................................... 487
18.9 An Application to Finance ................................................ 488
The Stock Price Process................................................ 489
Self-Financing Portfolios.............................................. 489
Call Options........................................................... 490
The Black-Scholes Option Price......................................... 490
IV Appendices 493
A Change of Variables Theorem 495
B Separate and Joint Continuity 501
References 505
List of Symbols 509
Index
511
|
| any_adam_object | 1 |
| author | Junghenn, Hugo D. 1939- |
| author_GND | (DE-588)1089835477 |
| author_facet | Junghenn, Hugo D. 1939- |
| author_role | aut |
| author_sort | Junghenn, Hugo D. 1939- |
| author_variant | h d j hd hdj |
| building | Verbundindex |
| bvnumber | BV044874647 |
| callnumber-first | Q - Science |
| callnumber-label | QA331 |
| callnumber-raw | QA331.5 |
| callnumber-search | QA331.5 |
| callnumber-sort | QA 3331.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 400 |
| ctrlnum | (OCoLC)1039837385 (DE-599)BVBBV044874647 |
| dewey-full | 515/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.8 |
| dewey-search | 515/.8 |
| dewey-sort | 3515 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV044874647 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:03:30Z |
| institution | BVB |
| isbn | 9781498773287 |
| language | English |
| lccn | 017061660 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030269004 |
| oclc_num | 1039837385 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XX, 520 Seiten |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| spelling | Junghenn, Hugo D. 1939- (DE-588)1089835477 aut Principles of analysis measure, integration, functional analysis, and applications Hugo D. Junghenn Principles of real analysis Boca Raton CRC Press, Taylor & Francis Group 2018 XX, 520 Seiten txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Functions of real variables Textbooks Mathematical analysis Textbooks Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030269004&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Junghenn, Hugo D. 1939- Principles of analysis measure, integration, functional analysis, and applications Functions of real variables Textbooks Mathematical analysis Textbooks Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Principles of analysis measure, integration, functional analysis, and applications |
| title_alt | Principles of real analysis |
| title_auth | Principles of analysis measure, integration, functional analysis, and applications |
| title_exact_search | Principles of analysis measure, integration, functional analysis, and applications |
| title_full | Principles of analysis measure, integration, functional analysis, and applications Hugo D. Junghenn |
| title_fullStr | Principles of analysis measure, integration, functional analysis, and applications Hugo D. Junghenn |
| title_full_unstemmed | Principles of analysis measure, integration, functional analysis, and applications Hugo D. Junghenn |
| title_short | Principles of analysis |
| title_sort | principles of analysis measure integration functional analysis and applications |
| title_sub | measure, integration, functional analysis, and applications |
| topic | Functions of real variables Textbooks Mathematical analysis Textbooks Analysis (DE-588)4001865-9 gnd |
| topic_facet | Functions of real variables Textbooks Mathematical analysis Textbooks Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030269004&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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