Theory of Random Sets

Some History......Page 7
 Central Topics of the Book......Page 8
 Plan......Page 9
 Second Edition......Page 10
 Acknowledgements......Page 12
Contents......Page 13
   Measurability Definition......Page 17
   Examples of Random Closed Sets......Page 20
   Random Variables Associated with Random Closed Sets......Page 21
   Definition......Page 22
   Complete Alternation......Page 24
   Extension of the Capacity Functional......Page 26
   Complete Alternation and Monotonicity of General Functionals......Page 27
   Complete Alternation and Positive Definiteness......Page 30
   Capacity Functionals and Probability Measures on F......Page 31
   Locally Finite Measures on F......Page 32
   Maxitive Capacity Functionals......Page 34
   Independence and Conditional Distributions......Page 35
   Measure-Theoretic Proof......Page 36
   Harmonic Analysis Proof......Page 41
   Definition......Page 43
   Continuity Sets and Extension of Functionals......Page 44
   Choquet\'s Theorem for Separating Classes......Page 45
   Alternative Proof of Choquet\'s Theorem......Page 46
   Avoidance, Containment and Inclusion Functionals......Page 47
   Covariance Functions......Page 49
   Contact Distribution Functions......Page 51
   Random Sets in Finite Spaces......Page 52
   Finite-Dimensional Distributions of the Indicator Process......Page 54
   Separability......Page 56
   P-Continuity......Page 59
  1.1.8 Hitting Processes......Page 60
   Trapping Systems......Page 62
   Deterministic Incidence Functions......Page 63
   Random Weak and Strong Incidence Functions......Page 64
  1.2.1 Basic Constructions......Page 65
  1.2.2 Existence of Measures on Partially Ordered Sets......Page 66
  1.2.3 Locally Finite Measures on Posets......Page 70
  1.2.4 Existence of Random Sets Distributions......Page 72
   Effros Measurability......Page 73
   The Fundamental Measurability Theorem......Page 74
   Measurability of Special Multifunctions......Page 77
   Measurability and Tightness......Page 78
   Distributions of Random Compact Sets in Polish Spaces......Page 79
   The Borel Interpretation of the Effros σ-Algebra......Page 80
   Approximability of Random Closed Sets......Page 81
  1.3.4 Distribution of Random Closed Sets in Polish Spaces......Page 83
   Set-Theoretic Operations......Page 85
   Inverse Functions and the Infimum......Page 87
   Graph Measurable Random Sets......Page 88
   Borel Measurable Multifunctions and Random Countable Sets......Page 90
   Random Measurable Sets......Page 91
   Random Open Sets......Page 92
   The Fundamental Selection Theorem......Page 93
   Distributions and Selections......Page 94
   The Steiner Point and Selection Operators......Page 97
   Selectionability......Page 98
   Results on Matching......Page 101
   Core Determining Classes......Page 102
   Elements of the Core......Page 103
   The Capacity Functional as an Upper Probability......Page 104
   Closedness and Continuity......Page 105
   Stationary and Isotropic Random Sets......Page 106
   Ergodicity and Mixing......Page 107
   Stationary Random Sets on the Line......Page 108
   Scale Invariance......Page 109
   p-Functions......Page 110
   The Lévy Measure and Subordinator......Page 111
   Robbins\' Theorem......Page 113
   Valuations......Page 114
   Moment Measures......Page 115
   Measurability......Page 116
   Bounds on Hausdorff Dimension......Page 117
   Intersection-Equivalence and Capacity-Equivalence......Page 119
   Comparison of Random Elements......Page 120
   Stochastic Order for Random Sets......Page 121
   Application to Selections......Page 123
  1.5.6 Transformation of Capacities......Page 124
  1.5.7 Rearrangement Invariance......Page 126
   Definition......Page 128
   Subadditivity......Page 131
   Continuity......Page 132
   Comonotonic Additivity......Page 134
   Equalised Capacity Functionals......Page 135
   The Indefinite Choquet Integral......Page 136
   Absolute Continuity......Page 137
   Definition......Page 138
   Derivatives of Capacity Functionals......Page 139
   The Derivative of the Lebesgue Measure......Page 140
   Differentiation of the Choquet Integral......Page 141
   Definition and Weak Relative Compactness......Page 143
   Continuity Sets......Page 144
   Pointwise Convergence of Capacity Functionals......Page 145
   Convergence Determining Classes......Page 146
   Weak Convergence of Random Sets in Polish Spaces......Page 148
   Convergence to a Singleton......Page 151
   Definition of Almost Sure Convergence......Page 152
   Deterministic Limits......Page 153
   Convergence in Probability......Page 154
   Probability Metrics for Random Compact Sets......Page 156
   Probability Metrics for Families of Selections......Page 157
   Probability Metrics Based on Capacity Functionals......Page 158
   Weak Convergence......Page 159
   Uniform Convergence......Page 160
   C-Additivity......Page 162
   Semi-Markov Random Sets......Page 164
   Random Convex Bodies......Page 166
   Expected Intrinsic Volumes......Page 167
   Weak Convergence of Random Convex Bodies......Page 168
   Selections of Random Convex Bodies......Page 171
   Unbounded Random Convex Closed Sets in Rd......Page 172
   Lipschitz Space of Random Convex Sets......Page 174
   Lower Random Sets......Page 175
   Random Convex Cones......Page 176
   Random Counting Measures......Page 177
   Application of Choquet\'s Theorem......Page 179
   Poisson Point Process......Page 181
   Weak Convergence of Point Processes......Page 183
   Point Processes on F and K......Page 186
   Decomposition Theorem......Page 187
   Germ-Grain Model and Cutouts......Page 189
   Random Measures Associated with Random Sets......Page 191
   Intersections of Random Sets......Page 192
  1.9.4 Random Capacities......Page 194
   Carathéodory\'s Extension......Page 197
   Carathéodory\'s Construction for Random Capacities......Page 199
   Capacity Version of Robbins\' Theorem......Page 200
   Intrinsic Density......Page 201
   Upper Bound on Hausdorff Dimension......Page 203
   The Core of a Non-additive Measure......Page 204
   Decomposition of Non-additive Measures......Page 205
   Choquet and Sugeno Integrals......Page 206
   Definitions......Page 208
   Symmetric Upper Probabilities......Page 209
   Upper and Lower Integrals......Page 213
   Belief and Plausibility Functions......Page 214
   Updating Belief Functions......Page 215
   Likelihood-Based Belief Function......Page 216
   The Neyman–Pearson Lemma......Page 217
   Application of Belief Functions......Page 219
 Notes to Chap.1......Page 220
 2.1 The Selection Expectation and Aumann Integral......Page 240
   Integrable Random Closed Sets......Page 241
   Properties of Integrable Selections......Page 242
   Decomposable Sets......Page 247
   Existence of Integrable Selections......Page 249
   Weak Compactness for Integrable Selections......Page 250
   Integral Functionals......Page 251
   Definitions......Page 253
   Convexification in Euclidean Space......Page 254
   Convexification in Banach Space......Page 258
   Properties of the Selection Expectation......Page 259
   The Debreu Expectation......Page 261
   Selection Expectation and the Support Function......Page 262
   The Aumann Integral......Page 263
   Selection Expectation in Rd......Page 265
   Degenerate Expectation......Page 266
   Selections with Given Properties......Page 267
   Characterisation of Distributions......Page 268
   Zonoids......Page 269
   Multivalued Measures......Page 273
   Reduced Selection Expectation......Page 274
   Translative Expectation......Page 276
   Firey p-Expectation......Page 277
   Fatou\'s Lemma for Integrably Bounded Random Sets in Rd......Page 278
   Fatou\'s Lemma for Unbounded Random Sets......Page 280
   Approximate Fatou\'s Lemma: Infinite-Dimensional Case......Page 283
   Monotone and Weak Convergence......Page 284
   Existence......Page 285
   Properties of the Conditional Expectation......Page 286
   Convergence of Conditional Expectations......Page 291
   Generalised Conditional Expectation......Page 292
   The Linearisation Approach......Page 293
   Evaluations and Expectations on Lattices......Page 295
   Properties of Expectations and Characterisation Results......Page 296
   Indicator and Coverage Functions......Page 297
   The Vorob\'ev Expectation as Minimiser......Page 299
   The Vorob\'ev Median......Page 300
   Distance Functions......Page 301
   Mean Distance Function and Distance Average......Page 302
  2.2.4 The Radius-Vector Expectation......Page 305
   The Fréchet Expectation......Page 306
   The Doss Expectation in Metric Spaces......Page 308
   The Doss Expectation for Random Sets and the Herer Expectation......Page 310
   Doss Convexity......Page 312
  2.2.6 Convex Combination Spaces......Page 313
   Sublinear and Superlinear Expectations of Random Variables......Page 314
   Set-Valued Nonlinear Expectations......Page 316
   The Support Function Approach......Page 317
   Superlinear Expectation of Lower Sets......Page 319
 Notes to Chap.2......Page 321
   The Shapley–Folkman–Starr Theorem......Page 332
   Convexification in Banach Spaces......Page 334
   Euclidean Case......Page 335
   SLLN in Banach Space......Page 338
   The Marcinkiewicz–Zygmund Law......Page 339
   Komlós\' Theorem......Page 340
   The Brunn–Minkowski Theorem......Page 341
   Translations of Quasiconcave Functions......Page 343
   Rounding of Deterministic Sets and Symmetrisation......Page 344
   Random Determinants......Page 345
   Convergence of Functionals......Page 346
   Convergence of Random Broken Lines......Page 347
   Allocation Problems......Page 348
   The Law of Large Numbers for Capacities......Page 349
  3.1.4 Non-identically Distributed Summands......Page 350
   Conditions of Uniform Integrability Type......Page 351
   Euclidean Case......Page 354
   SLLN in the Hausdorff Metric......Page 355
   SLLN in the Mosco and Wijsman Topologies......Page 356
   The Euclidean Case......Page 359
   Non-square Integrably Bounded Summands......Page 363
   The CLT in Banach Space......Page 364
   Lipschitz Functionals......Page 365
   A Characterisation Theorem in Rd......Page 366
   Non-compact Gaussian Random Sets......Page 367
   Compound Poisson Law and Lévy Measure......Page 369
   Characterisation of Infinite Divisibility......Page 370
   Definition and Characterisation......Page 372
   Strictly Stable Sets in Rd and the LePage Series Representation......Page 373
   Domain of Attraction......Page 375
  3.3.1 Convergence of Series......Page 376
   The Multivariate Renewal Theorem......Page 378
   The Containment Renewal Function for Random Sets......Page 379
   Further Renewal Functions......Page 381
  3.3.3 Ergodic Theorems......Page 382
   Uniformly Bounded Case......Page 384
   Large Deviation Principle......Page 385
   Regularly Varying Heavy-Tailed Case......Page 386
 Notes to Chap.3......Page 388
   Definition......Page 394
   Fixed Points of a Random Closed Set......Page 395
   A Characterisation Theorem......Page 396
   Representation by a Poisson Process......Page 399
  4.1.2 Scheme of Series for Unions of Random Closed Sets......Page 401
  4.1.3 Infinite Divisibility of Lattice-Valued RandomElements......Page 403
   Univariate Extreme Values......Page 407
   Definition of Stability for Unions......Page 408
   Characterisation of Distributions for Union-Stable Sets......Page 409
   Examples of Union-Stable Random Sets......Page 413
   LePage Series for Unions......Page 415
   Different Scaling Factors......Page 419
   Affine Normalisation......Page 420
   Additive Normalisation......Page 422
  4.2.1 Stability of Limits......Page 424
   Regular Variation on F......Page 425
   Random Points and Multifunctions......Page 430
  4.2.3 Necessary Conditions......Page 432
   Inequalities Between Metrics......Page 435
   Ideal Metrics and Convergence Rates......Page 437
  4.3.1 Regularly Varying Capacities......Page 439
  4.3.2 Almost Sure Convergence of Scaled Unions......Page 441
   Star-Shaped Limits......Page 444
   Singletons and Functions of Singletons......Page 446
  4.3.4 Functionals of Unions......Page 447
  4.4.1 Infinite Divisibility for Convex Hulls......Page 449
   Distribution and Containment Functionals......Page 452
   Containment Functionals and the LePage Series......Page 453
   Characterisation of Distribution......Page 455
   The Polar Transform of Convex-Stable Sets......Page 456
   Intersections of Random Half-Spaces......Page 457
 Notes to Chap.4......Page 459
   Definition and Main Properties......Page 465
   Convergence of Multivalued (Sub- and Super-)Martingales......Page 468
   Martingale Selections......Page 473
   Optional Sampling......Page 475
   Measurability, Separability, and Selections......Page 476
   Weak Convergence......Page 479
   Graphical Convergence......Page 481
   Set-Valued Step Functions......Page 482
   Uniform Laws of Large Numbers......Page 485
   Set-Valued Markov Processes......Page 487
   Stationary Processes......Page 488
   Increasing Set-Valued Processes......Page 489
   Random Differential Inclusions......Page 491
   Set-Valued Stochastic Integrals......Page 492
   Set-Valued Shot-Noise Processes......Page 494
   Upper Semicontinuous Functions......Page 496
   Random Open Domains......Page 497
   Multivalued Operators with Stochastic Domain......Page 498
   Allocation Problems......Page 499
   Smooth Random Functions on R......Page 500
   Gaussian Random Fields......Page 501
   Paths and Hitting Times......Page 503
  5.2.2 Random Subsets of the Positive Half-Line andFiltrations......Page 505
  5.2.3 Level Sets of Strong Markov Processes......Page 508
   Regenerative Sets......Page 509
   Subordinators and Local Time......Page 510
   Weak Convergence, Intersections and Embedding......Page 513
   Set-Indexed Filtration......Page 515
   Stopping Set......Page 516
   Epi-Convergence......Page 517
   Argmin Sets and Their Convergence......Page 519
   Normal Integrands......Page 521
   Conjugate and Subdifferential......Page 525
   Weak Epi-Convergence......Page 526
   The M_2-Topology......Page 528
   Epigraphical Representation of Additive Functionals......Page 530
   Epi-Convergence of Averages......Page 532
   Minimisation of Expectations......Page 534
   Convergence of Maximum Likelihood Estimators......Page 535
   Pointwise Extremes......Page 537
   Max-Stability of Functions......Page 538
   Argmax Sets......Page 540
   Continuous Choice Processes......Page 541
   Epi-Convergence of Support Functions......Page 542
  5.3.5 Increasing Set-Valued Processes of Excursion Sets......Page 546
   Operations with Epigraphs......Page 548
   Selection Expectation of Normal Integrands......Page 549
   Law of Large Numbers and Its Application to the Allocation Problem......Page 550
   Deterministic Level Sums and Convergence......Page 551
   Random Level Sums......Page 553
 Notes to Chap.5......Page 554
 A Topological Spaces and Metric Spaces......Page 567
 B Linear Spaces......Page 574
 C Space of Closed Sets......Page 580
 D Compact Sets and the Hausdorff Metric......Page 585
 E Multifunctions and Semicontinuity......Page 593
 F Measures and Probabilities......Page 597
 G Capacities......Page 604
 H Convex Sets......Page 609
 I Semigroups, Cones and Harmonic Analysis......Page 616
 J Regular Variation......Page 620
References......Page 626
Name Index......Page 661
Subject Index......Page 670
List of Notation......Page 685