Handbook of measure theory

Front cover......Page 1
Date-line......Page 2
Preface......Page 3
List of Contributors......Page 5
Contents......Page 7
CHAPTER 1 History of Measure Theory (Djura Paunic)......Page 11
1. Beginnings......Page 13
2. The Greeks......Page 14
3. Archimedes......Page 16
4. Infinitesimal methods......Page 18
5. Loss of measure......Page 24
6. New beginning......Page 28
7. Newly found measure......Page 31
References......Page 35
CHAPTER 2 Some Elements of the Classical Measure Theory (Endre Pap)......Page 37
Introduction......Page 39
1.1. Classes of sets......Page 40
1.2. Set functions......Page 42
1.4. Measurable functions......Page 44
2.1. Positive measures......Page 45
2.2. Measure spaces. Extension of measures......Page 46
2.3. Completion of a measure space......Page 48
2.4. Measures with finite variation......Page 49
2.5. The variation of real-valued measures on a ring......Page 50
2.6. Measurability with respect to a positive measure......Page 51
2.7. Examples of measures......Page 53
2.8. Lebesgue measure on $\mathbb{R}^r$......Page 57
3. Integration......Page 60
3.1. The immediate integral......Page 61
3.2. Integral of positive step functions......Page 63
3.3. Integral of positive functions......Page 64
3.4. The classical integral......Page 65
3.5. Lebesgue and Riemann integrals......Page 69
3.6. Integration with respect to a real-valued measure......Page 71
3.8. Absolutely continuous functions......Page 73
3.10. The Radon-Nikodym theorem......Page 75
3.11. Conditional expectation......Page 77
3.13. Transformation of coordinates......Page 78
4.1. Essentially bounded functions......Page 79
4.2. The space $L^p(\mu)$......Page 80
4.3. Convergence of sequences of functions......Page 81
5.1. Product measures......Page 82
5.2. Repeated (iterated) integral......Page 83
5.3. Infinite products......Page 85
5.5. Integral transforms......Page 87
6.1. Topological measures......Page 89
References......Page 91
CHAPTER 3 Paradoxes in Measure Theory (Miklos Laczkovich)......Page 93
Introduction......Page 95
1. Paradoxical sets......Page 96
2. Paradoxes in $\mathbb{R}^n$ for $n \geq 3$ and in non-euclidean spaces......Page 99
3. Invariant measures and amenable groups......Page 102
4. Decompositions and perfect matchings......Page 107
5. The type semigroup......Page 108
6. Nonamenable actions and local commutativity......Page 111
7. Marczewski's problem......Page 118
8. Tarski's circle-squaring problem......Page 120
9. The problem of equidecomposability with measurable pieces......Page 125
10. Countable equidecomposability and countably additive invariant measures......Page 127
11. The nonobstructive element in the paradoxes......Page 129
References......Page 130
CHAPTER 4 Convergence Theorems for Set Functions (Paolo de Lucia, Endre Pap)......Page 135
1. Introduction......Page 137
2.1. Exhaustive set functions......Page 139
2.2. Cafiero uniform exhaustivity theorem......Page 140
2.3. Nikodym convergence theorem......Page 142
2.4. Vitali-Hahn-Saks theorem......Page 143
2.5. Nikodym boundedness theorem......Page 145
2.6. Drewnowski lemma......Page 147
2.7. Cafiero theorem for additive set functions......Page 148
2.9. Vitali-Hahn-Saks theorem for additive case......Page 149
2.10. Counter-examples on algebras......Page 153
2.11. Algebras with SCP and SIP......Page 154
3.2. Orlicz-Pettis theorem......Page 156
3.3. Schur lemma and Phillips lemma......Page 159
3.4. Rosenthal's lemma......Page 160
3.5. Hewitt-Yosida theorem......Page 162
3.6. Biting Lemma......Page 163
4. The relation between boundedness and exhaustiveness......Page 164
5. Diagonal theorem for triangular set function......Page 166
5.1. $\alpha$-boundedness......Page 167
5.2. Diagonal theorems......Page 170
5.3. SCP and SIP algebras......Page 172
6. Dieudonne type theorems......Page 173
6.1. Triangular set functions......Page 174
6.2. Convergence theorem......Page 178
7.1. Nikodym type theorems for lattice-valued measures......Page 179
7.2. Convergences of measures on orthomodular posets......Page 180
7.3. General Nikodym type theorems......Page 181
References......Page 184
CHAPTER 5 Differentiation (Brian S. Thomson)......Page 189
2. Introduction......Page 191
3. Differentiation in $\mathbb{R}^n$......Page 193
4. Some motivation......Page 194
5.1. Covering relations......Page 196
5.2. Derivation bases......Page 197
5.3. The limit operation......Page 198
5.4. The dual basis......Page 200
5.5. Some properties of the dual......Page 202
5.6. The variation......Page 203
5.7. Growth estimates......Page 204
5.8. Differentiation under strong Vitali assumptions......Page 208
5.9. Topological considerations......Page 210
5.10. Differential equivalence......Page 212
6.1. A general construction......Page 213
6.3. Measure contraction......Page 214
6.6. Federer's scheme......Page 215
6.8. Bases associated with a lifting......Page 216
7. Derivation bases in a metric space......Page 217
7.1. Differentiation of the integral......Page 218
7.2. The density property......Page 222
8. Differentiation under strong Vitali assumptions......Page 228
9. Strong Vitali conditions......Page 232
9.1. Strong Vitali property......Page 233
9.2. Classical proofs of the Vitali theorem......Page 234
9.3. (Q)-Property......Page 236
9.4. Besicovitch-Morse property......Page 238
10. Weak Vitali covering properties......Page 240
11. Derivation bases in $\mathbb{R}$......Page 241
11.1. The ordinary derivative......Page 242
11.3. The sharp derivative......Page 247
11.5. The approximate derivative......Page 248
12.1. The cube basis......Page 249
12.3. The interval basis......Page 250
12.4. Rectangle basis......Page 251
12.6. Star bases......Page 252
13. De La Vallee Poussin theorem......Page 253
15. Some further remarks......Page 254
References......Page 255
CHAPTER 6 Radon-Nikodym Theorems (Domenico Candeloro, Aljosa Volcic)......Page 259
1. Introduction......Page 261
2. The $\sigma$-additive case......Page 265
3. The finitely additive case......Page 269
4. The Banach-valued case......Page 275
5. Finitely additive Banach-valued measures......Page 285
6. Further results......Page 292
Appendix. Singularity and decomposition theorems......Page 300
References......Page 302
CHAPTER 7 One-Dimensional Diffusions and Their Convergence in Distribution (James K. Brooks)......Page 305
Introduction......Page 307
1.2. Mappings of Brownian motion......Page 308
1.3. Markov times and the strong Markov property......Page 310
1.4. Further properties of Brownian motion......Page 311
2.1. The setting......Page 314
2.2. Definitions and properties......Page 315
2.3. The scale function......Page 316
2.4. Speed measure......Page 317
2.5. Local time for Brownian motion......Page 319
2.6. Time changing Brownian motion......Page 320
3.1. Preliminaries......Page 323
3.2. The general convergence theorem......Page 324
3.3. Convergence of diffusions......Page 331
4.1. Stretched Brownian motion......Page 340
4.2. Diffusions as a limit of stretched random walks......Page 345
4.3. Natural time......Page 350
References......Page 352
CHAPTER 8 Vector Integration in Banach Spaces and Application to Stochastic Integration (Nicolae Dinculeanu)......Page 355
Introduction......Page 357
1.2. Measurable functions......Page 358
1.3. Integral of step functions......Page 359
1.4. Measurability with respect to a positive measure......Page 360
2.1. The seminorm......Page 361
2.2. Bochner integrability......Page 362
2.3. The Bochner integral......Page 363
2.4. The spaces $L^p_F(\mu)$......Page 364
3.1. Measures with finite variation......Page 366
3.2. Integration with respect to a measure with finite variation......Page 367
3.3. The indefinite integral......Page 369
3.4. The Radon-Nikodym theorem......Page 371
4.1. The semivariation......Page 374
4.2. Semivariation and norming spaces......Page 375
4.3. Semivariation of $\sigma$-additive measures......Page 376
4.5. Extension of measures......Page 377
4.6. Extension of positive measures......Page 378
4.8. Canonical extensions......Page 379
4.9. Canonical extension of additive measures......Page 380
5.1. Measurability with respect to a vector measure......Page 381
5.2. The seminorm $\tilde{m}_{F,G}(f)$......Page 382
5.3. The space Point $\mathcal{F}_D(\tilde{m}_{F,G})$......Page 384
5.4. The integral......Page 385
5.5. Convergence theorems......Page 386
5.6. The indefinite integral of measures with finite semivariation......Page 387
5.7. Integral representation of linear operations on $L^p$-spaces......Page 388
6. The Stieltjes integral......Page 389
6.1. The variation and the semivariation of a function......Page 390
6.2. Semivariation and norming spaces......Page 391
6.3. The measure associated to a function......Page 392
6.4. The Stieltjes integral......Page 393
7.1. Notations and definitions......Page 396
7.2. The measure $I_X$. Summable processes......Page 397
7.3. The stochastic integral......Page 398
7.4. Convergence theorems......Page 400
7.6. Local summability and local integrability......Page 401
8.1. Processes with finite variation or semivariation......Page 402
8.2. Optional and predictable stochastic measures......Page 403
8.3. The measure $\mu_X$......Page 404
8.4. Summability of processes with integrable variation or integrable semivariation......Page 405
9. Martingales......Page 406
References......Page 408
CHAPTER 9 The Riesz Theorem (Joe Diestel, Johan Swart)......Page 411
Introduction......Page 413
1.1. A brief historical discussion......Page 414
1.3. The abstract characterization of $C(K)$ as a Banach algebra......Page 416
1.4. The Josefson-Nissenzweig Theorem......Page 418
2.1. The dual of the injective tensor product of two Banach spaces......Page 421
2.2. Choquet's representation theorem......Page 423
2.3. The Stone-Weierstrass theorem......Page 427
2.4. The Pietsch domination theorem......Page 429
3.1. Weakly compact operators on $C(K)$......Page 431
3.2. The Dunford-Pettis property and property V......Page 432
3.3. Absolutely summing, integral and nuclear operators on $C(K)$......Page 434
4.The Riesz theorem for vector-valued continuous function spaces......Page 440
5.Notes and remarks......Page 445
5.1. Notes and remarks to Section 1......Page 446
5.2. Notes and remarks to Section 2......Page 447
5.3. Notes and remarks to Section 3......Page 449
5.4. Notes and remarks to Section 4......Page 450
5.5. Notes and remarks on tensor products......Page 451
References......Page 454
CHAPTER 10 Stochastic Processes and Stochastic Integration in Banach Spaces (James K. Brooks)......Page 459
Introduction......Page 461
1.1. Notation and definitions......Page 462
1.2. The stochastic measure......Page 463
1.3. The summability question......Page 464
1.4. The stochastic integral......Page 468
1.5. Closures of subsets in $\mathcal{F}_{F,G}(X)$......Page 471
1.6. Completeness of the space $L^1_{F,G}(X)$......Page 472
1.7. Stopping the stochastic integral......Page 474
1.8. Convergence theorems in $L^1_{F,G}(X)$......Page 476
1.9. Weak completeness and compactness of $L^1_{F,G}(\mathcal{B},X)$......Page 477
1.10. Local summability and local integrability......Page 478
1.11. Ito's formula......Page 479
1.12. Further extensions of the stochastic integral......Page 487
Introduction......Page 488
2.1. Notation......Page 489
2.3. The mean variation of $X$ and quasimartingales......Page 490
2.4. Regularity of quasimartingales......Page 491
2.5. Cadlag modification without RNP......Page 503
2.6. The Doob-Meyer decomposition of quasimartingales......Page 506
A.1. The Doleans function......Page 507
A.2. Mean variation......Page 508
A.3. Quasimartingales......Page 509
A.4. Right continuous quasimartingales......Page 510
References......Page 511
CHAPTER 11 Daniell Integral and Related Topics (M. Diaz Carrillo)......Page 513
1.1. Daniell systems of functions and their first extensions......Page 515
1.2. Daniell upper and lower integrals. Summable functions......Page 516
1.3. Characterisations and extensions of integrable functions......Page 518
1.4. Generalisations and related topics......Page 521
2.1. Measures induced by integrals. Daniell-Stone theorem......Page 524
2.2. Related results and applications......Page 526
2.3. The interplay between measure and topology. Riesz representation theorem......Page 529
3. The abstract Fubini theorem......Page 531
4. The Radon-Nikodym theorem......Page 532
5. Integral norms. Local integral metrics and Daniell-Loomis integrals......Page 534
References......Page 536
CHAPTER 12 Pettis Integral (Kazimierz Musial)......Page 541
Introduction......Page 543
1. Preliminaries......Page 544
2. Measurable functions......Page 545
3. Scalar integrals, basic properties......Page 548
4. Pettis integral......Page 551
5. Limit theorems......Page 560
6. The range of the Pettis integral......Page 563
7. Universal integrability......Page 567
8. Pettis integral property......Page 568
9. Weak Radon-Nikody'm property and related properties......Page 570
10. Conditional expectation......Page 578
11. Differentiation......Page 580
12. Fubini theorem......Page 582
13.1. Space of all Pettis integrable functions......Page 583
13.2. Functions satisfying the strong law of large numbers......Page 585
13.3. Functions with integrals of bounded variation......Page 587
13.4. $LLN(\mu, X)$ equipped with the variation norm of integrals......Page 588
13.5. Bounded Pettis integrable functions......Page 589
References......Page 590
CHAPTER 13 The Henstock-Kurzweil Integral (Benedetto Bongiorno)......Page 597
1. Partitions and Riemann sums......Page 599
2. The Henstock-Kurzweil integral on the real line......Page 600
2.1. The primitives......Page 602
2.2. Convergence theorems......Page 603
3.2. The $C$-integral......Page 606
3.3. Integrals induced by differentiation bases......Page 607
4. Multidimensional Riemann-type integrals......Page 609
4.1. The Henstock-Kurzweil integral......Page 610
4.2. The $\alpha$-regular Henstock-Kurzweil integral......Page 611
4.4. The $\rho$-integral......Page 612
4.5. The divergence theorem......Page 613
5. The Henstock-Kurzweil integral of vector valued functions......Page 618
6. The Henstock-Kurzweil integral on general spaces......Page 619
References......Page 620
CHAPTER 14 Set-Valued Integration and Set-Valued Probability Theory: An Overview (Christian Hess)......Page 627
1. Introduction......Page 629
2. Notations and preliminaries......Page 630
3. Integration of strongly measurable multifunctions......Page 632
4. Weakly measurable multifunctions and graph-measurable multifunctions......Page 639
5. The Aumann integral......Page 644
6. The set-valued conditional expectation of closed valued multifunctions......Page 649
7. Set-valued measures......Page 653
8. The probability distribution of a measurable multifunction......Page 657
9.1. Convergence in the Hausdorff metric topology......Page 662
9.2. Convergence in the sense of Painleve-Kuratowski......Page 664
10. Set-valued martingales......Page 668
11. Gaussian multifunctions. The set-valued Central Limit Theorem......Page 670
12. Set-valued versions of the Fatou Lemma......Page 672
13. Epigraphical convergence......Page 673
14. Concluding remarks......Page 675
References......Page 676
CHAPTER 15 Density Topologies (Wladyslaw Wilczynski)......Page 685
Introduction......Page 687
1. Points of density of linear sets......Page 688
2. Density topology......Page 692
3. Approximately continuous functions......Page 696
4. Density topology in Euclidean space......Page 701
5. $\Psi$-density topologies on the real line......Page 703
References......Page 710
CHAPTER 16 FN-Topologies and Group-Valued Measures (Hans Weber)......Page 713
Introduction......Page 715
1. Definition and generation of FN-topologies......Page 716
2. Exhaustivity......Page 719
3. $\sigma$-submeasures and completeness......Page 723
4. $\sigma$-order continuity and order continuity......Page 726
5.1. Extension of $\sigma$-order continuous FN-topologies and $\sigma$-additive measures......Page 729
5.2. Extension of exhaustive FN-topologies and exhaustive measures......Page 734
6. The completion of topological Boolean rings and the system FN$_c(R)$ of exhaustive FN-topologies......Page 736
7. More on the $\mu$-topology......Page 742
8. Decomposition of exhaustive measures......Page 744
9. Connectedness......Page 748
10. Vitali-Hahn-Saks and Nikodym theorems......Page 749
References......Page 751
CHAPTER 17 On Products of Topological Measure Spaces (Stratos Grekas)......Page 755
1. A central problem in topological measure theory......Page 757
1.2. On product measures......Page 758
1.3. Fremlin's example and related results......Page 761
2. On the 'product-like' structure of the Haar measure on a compact group......Page 762
3. Topological liftings for product measures......Page 765
3.1. Baire, Borel and Strong liftings......Page 766
3.2. On the existence of Baire strong liftings......Page 767
References......Page 772
CHAPTER 18 Perfect Measures and Related Topics (Doraiswamy Ramachandran)......Page 775
0. Preliminaries......Page 777
1. Compact measures......Page 779
2. Perfect measures......Page 781
3. Measures on product spaces......Page 783
4. The marginal problem......Page 785
5. Monge-Kantorovich duality spaces......Page 786
6. Regular conditional probabilities......Page 788
7. Independence and Blackwell spaces......Page 789
8. Standard measure spaces......Page 790
10. Disintegrations......Page 791
References......Page 793
CHAPTER 19 Riesz Spaces and Ideals of Measurable Functions (Martin Vath)......Page 797
1. Preliminaries......Page 799
2.1. Riesz spaces......Page 800
2.2. Dedekind completeness and support......Page 802
2.3. Order convergence and topology......Page 806
2.4. Bibliographical remarks......Page 807
3.1. Ideals and bands in Riesz spaces......Page 808
3.2. Ideals of measurable functions......Page 810
3.3. Comparison of various sorts of convergence......Page 813
4.1. Preideal spaces......Page 815
4.2. Fatou property and perfectness......Page 818
4.3. Completeness......Page 820
4.4. Convergence theorems......Page 821
4.5. Bibliographical remarks......Page 823
5.1. The associate space......Page 825
5.2. Applications to ideal spaces on product measures......Page 827
5.3. Order duals of a Riesz space......Page 828
5.4. Duals of ideals of measurable functions......Page 830
References......Page 832
CHAPTER 20 Measures on Quantum Structures (Anatolij Dvurecenskij)......Page 837
1. Introduction......Page 839
2.1. Orthomodular lattices and orthomodular posets......Page 841
2.2. Orthoalgebras and effect algebras......Page 843
2.3. Decompositions of states......Page 846
3.1. Introduction......Page 850
3.2. Generalizations of Gleason's theorem......Page 852
3.3. Gleason's theorem and completeness criteria of inner product spaces......Page 858
4. States on MV-algebras......Page 863
5.1. Commutative BCK-algebras with the relative cancellation property......Page 866
5.2. Measures on commutative BCK-algebras with the relative cancellation property......Page 868
6.1. Pseudo MV-algebras......Page 870
6.2. States and pseudo MV-algebras......Page 873
6.3. Existence of states......Page 875
References......Page 876
CHAPTER 21 Probability on MV-Algebras (Beloslav Riecan, Daniele Mundici)......Page 879
Prologue: Operator algebras and MV-algebras......Page 881
1.1. MV-algebras: basic notions......Page 882
1.2. Representation of semisimple MV-algebras......Page 883
1.3. MV-algebras and $l$-groups......Page 884
1.4. Tribes and $\sigma$-complete MV-algebras......Page 886
1.5. The Loomis-Sikorski theorem for MV-algebras......Page 887
1.6. Bibliographical remarks......Page 888
2.1. States and observables in MV-algebras......Page 889
2.2. Independence......Page 893
2.3. Kolmogorov's construction......Page 894
2.4. Functions of several observables......Page 895
2.5. The MV-algebraic central limit theorem......Page 896
2.6. MV-algebraic laws of large numbers......Page 898
2.7. Bibliographical remarks......Page 899
3. MV-algebras with product......Page 900
3.1. Product and joint observables......Page 901
3.2. Conditional expectation......Page 903
3.3. Upper and lower limits......Page 905
3.4. Individual ergodic theorem......Page 907
3.5. Bibliographical remarks......Page 908
4. Finitely additive measures......Page 909
4.1. Basics......Page 910
4.2. Entropy of dynamical systems......Page 911
4.3. Entropy of full tribes......Page 913
4.4. Bibliographical remarks......Page 914
5. Open problems......Page 915
References......Page 916
CHAPTER 22 Measures on Clans and on MV-Algebras (Giuseppina Barbieri and Hans Weber)......Page 921
Introduction......Page 923
1.1. Basic properties of MV-algebras......Page 924
1.2. The centre of an MV-algebra......Page 926
1.3. Representation of MV-algebras......Page 928
1.5. Decomposition of complete MV-algebras......Page 929
2. Submeasures on MV-algebras......Page 931
3.1. The space $ba(L)$......Page 933
4.1. The lattice $\mathcal{LUA}(L)$......Page 937
4.2. Decomposition of order continuous uniform MV-algebras......Page 940
4.3. The lattice $\mathcal{LUA}(L,w)$......Page 943
4.4. The uniformity generated by a measure......Page 944
5.1. Representation of measures......Page 946
5.2. Decomposition theorems......Page 948
5.3. Hammer-Sobczyk's decomposition theorem and Lyapunov's theorem......Page 949
5.4. Vitali-Hahn-Saks-Nikodym theorem and Nikodym boundedness theorem......Page 951
5.5. Extension of measures......Page 953
References......Page 954
CHAPTER 23 Triangular Norm-Based Measures (Dan Butnariu, Erich Peter Klement)......Page 957
2. Triangular norms, fuzzy subsets......Page 959
3. $T$-tribes......Page 966
4. $T$-measures and their representation by Markov kernels......Page 968
5. Integral representation of $T_L$ -measures......Page 974
6. Integral representation of monotone $T_\lambda$-measures......Page 980
7. Decomposition of monotone $T_\lambda$-measures......Page 983
8. Jordan decomposition of bounded $T_L$-measures......Page 993
9. Jordan decomposition of finite $T_L$-measures......Page 998
10. Absolute continuity of $T_L$-measures......Page 1001
11. Vector $T_L$-measures with Darboux property......Page 1002
12. Nonatomic $T_L$-measures......Page 1007
13. A Liapounoff type theorem for $T_L$-measures......Page 1014
14. A Liapounoff type theorem for $T_\lambda$ -measures......Page 1016
References......Page 1017
CHAPTER 24 Geometric Measure Theory: Selected Concepts, Results and Problems (Miroslav Chlebik)......Page 1021
1. Preliminaries from measure theory......Page 1023
2. Structure theory for integral dimensional sets......Page 1027
3.1. Preiss's theorem......Page 1030
3.2. Related results and problems......Page 1032
3.3. Besicovitch $\frac{1}{2}$-problem......Page 1033
4. Sets of finite perimeter......Page 1035
5. Measure-theoretic calculus of variations......Page 1037
5.1. Normal and integral currents......Page 1038
5.2. Closure and compactness theorem......Page 1043
5.3. Regularity of minimal surfaces......Page 1044
References......Page 1045
CHAPTER 25 Fractal Measures (Kenneth J. Falconer)......Page 1047
2.1. Hausdorff measures......Page 1049
2.2. Packing measures......Page 1051
2.3. Calculation of Hausdorff and packing measures and dimensions......Page 1052
2.4. Geometric measure theory......Page 1054
2.5. Geometry of Hausdorff measures and dimensions......Page 1056
3.1. Local dimensions......Page 1057
3.2. Dimension decomposition......Page 1058
3.3. Multifractal measures......Page 1059
References......Page 1061
CHAPTER 26 Positive and Complex Radon Measures in Locally Compact Hausdorff Spaces (T.V. Panchapagesan)......Page 1065
1. Preliminaries......Page 1067
2. Regular extensions of (positive) measures......Page 1069
3. Complex Radon measures and their properties......Page 1072
4. Regular extensions of positive and complex measures......Page 1078
5. Bounded complex Radon measures......Page 1081
6. Characterizations of complex Radon measures......Page 1083
7. Isomorphic representations of $\mathcal{K}(T)^st$, $\mathcal{K}(T,\mathbb{R})^st$, $\mathcal{K}(T)_b^st$, $\mathcal{K}(T,\mathbb{R})_b^st$......Page 1087
8. Applications......Page 1091
9. Generalization to Radon vector measures......Page 1096
References......Page 1100
CHAPTER 27 Measures on Algebraic-Topological Structures (Piotr Zakrzewski)......Page 1101
Introduction......Page 1103
1. Invariant measures on arbitrary $G$-spaces......Page 1104
2. Nonmeasurable sets for invariant measures......Page 1112
3. Extensions of invariant measures......Page 1117
4. Invariant measures on Polish groups......Page 1119
5. Invariant measures on Polish $G$-spaces......Page 1124
6. Isometrically invariant measures on Euclidean spaces......Page 1133
References......Page 1137
CHAPTER 28 Liftings (Werner Strauss, Nikolaos D. Macheras, Kazimierz Musiat)......Page 1141
1. Terminology......Page 1143
2. Existence of liftings and densities......Page 1145
3. Liftings for functions......Page 1150
4. Liftings on topological spaces......Page 1157
5. Liftings on topological groups......Page 1164
6.1. Liftings in products......Page 1167
6.2. Strong liftings in products......Page 1171
6.3. Liftings on projective limits......Page 1173
6.4. Various Fubini products......Page 1176
6.5. Applications of Fubini products to stochastic processes......Page 1178
7. Liftings for abstract valued functions......Page 1180
8. Liftings and densities with respect to ideals of sets......Page 1183
9. Beyond $\mathcal{L}^\infty(\mu)$......Page 1184
10. Further applications......Page 1185
References......Page 1188
CHAPTER 29 Ergodic Theory (Frank Blume)......Page 1195
Introduction......Page 1197
1. Basic examples......Page 1199
2. Ergodic theorems......Page 1202
3. Ergodicity......Page 1206
4. Recurrence......Page 1210
5. Mixing......Page 1213
6. More about convergence in ergodic theory......Page 1220
7. Entropy and information......Page 1224
8. Constructions in ergodic theory......Page 1236
References......Page 1241
CHAPTER 30 Generalized Derivatives (Endre Pap, Arpad Takaci)......Page 1247
1.1. Introduction......Page 1249
1.2. Sobolev spaces......Page 1252
1.3. Imbedding theorems and traces of functions......Page 1254
2.1. Introduction......Page 1256
2.2. Distribution spaces......Page 1258
2.3. Fundamental solutions......Page 1261
3.2. Operators......Page 1264
References......Page 1269
CHAPTER 31 Real Valued Measurability, Some Set-Theoretic Aspects (Aleksandar Jovanovic)......Page 1271
Introduction......Page 1273
0. Notation......Page 1274
1. RVM - first equiconsistency......Page 1278
2. Cardinal monotony......Page 1284
3. Nonregular ultrafilter......Page 1287
4. Rudin-Keisler order......Page 1289
5. Measures and normal Moore spaces......Page 1291
6. Counting remarks......Page 1293
7. Some complementary results......Page 1296
8. Set-theoretic measure theory......Page 1298
References......Page 1300
CHAPTER 32 Nonstandard Analysis and Measure Theory (Peter A. Loeb)......Page 1305
2. Extending the real number system......Page 1307
3. Calculus......Page 1310
5. Set-theoretic measure theory......Page 1313
6. A lattice approach to measure theory......Page 1315
7. Internal functionals on continuous functions......Page 1322
9. Representing measures in potential theory......Page 1324
10. Poisson process......Page 1326
11. Anderson's Brownian motion......Page 1327
12. The martingale convergence theorem......Page 1329
13. On an infinite number of independent random variables......Page 1332
14. Exact law of large numbers and independence......Page 1336
References......Page 1337
CHAPTER 33 Monotone Set Functions-Based Integrals (Pietro Benvenutia, Radko Mesiarb, Doretta Vivonaa)......Page 1339
1. Introduction......Page 1341
2.1. Preliminaries......Page 1343
2.2. Basic and simple functions......Page 1344
2.3. Choquet integral......Page 1346
2.4. Sugeno integral......Page 1350
2.5. Similarity of Choquet and Sugeno integrals......Page 1352
3.1. Pseudo-addition $\oplus$......Page 1353
3.3. Pseudo-multiplication $\odot$......Page 1355
3.4. Additional conditions with pseudo-ring structures......Page 1358
4. General fuzzy integral......Page 1362
4.1. Integral of simple functions......Page 1364
4.2. Integral of measurable functions......Page 1368
4.3. Additional properties of integral with pseudo-ring operations......Page 1371
4.4. Extension of the integral to $\[-F, F\]$......Page 1373
5. Examples......Page 1374
6. Conclusions......Page 1386
References......Page 1387
CHAPTER 34 Set Functions over Finite Sets: Transformations and Integrals (Michel Grabisch)......Page 1391
2. Set functions over finite sets......Page 1393
3. Transformations of set functions......Page 1395
4. The Choquet and Sipos integrals......Page 1400
5. $k$-additive measures......Page 1402
6. The ordinal case: the Sugeno integral......Page 1405
References......Page 1410
CHAPTER 35 Pseudo-Additive Measures and Their Applications (Endre Pap)......Page 1413
Introduction......Page 1415
1.1. Pseudo operations......Page 1416
1.2. Pseudo-additive measures......Page 1418
1.3. Pseudo-integral......Page 1423
1.4. Pseudo-convolution......Page 1425
1.5. The Riesz type theorem......Page 1432
2.1. Probabilistic metric spaces......Page 1434
2.2. Fuzzy numbers......Page 1436
2.3. Information theory......Page 1438
2.5. Optimization and morphism with the probability......Page 1440
2.6. Pseudo-linear operators......Page 1443
3.1. Pseudo-operations......Page 1446
3.2. Measures and integrals......Page 1448
4.1. Hamilton-Jacobi equation with non-smooth Hamiltonian......Page 1450
4.2. Bellman differential equation for multicriteria optimization problems......Page 1453
4.3. Stochastic optimization......Page 1455
4.4. Option pricing......Page 1458
5. Non-commutative and non-associative pseudo-operations......Page 1459
5.1. Applications on nonlinear PDE......Page 1462
5.2. Corresponding pseudo-measure......Page 1464
6.2. Hybrid probability-possibilistic measure and integral......Page 1468
6.3. Hybrid utility function......Page 1473
References......Page 1475
CHAPTER 36 Qualitative Possibility Functions and Integrals (Didier Dubois and Henri Prade)......Page 1479
1. Introduction......Page 1481
2.1. Basic postulates of confidence relations......Page 1482
2.2. Basic examples of confidence relations......Page 1483
2.3. Representing confidence relations by set-functions......Page 1485
3. Qualitative possibility theory......Page 1487
3.1. Possibility distributions......Page 1488
3.2. Information content of a possibility distribution......Page 1489
3.3. Interpretation of possibility distributions......Page 1491
3.4. Refined possibility relations......Page 1493
4. Conditional possibility and plausible inference......Page 1496
4.1. Qualitative conditioning......Page 1497
4.2. Plausible inference with a possibility distribution......Page 1498
4.3. Universal possibilistic entailment......Page 1501
4.4. Probabilistic interpretations of plausible inference......Page 1504
5.1. The paradoxes of probabilistic independence......Page 1506
5.2. The minimum rule in possibility theory......Page 1508
5.3. Possibilistic independence between variables......Page 1509
5.4. Possibilistic event independence based on conditioning......Page 1512
5.5. Qualitative independence and belief revision......Page 1515
6. Qualitative integrals......Page 1516
6.1. Possibilistic integrals......Page 1517
6.2. Axiomatics for possibilistic integrals......Page 1520
6.3. Sugeno integrals......Page 1523
6.4. Comparing functions without commensurateness......Page 1525
References......Page 1528
CHAPTER 37 Measures of Information (Wolfgang Sander)......Page 1533
Introduction......Page 1535
1. Probabilistic and non-probabilistic information measures......Page 1536
2. Branching inset information measures......Page 1541
3. Recursivity and generalized additivity......Page 1544
4. Recursive measures of multiplicative type......Page 1546
5. Regular weighted $(l,m)$-additive measures of degree $(\alpha,\beta)$......Page 1550
6. Subadditive information measures for random vectors......Page 1553
7. Information measures in a theory of evidence......Page 1556
8. Measures of fuzziness......Page 1560
9. Weighted entropies......Page 1565
10. Summary......Page 1571
References......Page 1572
Author Index......Page 1577
Subject Index......Page 1597