Integral Representation Theory (eBook)

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Jaroslav Lukeš, Jan Malý, Ivan Netuka and Jiri Spurný, Charles University, Prague, Czech Republic.

Introduction 6
Contents 12
Prologue 18
1.1 The Korovkin theorem 18
1.2 Notes and comments 20
Compact convex sets 21
2.1 Geometry of convex sets 22
2.2 Interlude: On the space M ( K ) 39
2.3 Structures in convex sets 43
2.4 Exercises 57
2.5 Notes and comments 66
Choquet theory of function spaces 69
3.1 Function spaces 70
3.2 More about Korovkin theorems 81
3.3 On the H - barycenter mapping 83
3.4 The Choquet representation theorem 84
3.5 In-between theorems 87
3.6 Maximal measures 90
3.7 Boundaries and the Simons lemma 95
3.8 The Bishop–de Leeuw theorem 98
3.9 Minimum principles 101
3.10 Orderings and dilations 103
3.11 Exercises 112
3.12 Notes and comments 122
Affine functions on compact convex sets 124
4.1 Affine functions and the barycentric formula 124
4.2 Barycentric theorem and strongly affine functions 130
4.3 State space and representation of affine functions 137
4.4 Affine Baire-one functions on dual unit balls 144
4.5 Exercises 146
4.6 Notes and comments 150
Perfect classes of functions and representation of affine functions 152
5.1 Generation of sets and functions 153
5.2 Baire and Borel sets 159
5.3 Baire and Borel mappings 163
5.4 Perfect classes of functions 166
5.5 Affinely perfect classes of functions 167
5.6 Representation of H - affine functions 171
5.7 Exercises 176
5.8 Notes and comments 183
Simplicial function spaces 185
6.1 Basic properties of simplicial spaces 186
6.2 Characterizations of simplicial spaces 193
6.3 Simplicial spaces as L1-preduals 195
6.4 The weak Dirichlet problem and Ac(H)-exposed points 197
6.5 The Dirichlet problem for a single function 199
6.6 Special classes of simplicial spaces 202
6.7 The Daugavet property of simplicial spaces 213
6.8 Choquet simplices 215
6.9 Restriction of function spaces 221
6.10 Exercises 222
6.11 Notes and comments 230
Choquet theory of function cones 233
7.1 Function cones 233
7.2 Maximal measures 239
7.3 Representation theorem 241
7.4 Simplicial cones 244
7.5 Ordered compact convex sets and simplicial measures 249
7.6 Exercises 257
7.7 Notes and comments 260
Choquet-like sets 261
8.1 Split and parallel faces 261
8.2 H - extremal and H - convex sets 263
8.3 Choquet sets, M -sets and P -sets 267
8.4 H - exposed sets 274
8.5 Weak topology on boundary measures 276
8.6 Characterizations of simpliciality by Choquet sets 279
8.7 Exercises 285
8.8 Notes and comments 290
Topologies on boundaries 291
9.1 Topologies generated by extremal sets 291
9.2 Induced measures on Choquet boundaries 295
9.3 Functions continuous in ext and max topologies 301
9.4 Strongly universally measurable functions 305
9.5 Facial topology generated by M -sets 313
9.6 Exercises 320
9.7 Notes and comments 325
Deeper results on function spaces and compact convex sets 327
10.1 Boundaries 328
10.2 Isometries of spaces of affine continuous functions 337
10.3 Baire measurability and boundedness of affine functions 340
10.4 Embedding of `1 352
10.5 Metrizability of compact convex sets 355
10.6 Continuous affine images 368
10.7 Several topological results on Choquet boundaries 375
10.8 Convex Baire-one functions 382
10.9 Function spaces with continuous envelopes 387
10.10 Exercises 395
10.11 Notes and comments 401
Continuous and measurable selectors 406
11.1 The Lazar selection theorem 406
11.2 Applications of the Lazar selection theorem 411
11.3 The weak Dirichlet problem for Baire functions 415
11.4 Pointwise approximation of maximal measures 417
11.5 Measurable selectors 419
11.6 Exercises 429
11.7 Notes and comments 433
Constructions of function spaces 436
12.1 Products of function spaces 437
12.2 Inverse limits of function spaces 457
12.3 Several examples 472
12.4 Exercises 494
12.5 Notes and comments 503
Function spaces in potential theory and the Dirichlet problem 506
13.1 Balayage and the Dirichlet problem 508
13.2 Boundary behavior of solutions 513
13.3 Function spaces and cones in potential theory 521
13.4 Dirichlet problem: solution methods 534
13.5 Generalized Dirichlet problem and uniqueness questions 554
13.6 Exercises 563
13.7 Notes and comments 572
Applications 580
14.1 Representation of convex functions 581
14.2 Representation of concave functions 584
14.3 Doubly stochastic matrices 589
14.4 The Riesz–Herglotz theorem 590
14.5 Typically real holomorphic functions 592
14.6 Holomorphic functions with positive real part 597
14.7 Completely monotonic functions 603
14.8 Positive definite functions on discrete groups 606
14.9 Range of vector measures 610
14.10 The Stone–Weierstrass approximation theorem 612
14.11 Invariant and ergodic measures 614
14.12 Exercises 620
14.13 Notes and comments 622
Appendix 625
A.1 Functional analysis 625
A.2 Topology 632
A.3 Measure theory 641
A.4 Descriptive set theory 654
A.5 Resolvable sets and Baire-one functions 657
A.6 The Laplace equation 662
A.7 The heat equation 666
A.8 Axiomatic potential theory 669
Bibliography 686
List of symbols 712
Index 720

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"Incorporating many original results of the authors, the book is addressed both to students who can find a clear and thorough presentation of the basics of the integral theory, as well as for the advanced readers who find a substantial amount of recent results, appearing for the first time in book form."
S. Cobzas in: Studia Universitatis Babes-Bolyai LV/2010