Analysis on Polish Spaces and an Introduction to Optimal Transportation

D. J. H. Garling(2018), Analysis on Polish Spaces and an Introduction to Optimal Transportation, London Mathematical Society Student Texts 89, Cambridge University Press......Page 1
Introduction......Page 11
Contents......Page 6
Part I: Topological Properties......Page 17
1.1 Topological Spaces......Page 19
1.2 Compactness......Page 25
2.1 Metric Spaces......Page 28
2.2 The Topology of Metric Spaces......Page 31
2.3 Completeness: Tietze’s Extension Theorem......Page 34
2.4 More on Completeness......Page 37
2.5 The Completion of a Metric Space......Page 39
2.6 Topologically Complete Spaces......Page 41
2.7 Baire’s Category Theorem......Page 43
2.8 Lipschitz Functions......Page 45
3.1 Polish Spaces......Page 48
3.2 Totally Bounded Metric Spaces......Page 49
3.3 Compact Metrizable Spaces......Page 51
3.4 Locally Compact Polish Spaces......Page 57
4.2 Semi-continuity......Page 60
4.3 The Brézis–Browder Lemma......Page 63
4.4 Ekeland’s Variational Principle......Page 64
5.1 Uniform Spaces......Page 66
5.2 The Uniformity of a Compact Hausdorff Space......Page 69
5.3 Topological Groups......Page 71
5.4 The Uniformities of a Topological Group......Page 74
5.5 Group Actions......Page 76
5.6 Metrizable Topological Groups......Page 77
6.1 Càdlàg Functions......Page 81
6.2 The Space (D[0, 1], d_∞)......Page 82
6.3 The Skorohod Topology......Page 83
6.4 The Metric d_B......Page 85
7.1 Normed Spaces and Banach Spaces......Page 89
7.2 The Space BL(X) of Bounded Lipschitz Functions......Page 92
7.3 Introduction to Convexity......Page 93
7.4 Convex Sets in a Normed Space......Page 96
7.5 Linear Operators......Page 98
7.6 Five Fundamental Theorems......Page 101
7.7 The Petal Theorem and Daneš’s Drop Theorem......Page 105
8.1 Inner-product Spaces......Page 107
8.2 Hilbert Space; Nearest Points......Page 111
8.3 Orthonormal Sequences; Gram–Schmidt Orthonormalization......Page 114
8.4 Orthonormal Bases......Page 117
8.5 The Fréchet–Riesz Representation Theorem; Adjoints......Page 118
9.1 The Hahn–Banach Extension Theorem......Page 122
9.2 The Separation Theorem......Page 126
9.3 Weak Topologies......Page 128
9.4 Polarity......Page 129
9.5 Weak and Weak* Topologies for Normed Spaces......Page 130
9.6 Banach’s Theorem and the Banach–Alaoglu Theorem......Page 134
9.7 The Complex Hahn–Banach Theorem......Page 135
10.1 Convex Envelopes......Page 138
10.2 Continuous Convex Functions......Page 140
11.1 Differentials and Subdifferentials......Page 143
11.2 The Legendre Transform......Page 144
11.3 Some Examples of Legendre Transforms......Page 147
11.4 The Episum......Page 149
11.5 The Subdifferential of a Very Regular Convex Function......Page 150
11.6 Smoothness......Page 153
11.7 The Fenchel–Rockafeller Duality Theorem......Page 158
11.8 The Bishop–Phelps Theorem......Page 159
11.9 Monotone and Cyclically Monotone Sets......Page 161
12.1 Compact Polish Subsets of a Dual Pair......Page 165
12.2 Extreme Points......Page 167
12.3 Dentability......Page 170
13.1 The Contraction Mapping Theorem......Page 172
13.2 Fixed Point Theorems of Caristi and Clarke......Page 175
13.3 Simplices......Page 177
13.4 Sperner’s Lemma......Page 178
13.5 Brouwer’s Fixed Point Theorem......Page 180
13.6 Schauder’s Fixed Point Theorem......Page 181
13.7 Fixed Point Theorems of Markov and Kakutani......Page 183
13.8 The Ryll–Nardzewski Fixed Point Theorem......Page 185
Part II: Measures on Polish Spaces......Page 187
14.1 Measurable Sets and Functions......Page 189
14.2 Measure Spaces......Page 192
14.3 Convergence of Measurable Functions......Page 194
14.4 Integration......Page 197
14.5 Integrable Functions......Page 198
15.1 Riesz Spaces......Page 201
15.2 Signed Measures......Page 204
15.3 M(X), L^1 and L^∞......Page 206
15.4 The Radon–Nikodym Theorem......Page 209
15.5 Orlicz Spaces and L^p Spaces......Page 213
16.1 Borel Measures, Regularity and Tightness......Page 220
16.2 Radon Measures......Page 224
16.3 Borel Measures on Polish Spaces......Page 225
16.4 Lusin’s Theorem......Page 226
16.5 Measures on the Bernoulli Sequence Space \Omega(N)......Page 228
16.6 The Riesz Representation Theorem......Page 232
16.7 The Locally Compact Riesz Representation Theorem......Page 235
16.8 The Stone–Weierstrass Theorem......Page 236
16.9 Product Measures......Page 238
16.10 Disintegration of Measures......Page 241
16.11 The Gluing Lemma......Page 244
16.12 Haar Measure on Compact Metrizable Groups......Page 246
16.13 Haar Measure on Locally Compact Polish Topological Groups......Page 248
17.1 Borel Measures on R and R^d......Page 253
17.2 Functions of Bounded Variation......Page 255
17.3 Spherical Derivatives......Page 257
17.4 The Lebesgue Differentiation Theorem......Page 259
17.5 Differentiating Singular Measures......Page 260
17.6 Differentiating Functions in bv_0......Page 261
18.1 The Norm ||.||_{TV}......Page 267
18.2 The Weak Topology w......Page 268
18.3 The Portmanteau Theorem......Page 270
18.4 Uniform Tightness......Page 274
18.5 The β Metric......Page 276
18.6 The Prokhorov Metric......Page 279
18.7 The Fourier Transform and the Central Limit Theorem......Page 281
18.8 Uniform Integrability......Page 286
18.9 Uniform Integrability in Orlicz Spaces......Page 288
19.1 Barycentres......Page 290
19.2 The Lower Convex Envelope Revisited......Page 292
19.3 Choquet’s Theorem......Page 294
19.4 Boundaries......Page 295
19.5 Peak Points......Page 299
19.6 The Choquet Ordering......Page 301
19.7 Dilations......Page 303
Part III: Introduction to Optimal Transportation......Page 307
20.1 The Monge Problem......Page 309
20.2 The Kantorovich Problem......Page 310
20.3 The Kantorovich–Rubinstein Theorem......Page 313
20.4 c-concavity......Page 315
20.5 c-cyclical Monotonicity......Page 318
20.6 Optimal Transport Plans Revisited......Page 320
20.7 Approximation......Page 323
21.1 The Wasserstein Metrics W_p......Page 325
21.2 The Wasserstein Metric W_1......Page 327
21.3 W_1 Compactness......Page 328
21.4 W_p Compactness......Page 330
21.5 W_p-Completeness......Page 332
21.6 The Mallows Distances......Page 333
22.1 Strictly Subadditive Metric Cost Functions......Page 335
22.2 The Real Line......Page 336
22.3 The Quadratic Cost Function......Page 337
22.4 The Monge Problem on R^d......Page 339
22.5 Strictly Convex Translation Invariant Costs on R^d......Page 341
22.6 Some Strictly Concave Translation–Invariant Costs on R^d......Page 346
Further Reading......Page 349
References......Page 350
Index......Page 352