Analysis on Polish Spaces and an Introduction to Optimal Transportation (London Mathematical Society Student Texts, Series Number 89)

A large part of mathematical analysis, both pure and applied, takes place on Polish spaces: topological spaces whose topology can be given by a complete metric. This analysis is not only simpler than in the general case, but, more crucially, contains many important special results. This book provides a detailed account of analysis and measure theory on Polish spaces, including results about spaces of probability measures. Containing more than 200 elementary exercises, it will be a useful resource for advanced mathematical students and also for researchers in mathematical analysis. The book also includes a straightforward and gentle introduction to the theory of optimal transportation, illustrating just how many of the results established earlier in the book play an essential role in the theory.--Page 4 de la couverture 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 General topology -- Metric spaces -- Polish spaces, and compactness -- Semi-continuous functions -- Uniform spaces and topological groups -- Càdlàg functions -- Banach spaces -- Hilbert space -- The Hahn-Banach theorem -- Convex functions -- Subdifferentials and the legendre transform -- Compact convex Polish spaces -- Some fixed point theorems -- Abstract measure theory -- Further measure theory -- Borel measures -- Measures on Euclidean space -- Convergence of measures -- Introduction to Choquet theory -- Optimal transportation -- Wasserstein metrics -- Some examples This detailed account of analysis on Polish spaces contains results that apply to probability theory and a gentle introduction to optimal transportation. Containing more than 200 elementary exercises, it is a useful resource for advanced mathematical students and also for researchers in mathematical analysis.