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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.
Table des matières
Introduction; Part I. Topological Properties: 1. General topology; 2. Metric spaces; 3. Polish spaces and compactness; 4. Semi-continuous functions; 5. Uniform spaces and topological groups; 6. Càdlàg functions; 7. Banach spaces; 8. Hilbert space; 9. The Hahn-Banach theorem; 10. Convex functions; 11. Subdifferentials and the legendre transform; 12. Compact convex Polish spaces; 13. Some fixed point theorems; Part II. Measures on Polish Spaces: 14. Abstract measure theory; 15. Further measure theory; 16. Borel measures; 17. Measures on Euclidean space; 18. Convergence of measures; 19. Introduction to Choquet theory; Part III. Introduction to Optimal Transportation: 20. Optimal transportation; 21. Wasserstein metrics; 22. Some examples; Further reading; Index.
A propos de l'auteur
D. J. H. Garling is a Fellow of St John's College, Cambridge, and Emeritus Reader in Mathematical Analysis at the University of Cambridge. He has written several books on mathematics, including Inequalities: A Journey into Linear Algebra (Cambridge, 2007) and A Course in Mathematical Analysis (Cambridge, 2013).
Résumé
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.
