Analysis on Polish Spaces and an Introduction to Optimal Transportation
Analysis on Polish Spaces and an Introduction to Optimal Transportation
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.
- Includes results that apply to probability theory
- Contains a gentle introduction to optimal transportation
- Brings together many results previously scattered across different texts
Reviews & endorsements
'This book provides a detailed and concise account of analysis and measure theory on Polish spaces, including results about probability measures. Containing more than 200 elementary exercises, it will be a useful resource for advanced mathematical students and also for researchers in analysis.' Luca Granieri, Mathematical Reviews
Product details
No date availablePaperback
9781108431767
356 pages
227 × 150 × 20 mm
0.51kg
2 b/w illus. 210 exercises
Table of Contents
- 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. Cà dlà g 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.
