Analysis on Polish Spaces and an Introduction to Optimal Transportation

Detailed account of analysis on Polish spaces with a straightforward introduction to optimal transportation.

D. J. H. Garling (Author)

9781108431767, Cambridge University Press

Paperback / softback, published 21 December 2017

356 pages, 2 b/w illus. 210 exercises
22.7 x 15 x 2 cm, 0.51 kg

'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

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.

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.

Subject Areas: Topology [PBP], Geometry [PBM], Numerical analysis [PBKS]