Reference request: Choquet theory

Question

Recently I realized that many integral representation theorems (such as Herglotz' theorem, Bernstein's theorem, Riesz representation theorem, etc) may be systematically understood under the Choquet theory.

I have never been explicitly exposed to this subject, however, thus I would like to have some good introductory material on it. Any reference that leads to Choquet theorem is fine, but it will be much nicer if it contains some criteria for uniqueness of representation (if any such thing exists) as well as application to some well-known theorems.

Thank you for reading!

Answer 1

One book which seems to be related to the topic you are interested in is

Phelps, Robert R. Lectures on Choquet’s theorem. Lecture Notes in Mathematics. 1757. Berlin: Springer. 124 p. (2001). Google Books, DOI: 10.1007/b76887.

This book has been mentioned several times [in the main chatroom]. (Just try to search for choquet or phelps). You can see from those conversations that there are at least a few people on this site who know some stuff about this topic.

See also ZentralBlatt review and MathSciNet review.

Answer 2

Besides Phelps' book, which offers a very well rounded introduction to the topic, as well as the finite dimensional motivation behind it, I strongly recommend two more:

1. Alfsen E. M., Compact Convex Sets and Boundary Integrals (1971) and
2. Lukes, J., Maly, J., Netuka I., Spurny J. Integral Representation Theory: Applications to Convexity, Banach Spaces and Potential Theory (2010).

The second one, especially, is a class of its own. In its 700+ pages you can find a very modern treatise of Choquet's theory. For an introduction, I recommend reading Chapter 2 and then depending on your interests a selection of the rest of the topics of the book. In Chapter 14 you will find a lot of Applications, including the ones you mentioned in the original post.

Keep in mind that Choquet's Theory is demanding, so you already need to be familiar with some advanced topics from Functional Analysis and Measure Theory. Lukes' et al book contains a very detailed appendix which makes the book as self contained as it can get. Additionally, it provides references to most topics it didn't cover, which is a huge plus if you want to continue to something more specialized.

I've personally used all three of them during a couple of projects, having no prior knowledge of Choquet Theory, and they all helped me a lot.