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Section: New Results

Choquet-Kendall-Matheron Theorems

Participant : Jean Goubault-Larrecq.

One of the results obtained by Jean-Goubault-Larrecq in his theory of semantics for mixed non-deterministic and probabilistic choice [60] is that there is a one-to-one correspondence between continuous credibilities over some (state) space $X$ and certain compact subsets of the space of all continuous valuations over $X$, under mild assumptions on $X$. Similar theorems were produced by Choquet in the 1950s, refined by Kendall, then by Matheron in the 1970s, with applications in random set theory, among others.

Klaus Keimel and Jean Goubault-Larrecq produced an extremely simple proof of this fact [22] , based on a simple special case of Groemer's integral theorem. This proof also produces a much more general result than what was known earlier, as it does not assume that $X$ is second-countable or Hausdorff, and only local compactness.

A domain-theoretic view is that this is a representation theorem for mixed demonic choice and probabilistic choice; the angelic and erratic cases are also covered by Goubault-Larrecq and Keimel.

These results had been presented at Dagstuhl Seminar 10232, June 2010.