P. Terán (2008). Test 17, 505-514.
Trying to understand when the Herer expectation of a random set falls into the abstract definition of expectation in A law of large numbers in a metric space with a convex combination operation unexpectedly triggered a line of research on the Herer expectation itself.
This paper characterizes those separable Banach spaces where the Herer expectation is identical with the Aumann expectation, thus solving an open problem in Ilya Molchanov's book Theory of random sets as you can read here.
A few other results are presented: for instance, if the dual space is separable then the Aumann expectation is the intersection of the Herer expectations with respect to all equivalent norms. Nice!
It must be mentioned that the proof of the main theorem contains a gap and so applies only to Hausdorff approximable random sets. That does not compromise the solution of the problem as stated in Ilya's book. A different proof will be presented in forthcoming work.
Up the line:
·A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). Downloadable from Ilya's website.
Down the line:
·Intersections of balls and the ball hull mapping (2010).
·Expectations of random sets in Banach spaces (accepted 2013).
To download, click on the title or here. This preprint differs in minor details from the published version.