ABSTRACT
Monetary utility functions are -- except for the expected value -- not of von Neumann-Morgenstern type.  In case the utility function has convex level sets in the set of probability measures on the real line, we can give some characterisation that comes close to the vN-M form.  For coherent utility functions this was solved by Ziegel.  The general concave case under the extra assumptions of  weak compactness, was solved by Stephan Weber. In the general case the utility functions are only semi continuous. Using the fact that law determined utility functions are monotone with respect to convex ordering, we can overcome most of the technical problems.    The characterisation is similar to Weber's theorem except that we need vN_M utility functions that take the value $-\infty$. Having convex level sets can be seen as a weakened form of the independence axiom in the vN-M theorem.
This is joint work with Bellini, Bignozzi and Ziegel.