Both of your conjectures are correct. In the measurable / strong topology case, u will necessarily be bounded (from both directions), though it does not follow that the bounds are achievable by any probability distribution.
I described the VNM theorem as failing on sigma-algebras because the preference relation being closed (in the weak or strong topologies) is an additional assumption, which seems much more poorly motivated than the VNM axioms (in Abram’s terminology, the assumption is purely structural).
I think that one can argue that a computationally bounded agent cannot reason about probabilities with infinite precision, and that therefore preferences have to depend on probabilities in a way which is in some sense sufficiently regular, which can justify the topological condition. It would be nice to make this idea precise. Btw, it seems that the topological condition implies the continuity axiom.
Both of your conjectures are correct. In the measurable / strong topology case, u will necessarily be bounded (from both directions), though it does not follow that the bounds are achievable by any probability distribution.
I described the VNM theorem as failing on sigma-algebras because the preference relation being closed (in the weak or strong topologies) is an additional assumption, which seems much more poorly motivated than the VNM axioms (in Abram’s terminology, the assumption is purely structural).
I think that one can argue that a computationally bounded agent cannot reason about probabilities with infinite precision, and that therefore preferences have to depend on probabilities in a way which is in some sense sufficiently regular, which can justify the topological condition. It would be nice to make this idea precise. Btw, it seems that the topological condition implies the continuity axiom.