Here’s a quick issue I only just noticed but which fortunately is easily fixed:
Above I mentioned you probably want to restrict to a sigma-algebra of events and only allow measurable functions as actions. But, what does measurable mean here? Fortunately, the ordering on outcomes (even without utility) makes measurability meaningful. Except this puts a circularity in the setup, because the ordering on outcomes is induced from the ordering on actions.
Fortunately this is easily patched. You can start with the assumption of a total preorder on outcomes (considering the case of decisions without uncertainty), to make measurability meaningful and restrict actions to measurable functions (once we start considering decisions under uncertainty); then, for P3, instead of the current P3, you would strengthen the current P3 by saying that (on non-null sets) the induced ordering on outcomes actually matches the original ordering on outcomes. Then this should all be fine.
Here’s a quick issue I only just noticed but which fortunately is easily fixed:
Above I mentioned you probably want to restrict to a sigma-algebra of events and only allow measurable functions as actions. But, what does measurable mean here? Fortunately, the ordering on outcomes (even without utility) makes measurability meaningful. Except this puts a circularity in the setup, because the ordering on outcomes is induced from the ordering on actions.
Fortunately this is easily patched. You can start with the assumption of a total preorder on outcomes (considering the case of decisions without uncertainty), to make measurability meaningful and restrict actions to measurable functions (once we start considering decisions under uncertainty); then, for P3, instead of the current P3, you would strengthen the current P3 by saying that (on non-null sets) the induced ordering on outcomes actually matches the original ordering on outcomes. Then this should all be fine.