Note that if P dominates Q in the sense that there is a c>0 such that for all events E, P(E)>c⋅Q(E), U is integrable wrt P, then I think U is integrable wrt Q. I propose the space of all probability distribution dominated by a given distribution P.
Conveniently, if we move to semi-measures, we can take P to be the universal semi-measure. I think we can have our space of utility functions be anything integrable WRT the universal semi-measure, and our space of probabilities be anything lower semi-computable, and everything will work out nicely.
I think bounded functions are the only computable functions that are integrable WRT the universal semi-measure. I think this is equivalent to de Blanc 2007?
The construction is just the obvious one: for any unbounded computable utility function, any universal semi-measure must assign reasonable probability to a St Petersburg game for that utility function (since we can construct a computable St Petersburg game by picking a utility randomly then looping over outcomes until we find one of at least the desired utility).
Note that if P dominates Q in the sense that there is a c>0 such that for all events E, P(E)>c⋅Q(E), U is integrable wrt P, then I think U is integrable wrt Q. I propose the space of all probability distribution dominated by a given distribution P.
Conveniently, if we move to semi-measures, we can take P to be the universal semi-measure. I think we can have our space of utility functions be anything integrable WRT the universal semi-measure, and our space of probabilities be anything lower semi-computable, and everything will work out nicely.
I think bounded functions are the only computable functions that are integrable WRT the universal semi-measure. I think this is equivalent to de Blanc 2007?
The construction is just the obvious one: for any unbounded computable utility function, any universal semi-measure must assign reasonable probability to a St Petersburg game for that utility function (since we can construct a computable St Petersburg game by picking a utility randomly then looping over outcomes until we find one of at least the desired utility).