I agree with with your first question, the utilitarian needs a measure (they don’t need a separate utility function from their measure, but there may be other natural measures to consider in which case you do need a utility function).
With respect to your second question, I think you can either give up on the infinite cases (because you think they are “metaphysically” impossible, perhaps) or you demand that a regularization must exist (because with this the problem is “metaphysically” underspecified). I’m not sure what the correct approach is here, and I think it is an interesting question to try and understand this in more detail. On the latter case you have to give up impartiality, but only in a fairly benign way, and that our intuitions about impartiality are probably wrong here (analogous situations occur in physics with charge conservation, as I noted in another comment).
With respect to your third question, I think it is likely that problems with no regularization are non-sensical. This is not to say that all problems involving infinities are themselves non-sense, nor to say that correct choice of regularization is obvious.
As an intuition pump maybe we can consider cases that don’t involve infinities. Say we are in (rather contrived world) in which utility is literally a function of space-time, and we integrate to get the total utility. How should I assign utility for a function which has support on a non-measurable set? Should I even think such a thing is possible? After all, the existence of non-measuarable sets follows not from ZF alone, but requires also the axiom of choice. As another example, maybe my utility function depends on whether or not the continuum hypothesis is true or false. How should I act in this case?
My own guess is that such questions likely have no meaningful answer, and I think the same is true for questions involving infinities without specified ways to operationalize the infinities. I think it would be odd to give up on the utilitarian dream due to unmeasurable sets, and that the same is true for ill-defined infinities.
I agree with with your first question, the utilitarian needs a measure (they don’t need a separate utility function from their measure, but there may be other natural measures to consider in which case you do need a utility function).
With respect to your second question, I think you can either give up on the infinite cases (because you think they are “metaphysically” impossible, perhaps) or you demand that a regularization must exist (because with this the problem is “metaphysically” underspecified). I’m not sure what the correct approach is here, and I think it is an interesting question to try and understand this in more detail. On the latter case you have to give up impartiality, but only in a fairly benign way, and that our intuitions about impartiality are probably wrong here (analogous situations occur in physics with charge conservation, as I noted in another comment).
With respect to your third question, I think it is likely that problems with no regularization are non-sensical. This is not to say that all problems involving infinities are themselves non-sense, nor to say that correct choice of regularization is obvious.
As an intuition pump maybe we can consider cases that don’t involve infinities. Say we are in (rather contrived world) in which utility is literally a function of space-time, and we integrate to get the total utility. How should I assign utility for a function which has support on a non-measurable set? Should I even think such a thing is possible? After all, the existence of non-measuarable sets follows not from ZF alone, but requires also the axiom of choice. As another example, maybe my utility function depends on whether or not the continuum hypothesis is true or false. How should I act in this case?
My own guess is that such questions likely have no meaningful answer, and I think the same is true for questions involving infinities without specified ways to operationalize the infinities. I think it would be odd to give up on the utilitarian dream due to unmeasurable sets, and that the same is true for ill-defined infinities.