In order to avoid the procrastination paradox, we want our utility function to be upper semicontinuous in the natural topology on histories. Given such a utility function U on the space of infinite histories, there is a natural way to extend it to the space of finite & infinite histories preserving semicontinuity. Namely, we define U(x) = inf U(xy) where x is a finite history and y is an infinite continuation.
However, this prescription is not necessary: we can have an upper semicontinuous function in the space of finite & infinite histories which doesn’t arise in this way. Coming to think about it, it isn’t very attractive since intuitively the universe coming to end is a better outcome than the universe turning into hell.
In order to avoid the procrastination paradox, we want our utility function to be upper semicontinuous in the natural topology on histories. Given such a utility function U on the space of infinite histories, there is a natural way to extend it to the space of finite & infinite histories preserving semicontinuity. Namely, we define U(x) = inf U(xy) where x is a finite history and y is an infinite continuation.
However, this prescription is not necessary: we can have an upper semicontinuous function in the space of finite & infinite histories which doesn’t arise in this way. Coming to think about it, it isn’t very attractive since intuitively the universe coming to end is a better outcome than the universe turning into hell.