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.