The short answer? No. You can't put anything like a uniform distribution over a countable set. In order for the probability of the full space to be less than infinity, you would need to assign probability 0 to all the singletons. But the space itself can be expressed as the union of all the singletons, each of which has probability 0, so by countable additivity the whole space would have probability 0, which is a contradiction.

As far as I know there isn't a canonical way of bending the rules so that you can get away with things like this. I suppose a Bayesian might put a unit mass on each singleton and use it as an improper distribution for some purpose. But it wouldn't be a bona fide probability distribution.