I meant a distribution with "discrete" probability, i.e. a distribution where the probabilities of singletons are all equal and nonzero, so that a simple Bayesian argument could possibly be extended. My bad for not being precise enough.

Perhaps I should have stuck to natural numbers in my previous comment, otherwise yes, you can have uniform distributions with respect to some additional structure of the probability space (like [0,1] with the Lebesgue measure you suggest).