[xxxyy]

It still does hold up, but becomes more complicated. In general, there is no uniform distribution over any infinite set. We might: a) think about a pipe spitting out objects that have two possible features - being a raven and being black, or b) use a nonuniform distribution over an infinite set of objects and integrate (sum) to get a similar Bayesian result.

[x0x0]

What do you mean there is no uniform distribution over any infinite set? There is the uniform distribution over [0,1] which is both infinite and not even countable.

I'm confused.

[xxxyy]

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).