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Mcxz comments on LINK: Infinity, probability and disagreement - Less Wrong
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Minor nitpick: You mean infinite set. Any finite set is, of course, both countable and has a uniform probability distribution, and your point is correct for all (measurable) infinite sets.
It depends on your conventions. Some people use "countable" and "countably infinite" to refer to what I refer to as "at most countable" and "countable."
Well, using infinite set would be better either way as this is a property of all infinite sets, not just countable (infinite) sets. It would also avoid any confusion due to this difference of convention.
That said, I didn't actually know about your convention*, so thank you for making me aware of it.
*probably because any easy way of saying "at most countable" in my first language would be confused with "very countable" or even "mostly countable", so I'm guessing none of my professors thought about/remembered this other convention when defining countable sets.