[baduser]

I guess, but, the set is not even countably infinite. "Selecting at random" is something that happens in the real, non-infinite world, not in the mathematically rigorous would where infinities can exist. So, no, probability-zero events do not happen in either.

[rbanffy]

Not necessarily - you might just come up with a number you know is in the set, say pi/4. I know it's in the set because it satisfies the conditions that define it. Still, the odds of that particular number being picked up are zero.

[baduser]

If you picked that, you did not pick uniformly from the unit interval. You picked from something else, which was much smaller.