I'm drawing one single sample from an uncountable sample space. I know the probability of sampling any given single point is zero.
Now, what if I draw samples again and again and again, to infinity? What is the probability for any single point to be eventually drawn (as time goes to infinity)?
My guess is that it is still zero, since I cannot touch all the points in an continuum even if you give me infinite time. At infinity, I will still have infinite points left to touch. But is this reasoning correct? Every now and then a math professor starts talking about subtleties of this kind, but where do I find related bibliography?
In this case, what difference does it make to sample with replacement (after sampling a point, that point is not removed from the sample space) or without replacement (the sampled point is removed from the sample space)?