In infinite sets. Is there a difference between say, the natural numbers and the real numbers?
I know that there is a problem when trying to add the probabilities of hitting a certain real number which is P=zero so all probabilities are added to zero. But the same problem occurs in natural numbers as well. So they both aren't distributed uniformly.
My question is, is there any difference between the two sets in terms of the problem they both pose to probability? Are they both equally difficult to be dealt with in probability? Or is the set of natural numbers better in any regard and what is the implications of its advantage?