Question about random selection.

[Mano]

All this talk of infinite sets got me to thinking. Is it possible to truly make a random selection from an infinite set? It seems like this is equivalent to the axiom of choice, or something similar. But I keep thinking of examples which seem to say otherwise. Here is an example:

Suppose you are to select a positive integer completely at random. Think up as large of an integer as you can imagine, say something like 10^(10^100) and call it X. The probability that our randomly selected integer is less than X is ZERO, since there are a finite number of integers less than X and an infinite number greater than X. Therefore, can such a random selection be made?