Question

Show that it is not possible to choose a uniform positive integer at random. (In other words, we cannot define a probability measure on the positive integers that can be considered uniform). Hint: What would be the probability of choosing a particular number?

Answer 1

Answer: Think it this way:

If i told you to chose a positive integer at random, then all the positive integers have the same probability of being chosen.

this probability for each number is one (because you are choosing one number ) divided by the total of options in the set, and in this particular set, you have infinite options to chose.

so the probability of choosing one particular number is $\frac{1}{infinity}  = 0$

if this is counterintuitive for you, think that numbers like 124125194104198172918274 are integers, when we think on integers the numbers that come to mind are 1, 2, 3.... etc, but there are a infinity of them.