I think we can make an even stronger statement: the only way that the usual probability rules (normalizability, additivity) can be satisfied on an infinite sample space is if all but a finite number of simple outcomes have probability zero. — SophistiCat

Not so (duh!)

I can try to give you an example of "an algorithm that gives all integers (without limit) SOME probability of happening" if you like. — flannel jesus

I am not sure I understand your algorithm and what probability distribution it gives to the integers, but clearly, there is any number of distributions that one can come up with for a countable set of disjoint events that gives all events a non-zero probability. For example, {1/2, 1/4, 1/8, ...}

This algorithm is theoretically capable of randomly producing ANY of the infinite sequence of integers, but it preferentially chooses lower numbers. — flannel jesus

As for preferentially choosing lower numbers, I take it that you mean something like: "for any numbered event, there is a higher-numbered event with a lower probability," right? That has to be true, because otherwise there would be a lower bound on the probability of an unlimited number of events.

I am not sure I understand your algorithm and what probability distribution it gives to the integers — SophistiCat

There's a 50% chance that my algorithm ends with one digit. There's a 25% chance my algorithm ends with two digits. There's a 1/8 chance my algorithm ends with 3 digits. Etc. Each additional digit is another 50% chance of stopping there.

And each individual digit has 50% chance of being a 1 or a 0.

But it looks like you agree that there's a way to generate a random number without it being true that "all but a finite number of simple outcomes have probability zero.", which is what my algorithm was trying to illustrate