You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>28 All the W_i have the same external measure (In fact, they are constructed as translations modulo 1 IIRC). In that way you have the 'same probability' of being chosen, just as the OP said.