Cosider sample space, S, defined as
S = {1,2,3,4,5,6,7,8,9,................}
Experiment is to pick a number randomly from this set (with equal probability).
Question: How do you define probability measure for this?
Confusion I have -
Consider sets A1 = {1}, A2 = {2}, A3 = {3}, A4 = {4}, ..., An = {n},.... so on
Now let P(An) = x
P(S) = P(A1)+P(A2)+P(A3)+....+P(An)+..... (under the axiom of countable additivity)
LHS is 1 but RHS = 0 or infinite, so where is that I'm going wrong?
The flaw in my opinion is in assuming that it is possible to "randomly" choose an element of an infinitely countable set such that any element is equally likely to be chosen as any other. The arguments presented show that this assumption is false. Maybe in terms of measure theory there is a better way to say this; I have not studied measure theory.
I also found this thread on another forum; post #16, which seems to be by the same HallsofIvy who posts on this forum, might be helpful to you.