A measure is generally assumed to be countably additive. If the probability of an integer n is 0, then the total measure is 0. If the probability of n is epsilon, then the total measure is infinity, so it is not a probability measure. ====== Non countably-additive measures are legitimate structures, and provide one of the ways to define amenability for a group, but they are totally weird and I don't think you really want to go into that territory. In theory, there are translation-invariant additive but non-countaby-additive measures on Z, but it is known to be impossible to actually define any particular instance of one. Bill Thurston On Jun 5, 2011, at 9:22 PM, Fred lunnon wrote:
I'm getting lost here.
Exactly why can't a countable set --- eg. rationals x with 0 <= x < 1 --- be assigned uniform probability? [Admittedly, I can't offhand see quite how to do it!]
What has denumerability to do with this anyway? It's also impossible to assign uniform probability to the unbounded reals.