Infinite linear probability distributions
A number of people have noticed that the problem about probability I mentioned last week is not really a problem about infinite probability spaces, but rather about possible distributions over such spaces. For instance, a Gauss distribution over the reals will yield well-defined probabilities at every interval. But in the case of the arbitrary real number, the distribution would have to be a line parallel to the x-axis, and how could the segments of the area under this line possibly add up to 1?
I'm still not sure if talk about probability really does not make sense in such cases or if it does, but we (or at least I) lack an adequate mathematical treatment. For example, are the following three conditions logically inconsistent?
1. Atom A will decay at some time in the future.
2. The probability of decay at any day is the same as at the preceeding day.
3. Future is infinite.