Hyperreal probability density? I'm fairly new to the subject of hyperreal numbers and I'm wondering if there exists an infinitesimal number $a$ such that (in some reasonable sense) 
$$\sum_{n=1}^\infty a=1$$
?
In other words: Is there a uniform (hyperreal valued) probability distribution on the natural numbers?
I hope the question is posed clearly enough.

You might want to take a look at chapter 5 of the paper "Non-Archimedean Probability": http://arxiv.org/pdf/1106.1524v1.pdf
In section 5.2 you can find a non-archimedean uniform probability for the natural numbers defined in the setting of nonstandard analysis. Since that probability is uniform, every singleton has infinitesimal probability $a>0$, and the "sum over the natural numbers" of $a$ is indeed equal to $1$.
The notions of nonstandard analysis and the concept of "sum over the natural numbers" are introduced in chapters 3 and 4 of the same paper.
