TITLE: Is there a notion of integration over the algebraic numbers?
QUESTION [11 upvotes]: For reasons which are hard to articulate (due to they not being very clear in my mind), but having to do with the eprint From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group by  Robert de Mello Koch and Sanjaye Ramgoolam, I have been wondering whether there is a notion of integration over the algebraic numbers $\overline{\mathbb{Q}}$, as there is over the p-adic completions of the rationals.

REPLY [2 votes]: Pick an algebraic number at random.  What is its expected degree?  As noted, since the set is countable, we cannot expect to do conventional integrals on it.  But we still have things like this:  pick a natural number at random---then it is square-free with probability $6/\pi^2$.
So, maybe the question is:  What are "natural" Folner sets for the algebraic numbers?  The usual Folner sets for $\{1,2,3,\dots\}$  are the sets $\{1,2,\dots,n\}$, which define the "density" (not an actual measure)
$$
\delta(A) = \lim_{n\to\infty}\frac{\\#(A \cap \{1,\dots,n\})}{n}
$$
for sets $A \subseteq \{1,2,\cdots\}$.  See, for example, http://en.wikipedia.org/wiki/F%F8lner%5Fsequence for information on Folner sequences.