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/Følner_sequence http://en.wikipedia.org/wiki/F%F8lner%5Fsequence for information on Folner sequences.
