On Jun 6, 2011, at 1:35 PM, Andy Latto wrote:
I believe that what Bill is saying is that density can be extended to a finitely additive translation-invariant measure defined on all sets of integers. But there's no canonical way to do this, and no such way can be explicitly specified, and proving the existence of this extension requires use of AC.
That is correct. In contrast, it is impossible to define a finitely additive measure on the free group on two generators that is invariant under left multiplication by an element of the group. This is related to the Banach-Tarski paradox. A group admits such a measure if and only if it is amenable. A more geometric and more constructive definition of amenability is this: a finitely generated group is amenable if and only if for every epsilon, there is a finite subset S of the group such that for each generator g, the symmetric difference of S and gS has cardinality less than epsilon times the cardinality of S. For many amenable groups (including Z or Z^n) but not all amenable groups, you can take S to be the set of all elements of the group expressible with word length less than R, for some large R. There are other nice characterizations of amenability as well, there is a large literature on amenability, and there are a number of interesting and challenging open problems. Amenability is an interesting topic, but it's wandering far from the original question and not necessarily in the spirit of math-fun. Bill