Thank you for the clarification, Andy and Bill. I believe there are subsets of the integers that really should not be assigned a density. I'm interested in giving a density to a maximum -- or at least a maximal -- collection of subsets of Z that *ought* to have a density. Such a thing should of course be an extension of the usual density to a wider class of subsets. I'm not sure such a thing exists, but it would be nice to think it did. (Certainly subsets of Z whose density can be defined by iterated Cesaro summation should be included.) Auxiliary question: If the power set of the integers is identified with {0,1}^Z and given its product measure, then what is the measure of the collection of those subsets of Z that "ought" to have a density? --Dan Andy L. 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. Bill T. wrote: << That is correct. . . .

...

Sometimes the brain has a mind of its own.