The "classical Beurling density" of a subset of the natural numbers is $d(A)=lim_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$, when it exists. It defines a finitely additive probability measure on the natural numbers which is invariant with respect to the sum. Here is my question: does there exist a "nice formula" to describe a finitely additive probability measure on $\mathbb N$ which is invariant with respect to the multiplication?

A couple of remarks: I don't know if it is trivial that such a measure exists, but anyway it follows from the application of a general result of Vern Paulsen (on arxiv "Syndetic sets and amenability"). Another problem would be that of finding the measure of particular sets. What about the measure of {$1!,2!,3!,4!...$}? Sets with measure differente from 0 and 1? For example the set of numbers whose first digit through 4 to 9 seems to have measure $=\log_{10}4$... any other?

Thanks in advance, Valerio

content) is defined on a field. But the collection of subsets $A$ of the natural numbers having a density $d(A)$ is not a field. $\endgroup$ – Did Mar 21 '11 at 22:37