A density on the natural numbers invariant with respect to the multiplication 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
 A: The natural thing to do here is to replace the intervals $[1,n]$ (and $n = |[1,n]|$) in the definition of $d(A)$ with a sequence $F_n$ of subsets of $\mathbb{N}$ which is multiplicatively asymptotically invariant (or, in other words, a Folner sequence for the semigroup $(\mathbb{N},\cdot)$). For an exploration of this idea, as well as applications, see for instance this article by Vitaly Bergelson:
Multiplicatively large sets and ergodic Ramsey theory, Israel Journal of Mathematics 148 (2005), 23-40.
EDIT: One particular example (mentioned in the article) is to take $F_n$ to be the set of all positive integers which can be written as a product of powers of the first $n$ primes, where the powers are allowed to be any non-negative integer which is less than or equal to $n$. The motivation for choosing $F_n$ this way is that just as $1$ generates the additive semigroup $(\mathbb{N},+)$, the primes generate the multiplicative semigroup. Think of balls in the corresponding Cayley graphs with radius getting larger and larger. The article contains many other examples of such $F_n$, and each of them gives a notion of "multiplicative density" by setting $d(A) = \lim_{n \to \infty} \frac{|A \cap F_n|}{|F_n|}$ (if the limit exists. Otherwise one usually considers the limsup and liminf).
A: Here is an example.  Let $d(A)$ be as you define it in the question, namely $$d(A)=\lim_{n\to \infty} \frac{|A\cap [1,n]|}{n}.$$
Let $U_n=\{k\in\mathbb{N} \text{ such that } k \text{ is a mutiple of } n!\}$.  Then define $$\mu(A)=\lim_{n\to \infty} n!\cdot d(A\cap U_n).$$
Then I claim that $\mu$ is finitely additive and multiplicatively invariant.  Finite additivity is obvious.  For multiplicative invariance, note that for $s>k$, we have $(s+k)!\cdot d(kA\cap U_{s+k})=s!\cdot d(A\cap U_{s})$, unless I've screwed something up. 
EDIT:  Note by the way that one can replace the limit in the definition with $d$ with the Cesaro mean, for example, giving a much broader class of sets with defined measure.  For example, with this addition, the set of natural numbers with a fixed leading digit in a fixed prime base $p$ has density $1/(p-1)$.  
A: How about:
$$
\lim_{k\to\infty} \frac{|A \cap [1,k!]|}{k!}
$$
when it exists?
