This is a follow-up to Question #47134. There is obviously no uniform probability distribution on $\mathbf{N}$ (or $\mathbf{Z}$); however, using the notion of amenability, you can show that any positive linear form on $\mathrm{C_b}(\mathbf{Z})$ (the set of real bounded functions on $\mathbf{Z}$, endowed with the supremum norm) which is invariant by translation and maps $1$ (the constant function) to $1$ (the real number) must obey certain properties: for instance, the set of even integers must have “probability” $1/2$, in the sense that the linear form must map its indicator function to $1/2$; any finite set must have “probability” $0$, etc. So we have a canonical notion of “probability” for such sets of integers.
This notion can be further extended into the notion of natural density (this time it is more convenient to define it on $\mathbf{N}$): a set of integers $A$ is said to have natural density $p$ whenever $$ \lim_{n \to \infty} \frac{\operatorname{card}(A \cap [0, n))}{n} = p \text.$$ Natural density extends the “amenability density”, in the sense that any subset of $\mathbf{N}$ whose “probability” is imposed by the amenability criterion also has a natural density, which value is equal to that of the aforementioned “probability”. It also allows to give a “probability” to new subsets: for instance, the probability that an integer is square-free is $6/\pi^2$. So, natural density seems to be again a canonical notion of “uniform probability” for a set of integers.
Still one step further, you have the notion of logarithmic density, which allows to define a probability for some sets not having a natural density (and coincides with natural density whenever the latter in defined): the logarithmic density of $A$ is (if it exists) $$ \lim_{x \to \infty} \frac{\sum_{n = 1}^x \mathbf{1}_{n \in A} n^{-1}}{\sum_{n = 1}^x n^{-1}} \text,$$ and then you get e.g. that the probability that a natural number's decimal writing starts with ‘$1$’ is $\lg 2$ (this is the famous “Benford's law”). This definition is invariant by replacing “$n^{-1}$” by any “$(n + k)^{-1}$”; so, again, it is arguably canonical, and we have further extended the notion of “uniform probability on $\mathbf{N}$” in a canonical way.
Still still one step further, you may consider a “doubly logarithmic density” by replacing “$n^{-1}$” in the formula of the logarithmic density by $1/n \ln n$ (restricted to $n > 1$ of course); although, personally, I have never met such a definition in the literature. This way, the proportion of numbers whose numbers of digits (in base ten) starts with a ‘$1$’ would be $\lg 2$.
And one could go even further with triply, fourthly, … logarithmic densities, and even define a “limiting multi-logarithmic density” (if none of these densities yields a convergent sequence, but their upper and lower limits converge as the level of logarithmic mutliplicity increases); and then go even further…
And then I come to my question. Until which point can this process be considered to be “canonical”? Is there a concept of “uniform probability” which would extend all the cases above, but for which there would be strong reasons not to wish to extend it further? I guess that such a concept might involve ideas of information theory: maybe something in the flavor of the Solomonoff prior?… (But not the Solomonoff prior itself, as it is ill-defined and tends to favors even numbers over odd numbers! Actually the Solomonoff priori seems to have a good behaviour at large scale, but not at small scale! :-\ ).
I guess that this question is natural enough so that someone already asked themselves, and maybe even provided an answer. Do you have heard of such an answer?…
