A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| doesn't approach a limit. The "analytic" or "Dirichlet" density exists whenever the naive density does, and is equal to it -- but sometimes is well-defined even when the naive density is not, for instance when S is the "Benford set" of integers whose first digit is 1. (See this related MO question.)

Anyway, the philosopher asked: why stop at Dirichlet density? Is there a sequence of probability measures p_1, p_2, p_3, .... on the natural numbers such that

Whenever S has a Dirichlet density, the limit of p_i(S) exists and is equal to the Dirichlet density, but

there are subsets of the natural numbers such that p_i(S) approaches a limit but S has no Dirichlet density?

(Possibly clarifying addition: to obtain naive density, one can take p_i to be the measure assigning probability 1/i to each integer in [1..i] and 0 to integers greater than i. To get Dirichlet density, take p_i to be the measure with p_i(n) = 1/(n^{1+1/i}zeta(1+1/i)). In either case, the corresponding density of S is the limit in i of p_i(S), whenever this limit exists. So what I have in mind is densities which can be thought of as limits of probability measures, though perhaps there are reasons to have yet more general entities in mind?)

If so, are there any examples which are interesting or which are used for anything in practice?