Following Rosenblatt and Yang, I say that a subset $A$ of $\mathbb Z$ has a unique mean value if for all invariant means $\lambda_1,\lambda_2$ on $\mathbb Z$, one has $\lambda_1(A)=\lambda_2(A)$.
Notice that the set of subsets having a unique mean value has the same cardinality as the set of subsets non-having a unique mean value, thanks to Andreas Thom's answer to Intrinsically measurable subsets of amenable semigroups..
Nevertheless, roughly speaking, it should be clear that most subsets of $\mathbb Z$ should not have a unique mean value. Indeed, to a have a unique mean value one needs a very particular structure, as almost-periodicity.
Question: Is there any probabilistic way to formalize the intuition that generic subsets of $\mathbb Z$ does not have a unique mean value?
Thank you in advance,