This is a follow-up to this classical question asked recently here: we know (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ has mass $\frac1n$. The argument I know of makes use of an independence property induced by the formulation of the problem: $p_1\dots p_k \mathbb{Z} = p_1\mathbb{Z} \cap\dots\cap p_k\mathbb{Z}$ has measure $1/(p_1\dots p_k)= (1/p_1)\dots(1/p_k)$.

This question was looking to a weakened notion of "uniform" measure on $\mathbb{Z}$: one would even like better that every class $\mod n$ has mass $\frac1n$, but this is trivially impossible.

Here comes my question:

Does there exist a probability measure on $\mathbb{Z}$, such that for all $n\in\mathbb{N}$ there is at least one class $\mod n$ which has mass $\frac1n$?

this weakening should destroy the independence above, and thus prevents applying Borel-Cantelli.