Suppose, $F$ is the minimal $\sigma$-algebra on an infinite cyclic group $C_\infty = \langle a\rangle_\infty$, that contains all cosets of non-trivial subgroups of $C_\infty$. Does there exist such a measure $\mu$ on $(C_\infty, F)$, such that $\forall k, n \in \mathbb{N}$ $\mu(a^k \langle a^n \rangle) = \frac{1}{n}$?
I found two proofs, that prove opposite answers to this question. The problem is to determine, what one of them is wrong (or probably both).
The proof that such measure exists
The union of set of all cosets of non-trivial subgroups of $C_\infty$ and of a singleton containing empty set forms a semiring. The equalities from the question define a measure on that semiring. This measure is $\sigma$-additive (because it is additive and $C_\infty$ is Noetherian). Thus, by Caratheodory’s Extension Theorem this measure can be extended onto $F$.
The proof that such measure does not exist
$$\{a^n\} = (a^{n - 1}\langle a \rangle \setminus \bigcup_{i = 2}^\infty a^{n - 1}\langle a^i \rangle) \setminus a^{n - 2}\langle a^3 \rangle$$
From that it follows, that every singleton subset of $C_\infty$ belongs to $F$ and that their measure is equal. Now, two cases are possible. The first one is that the measure of a singleton is positive. Then by $\sigma$-additivity of measure the measure of $C_\infty$ is infinite. The second one is that the measure of a singleton is $0$. Then by $\sigma$-additivity of measure the measure of $C_\infty$ is also $0$. But we already know that the measure of $C_\infty$ is $1$, which gives us a contradiction.
The first proof is wrong: the measure is not $\sigma$-additive. Indeed, you can one by one pick disjoint cosets whose union is all of $C_\infty$ and in each step you can choose the new coset to have arbitrarily small measure, so that the sum of their measures is less than $1$.
(The second proof is also incomplete, in that it is not clear to me how you are concluding that all singletons have the same measure. However, it is trivial to just show directly that each singleton has measure $0$, by writing it as an intersection of cosets containing it whose measures converge to $0$.)