This should be possible for any group, assuming I have data on the correct extent of the measures.

Leave $ mathcal {B} $ be the Boolean algebra generated by cosets of finite index subgroups $ G $. Then there is a finite additive (in fact $ G $-invariant) measure $ mu $ in $ G $ so that if $ C $ It is a coset of $ H $Y $[G:H]= n < infty $, so $ mu (C) = 1 / n $. Specifically, if $ A in mathcal {B} $ then there is a finite index subgroup $ H $ of $ G $, such that $ A $ is a union of cosets of $ H $. Yes $ n $ is the index, and $ m $ is the number of cosets in the union then it is established $ mu (A) = m / n $. You can directly verify that this is a well-defined measure as desired.

Now one must be able to extend $ mu $ arbitrarily to a finely additive measure in $ mathcal {P} (G) $ (which will not be $ G $-invariant necessarily). I think this follows from Section 457 of Fremlin's "Theory of Measure".

Observation 1: the initial measure $ mu $ it is in fact the only $ G $-invariant finite probability measure additive in $ mathcal {B} $, and can be constructed from Haar's measure in the profinite completion of $ G $. In particular, if $ mathcal {N} $ Denotes the collection of normal subgroups of finite index of $ G $, then the profinite ending is $ hat {G} = varprojlim _ { mathcal {N}} G / N $. We can write elements of $ hat {G} $ as $ (C_N) _ {N in mathcal {N}} $, where $ C_N $ It is a coset of $ N $. Given a set $ A $ in $ mathcal {B} $define $ X_A $ to be the set of $ (C_N) _ {N in mathcal {N}} in hat {G} $ such that $ C_N cap A neq emptyset $ for all $ N en mathcal {N} $. So $ X_A $ is closed and you can check that $ mu (A) $ Make the measure of $ X_A $.

Observation 2: Perhaps it is also worth mentioning that if $ G $ is susceptible (for example, abelian) then there is a $ G $-invariant finite probability measure additive in $ mathcal {P} (G) $, which must satisfy the desired conditions directly by finite additivity.