Multiplicative measures on free groups

We introduce a family of multiplicative distributions {μ_s|s ∈ (0, 1)} on a free group F and study it as a whole. In this approach, the measure of a given set R ⊆ F is a function μ(R): s → μ _s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.