  We introduce a family of atomic measures on free groups generated by
no-return random walks. These measures are shown to be very convenient for
comparing "relative sizes" of subgroups, context-free and regular subsets
(that, subsets generated by finite automata) of free groups. Many asymptotic
characteristics of subsets and subgroups are naturally expressed as analytic
properties of related generating functions. We introduce an hierarchy of
asymptotic behaviour "at infinity" of subsets in the free groups, more
sensitive than the traditionally used asymptotic density, and apply it to
normal subgroups and regular subsets.
