Given a q-integrable function f on [0, infinity), we define s(x) = integral(x)(0) f (t)d(q)t and sigma(s(x)) = 1/x integral(x)(0) s(t)d(q)t for x > 0. It is known that if lim(x ->infinity) s(x) exists and is equal to A, then lim(x ->infinity) sigma(s(x)) = A. But the converse of this implication is not true in general. Our goal is to obtain Tauberian conditions imposed on the general control modulo of s(x) under which the converse implication holds. These conditions generalize some previously obtained Tauberian conditions.