A Diophantine Property of Some Summable Functions

Let g(t) denote a function of the class L² in O ≤ t ≤ I with period I. Let {f_n (x) } (n = 1, 2,...) denote a sequence of real functions of x (a≤x<b). In a former paper [5] I proved that if f_n (x) satisfies some general conditions, the relation

lim I/n Σ g(f_n(x)) = ∫g(t)dt                (1).