Absolute equal distribution of families of finite sets

Abstract

Let U ={{u in} i=1 d n } n⩾n 0 and V ={{v in} i=1 d n } n⩾n 0 , where u 1 n ⩽ u 2 n ⩽⋯⩽ u d n , n , v 1 n ⩽ v 2 n ⩽⋯⩽ v d n , n , n⩾ n 0, and lim n→∞ d n =∞. Let F be a set of continuous real-valued functions on R . Then U and V are equally distributed with respect to F if ∑ i=1 d n (F(u in)−F(v in))= o(d n), F∈ F, or absolutely equally distributed with respect to F if ∑ i=1 d n |F(u in)−F(v in)|= o(d n), F∈ F. We show that these definitions are equivalent if F= F∈C( R) | lim x→∞F(x) and lim x→−∞F(x) exist ( finite) , and we give sufficient conditions for U and V to be absolutely equally distributed with respect to F={F | F is bounded and uniformly continuous on R} and F={F∈C( R) | F is bounded } .