Let U={{u i n } i = 1 d n } n > = n 0 and V={{v i n } 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 i n )-F(v i n ))=o(d n ),F F,or absolutely equally distributed with respect to F if i = 1 d n |F(u i n )-F(v i n )|=o(d n ),F F.We show that these definitions are equivalent ifF=F C(R)|lim x - > ~ F(x)andlim x - > - ~ F(x)exist(finite),and we give sufficient conditions for U and V to be absolutely equally distributed with respect toF={F|FisboundedanduniformlycontinuousonR}and F={F C(R)|Fisbounded}.