Absolute equal distribution of families of finite sets Original Research Article

Let U={{uin}i=1dn}ngreater-or-equal, slantedn0 and V={{vin}i=1dn}ngreater-or-equal, slantedn0, where u1nless-than-or-equals, slantu2nless-than-or-equals, slantcdots, three dots, centeredless-than-or-equals, slantudn,n, v1nless-than-or-equals, slantv2nless-than-or-equals, slantcdots, three dots, centeredless-than-or-equals, slantvdn,n, ngreater-or-equal, slantedn0, and limn→∞dn=∞. Let image be a set of continuous real-valued functions on image. Then U and V are equally distributed with respect to image ifimageor absolutely equally distributed with respect to image ifimageWe show that these definitions are equivalent ifimageand we give sufficient conditions for U and V to be absolutely equally distributed with respect toimageand image.