In a previous paper [1] an Euler angle parameterization for SU (N) was given. Here we present a generalized Euler angle parameterization for U (N). The formula for the calculation of the volume for U (N), CP N as well as other SU (N) and U (N) cosets, normalized to this parameterization, will also be given. In addition, the mixed and pure state product measures for N-dimensional density matrice...