Frame measures for infinitely many measures

For every frame spectral measure $ \mu $, there exists a discrete measure $ \nu $ as a frame measure. If $ \mu $ is not a frame spectral measure, then there is not any general statement about the existence of frame measures $ \nu $ for $ \mu $. This motivated us to examine Bessel and frame measures. We construct infinitely many measures $ \mu $ which admit frame measures $ \nu $, and we show that there exist infinitely many frame spectral measures $ \mu $ such that besides having a discrete frame measure, they admit continuous frame measures too.