Frame measures for infinitely many measures

For every frame spectral measure $\mu$, there exists a discrete measure $\nu$ as a frame measure. Since if $\mu$ is not a frame spectral measure, then there is not any general statement about the existence of frame measures $\nu$ for $\mu$, we were motivated 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.