  We examine Fourier frames and, more generally, frame measures for different
probability measures. We prove that if a measure has an associated frame
measure, then it must have a certain uniformity in the sense that the weight is
distributed quite uniformly on its support. To be more precise, by considering
certain absolute continuity properties of the measure and its translation, we
recover the characterization on absolutely continuous measures $g\, dx$ with
Fourier frames obtained in \cite{Lai11}. Moreover, we prove that the frame
bounds are pushed away by the essential infimum and supremum of the function
$g$. This also shows that absolutely continuous spectral measures supported on
a set $\Omega$, if they exist, must be the standard Lebesgue measure on
$\Omega$ up to a multiplicative constant. We then investigate affine iterated
function systems (IFSs), we show that if an IFS with no overlap admits a frame
measure then the probability weights are all equal. Moreover, we also show that
the {\L}aba-Wang conjecture \cite{MR1929508} is true if the self-similar
measure is absolutely continuous. Finally, we will present a new approach to
the conjecture of Liu and Wang \cite{LW} about the structure of non-uniform
Gabor orthonormal bases of the form ${\mathcal G}(g,\Lambda,{\mathcal J})$.
