  Let $\mu$ be a Borel probability measure with compact support. We consider
exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We
show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure
type. We also classify various $\mu$ that admits either kind of exponential
bases, in particular, the discrete measures and their connection with integer
tiles. By using this and convolution, we construct a class of singularly
continuous measures that has an exponential Riesz basis but no exponential
orthonormal basis. It is the first of such kind of examples.
