  In this paper we prove that given a weakly almost periodic measure $\mu$
supported inside some model set $\Lambda(W)$ with closed window $W$, then the
strongly almost periodic component $\mu_S$ and the null weakly almost periodic
component $\mu_0$ are both supported inside $\Lambda(W)$. As a consequence we
prove that given any translation bounded measure $\omega$, supported inside
some model set, then each of the pure point diffraction spectrum
$\hat{\gamma}_{pp}$ and the continuous diffraction spectrum $\hat{\gamma}_c$ is
either trivial or have a relatively dense support.
