  We consider averages $\kappa$ of spectral measures of rank one perturbations
with respect to a $\sigma$-finite measure $\nu$. It is examined how various
degrees of continuity of $\nu$ with respect to $\alpha$-dimensional Hausdorff
measures ($0 \leq \alpha \leq 1$) are inherited by $\kappa$. This extends
Kotani's trick where $\nu$ is simply the Lebesgue measure.
