  Operators with zero dimensional spectral measures appear naturally in the
theory of ergodic Schr\"odinger operators. We develop the concept of a complete
family of Hausdorff measure functions in order to analyze and distinguish
between these measures with any desired precision. We prove that the dimension
of spectral measures of half-line operators with positive upper Lyapunov
exponent are at most logarithmic for every possible boundary phase. We show
that this is sharp by constructing an explicit operator whose spectral measure
obtains this dimension. We also extend and improve some basic results from the
theory of rank one perturbations and quantum dynamics to encompass generalized
Hausdorff dimensions.
