  The upper and lower Assouad dimensions of a metric space are local variants
of the box dimensions of the space and provide quantitative information about
the `thickest' and `thinnest' parts of the set. Less extreme versions of these
dimensions for sets have been introduced, including the upper and lower
quasi-Assouad dimensions, $\theta $-Assouad spectrum, and $\Phi $-dimensions.
In this paper, we study the analogue of the upper and lower $\Phi $-dimensions
for measures. We give general properties of such dimensions, as well as more
specific results for self-similar measures satisfying various separation
properties and discrete measures.
