  In analogy with the lower Assouad dimensions of a set, we study the lower
Assouad dimensions of a measure. As with the upper Assouad dimensions, the
lower Assouad dimensions of a measure provide information about the extreme
local behaviour of the measure. We study the connection with other dimensions
and with regularity properties. In particular, the quasi-lower Assouad
dimension is dominated by the infimum of the measure's lower local dimensions.
Although strict inequality is possible in general, equality holds for the class
of self-similar measures of finite type. This class includes all self-similar,
equicontractive measures satisfying the open set condition, as well as certain
`overlapping' self-similar measures, such as Bernoulli convolutions with
contraction factors that are inverses of Pisot numbers.
  We give lower bounds for the lower Assouad dimension for measures arising
from a Moran construction, prove that self-affine measures are uniformly
perfect and have positive lower Assouad dimension, prove that the Assouad
spectrum of a measure converges to its quasi-Assouad dimension and show that
coincidence of the upper and lower Assouad dimension of a measure does not
imply that the measure is $s$-regular.
