  We show that if $X$ is a uniformly perfect complete metric space satisfying
the finite doubling property, then there exists a fully supported measure with
lower regularity dimension as close to the lower dimension of $X$ as we wish.
Furthermore, we show that, under the condensation open set condition, the lower
dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower
dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is
the maximum of the Assouad dimensions of the corresponding self-similar set $E$
and the condensation set $C$. If the Assouad dimension of $C$ is strictly
smaller than the Assouad dimension of $E$, then the upper regularity dimension
of any measure supported on $E_C$ is strictly larger than the Assouad dimension
of $E_C$. Surprisingly, the corresponding statement for the lower regularity
dimension fails.
