  It is known that the $k$-dimensional Hausdorff measure on a $k$-dimensional
submanifold of $\mathbb{R}^n$ is closely related to the Lebesgue measure on
$\mathbb{R}^n$. We show that the Ashtekar-Lewandowski measure on the space of
generalized $G$-connections for a compact, semi-simple, Lie group $G$, is
analogously related to the product measure on the set of all $G$-valued
functions on the group of loops. We also show that, the Ashtekar-Lewandowski
measure is, under very mild conditions, supported on nowhere-continuous
generalized connections.
