  If $X$ is an analytic metric space satisfying a very mild doubling condition,
then for any finite Borel measure $\mu$ on $X$ there is a set $N\subseteq X$
such that $\mu(N)>0$, an ultrametric space $Z$ and a Lipschitz bijection
$\phi:N\to Z$ whose inverse is nearly Lipschitz, i.e., $\beta$-H\"older for all
$\beta<1$. As an application it is shown that a Borel set in a Euclidean space
maps onto $[0,1]^n$ by a nearly Lipschitz map if and only if it cannot be
covered by countably many sets of Hausdorff dimension strictly below $n$. The
argument extends to analytic metric spaces satisfying the mild condition.
Further generalization replaces cubes with self-similar sets, nearly Lipschitz
maps with nearly H\"older maps and integer dimension with arbitrary finite
dimension.
