  The classical Painlev\'e theorem tells that sets of zero length are removable
for bounded analytic functions, while (some) sets of positive length are not.
For general $K$-quasiregular mappings in planar domains the corresponding
critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly
one has improved removability. More precisely, we prove that sets $E$ of
$\sigma$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded
$K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not
enough to guarantee this property.
  We also study absolute continuity properties of pull-backs of Hausdorff
measures under $K$-quasiconformal mappings, in particular at the relevant
dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$,
$0 < t < 2$, we reduce the absolute continuity properties to an open question
on conformal mappings.
