  An important theorem of geometric measure theory (first proved by Besicovitch
and Davies for Euclidean space) says that every analytic set of non-zero
$s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of
non-zero (and indeed finite) $\mathcal H^s$-measure. We investigate the
question how hard it is to find such a set, in terms of the index set
complexity, and in terms of the complexity of the parameter needed to define
such a closed set. Among other results, we show that given a (lightface)
$\Sigma^1_1$ set of reals in Cantor space, there is always a
$\Pi^0_1(\mathcal{O})$ subset on non-zero $\mathcal H^s$-measure definable from
Kleene's $\mathcal O$. On the other hand, there are $\Pi^0_2$ sets of reals
where no hyperarithmetic real can define a closed subset of non-zero measure.

Logic