Finding subsets of positive measure

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 Kleenes $mathcal O$. 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.