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 Hs contains a closed subset of
non-zero (and indeed finite) Hs-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)
Σ11 set of reals in Cantor space, there is always a
Π10(O) subset on non-zero Hs-measure definable from
Kleene's O. On the other hand, there are Π20 sets of reals
where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The
Strength of the Besicovitch--Davies Theorem". The final publication of that
paper is available at Springer via
http://dx.doi.org/10.1007/978-3-642-13962-8_2