Finding Subsets of Positive Measure∗

An important theorem of geometric measure theory (first proved by Besicov-itch 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 in-deed 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 Π 0 1(O) sub-set on non-zero Hs-measure definable from Kleene’s O. On the other hand, there are Π02 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.