Finding subsets of positive measure [article]

Bjørn Kjos-Hanssen, Jan Reimann
2014 arXiv   pre-print
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 H^s contains a closed subset of non-zero (and indeed finite) 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)
more » ... 1 set of reals in Cantor space, there is always a Π^0_1(O) subset on non-zero H^s-measure definable from Kleene's O. On the other hand, there are Π^0_2 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.
arXiv:1408.1999v1 fatcat:zi4s6nw2azatvapyheo7nab6uy