2014/08/09 by Bjørn Kjos-Hanssen, Kjos-Hanssen, Bjørn, Jan Reimann +1
Computer Science · Mathematics · #03D #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1408.1999
openalex publication_date 2014/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Hs contains a closed subset of non-zero (and indeed finite) \mathcal 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 Π01(O) subset on non-zero \mathcal Hs-measure definable from Kleene's \mathcal O. On the other hand, there are Π02 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.