Finding subsets of positive measure

Abstract

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 ${\Pi }_1^0(\mathcal{O})$ subset on non-zero ${\mathcal{H}}^s$-measure definable from Kleene's $\mathcal{O}$. On the other hand, there are ${\Pi }_2^0$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.

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_26

References in corpus (1)

• Martin-Löf randomness and Galton-Watson processes