Capacitability for Co-Analytic Sets

Abstract

It follows from a theorem of Davies (1952) that if A is an analytic subset of the Cantor middle third set, λ is positive and the Hausdorff s-measure of A is greater than λ, then there is a compact subset C of A such that the Hausdorff s-measure of C is greater than λ. We exhibit a counterpoint to Davies’s theorem: In Gödel’s universe of sets, there is a co-analytic subset B of the Cantor set which has full Hausdorff dimension such that if C is a closed subset of B then C is countable.