Finding subsets of positive measure

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

Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

We analyze the pointwise convergence of a sequence of computable elements of L^1(2^omega) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA_0, each is equivalent to the assertion that every G_delta subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak K\"onig's lemma relativized to the Turing jump of any set. It is also equivalent to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Sigma_2 collection