A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an F$_{\sigma}$ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measure-theoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some $\omega$-models of RCA$_0$which are relevant for the reverse mathematics of measure-theoretic regularity.

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

In this work we investigate the Weihrauch degree of the problem Decreasing Sequence of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$, despite being hard to solve, is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then (...) generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level. (shrink)

We survey the theory of Mucnik and Medvedev degrees of subsets of $^{\omega}{\omega}$with particular attention to the degrees of $\Pi_{1}^{0}$ subsets of $^{\omega}2$. Sections 1-6 present the major definitions and results in a uniform notation. Sections 7-6 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.