Mass problems and measure-theoretic regularity

Abstract

A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an F_σ 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 ω-models of RCA₀ which are relevant for the reverse mathematics of measure-theoretic regularity.